Soft Gluon Resummation for associated t tH Production at the LHC - - PowerPoint PPT Presentation

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Soft Gluon Resummation for associated t tH Production at the LHC - - PowerPoint PPT Presentation

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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

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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¯ tH 1

  • V. Theeuwes
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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

[TeV] s 6 7 8 9 10 11 12 13 14 15 H+X) [pb] → (pp σ

2 −

10

1 −

10 1 10

2

10 M(H)= 125 GeV

LHC HIGGS XS WG 2016

H ( N N L O + N N L L Q C D + N L O E W ) → p p qqH (NNLO QCD + NLO EW) → pp W H ( N N L O Q C D + N L O E W ) → p p Z H ( N N L O Q C D + N L O E W ) → p p t t H ( N L O Q C D + N L O E W ) → p p

bbH (NNLO QCD in 5FS, NLO QCD in 4FS) → pp

tH (NLO QCD) → pp

[LHCHXSWG]

Threshold resummation for t¯ tH 2

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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¯ tH 3

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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¯ tH 4

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SLIDE 5

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 ∝ β4)
  • Small corrections from near absolute threshold

Threshold resummation for t¯ tH 4

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Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results

Definition of Threshold

Q-approach Threshold variable ˆ τQ = Q2

ˆ s Q2: the invariant mass final state particles

1 − ˆ τQ = 1 − Q2 ˆ s ∼ energy of the emitted gluons total available energy M-approach (absolute threshold) Threshold variable ˆ τM = M 2

ˆ s M: the sum of final state masses

1 − ˆ τM = 1 − M 2 ˆ s ∼ maximum energy of the emitted gluons total available energy

√ ˆ s: the partonic center of mass energy

Threshold resummation for t¯ tH 5

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Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results

Logarithms

Q-approach The IR divergences lead to logarithms: (1 − ˆ τQ)−1−2ǫ = − 1 2ǫδ (1 − ˆ τQ) +

  • 1

1 − ˆ τQ

  • +

− 2ǫ log(1 − ˆ τQ) 1 − ˆ τQ

  • +

✎ ✍ ☞ ✌ αn

s

logm(1 − ˆ τQ) 1 − ˆ τQ

  • +

In general logarithms of 1 − ˆ τQ M-approach After integration over ˆ τQ: logarithms of 1 − ˆ τM: ✞ ✝ ☎ ✆ αn

s logm(1 − ˆ

τM) Logarithms become large in threshold limit: ˆ τ → 1

Threshold resummation for t¯ tH 6

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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): ✤ ✣ ✜ ✢ ˜ σpp→t¯

tH(N)

≡ 1 dτ τ N−1σpp→t¯

tH(τ, µR, µF )

=

  • i,j

˜ fi/p(N + 1, µF ) ˜ fj/p(N + 1, µF )˜ ˆ σij→t¯

tH(N, µR, µF )

  • ˜

fi/p(N + 1, µF ): Mellin transform with respect to x

  • ˜

ˆ σij→t¯

tH(N, µR, µF ): Mellin transform with respect to ˆ

τ logn(1 − ˆ τ) ⇒ logn N and threshold ˆ τ → 1 ∼ N → ∞ First application for 2 → 3 in Mellin space

Threshold resummation for t¯ tH 7

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Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results

General factorization

In general cross section factorizes into: ˆ σij→kl... = Hij→kl... ,IJ ⊗ ψi ⊗ ψj ⊗ SJI ⊗ Jk ⊗ Jl . . .

  • Hij→kl,IJ Hard function
  • ψi,j Initial state collinear emission
  • Jk,l,... Final state collinear

emission

  • SJI Soft emission

ψi ψj

H H∗ S Jl Jk

Each of these functions is computed through renormalization group equations

Threshold resummation for t¯ tH 8

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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 ωPIM = z = 1 − Q2/ˆ

s

  • One particle inclusive ω1PI = s4 = ((p1 + p2 − p3)2 − m2

4)/ˆ

s

Threshold resummation for t¯ tH 9

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Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results

Resummation for differential distributions

✞ ✝ ☎ ✆ W = ω1c1 + ω2c2 + ωs + ωk + ωl + . . .

  • dWe−NW dσij→kl...

dˆ Πn =

  • dWe−NW HIJ
  • ˆ

Πn dω1dω2dωsdωkdωl . . . δ (W − ω1c1 − ω2c2 − ωs − ωk − ωl − . . . ) ψi/i (ω1) ψj/j (ω2) SJI (ωs) Jk (ωk) Jl (ωl) . . . = HIJ

  • ˆ

Πn

  • ˜

ψi/i (Nc1) ˜ ψj/j (Nc2) ˜ SJI (N) ˜ Jk (N) ˜ Jl (N) . . .

  • ω1/2 Initial state collinear weights
  • ωs Soft-wide angle weight
  • ωk/l... Final state collinear weights

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Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results

2 → 3 weights for differential notation

pp → Q ¯ QB + X ⇒ Q2 z5 = ˆ s − s345 ˆ s

Threshold resummation for t¯ tH 11

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Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results

2 → 3 weights for differential notation

pp → Q ¯ QB + X ⇒ Q2 z5 = ˆ s − s345 ˆ s pp → B + X[Q ¯ Q] ⇒ PT,5, y5 v5 = (p1 + p2 − p5)2 − s34 ˆ s = ˆ s + ˜ t15 + ˜ t25 + m2

5 − s34

ˆ s

Threshold resummation for t¯ tH 11

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Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results

2 → 3 weights for differential notation

pp → Q ¯ QB + X ⇒ Q2 z5 = ˆ s − s345 ˆ s pp → B + X[Q ¯ Q] ⇒ PT,5, y5 v5 = (p1 + p2 − p5)2 − s34 ˆ s = ˆ s + ˜ t15 + ˜ t25 + m2

5 − s34

ˆ s pp → Q ¯ Q + X[B] ⇒ PT,34, y34, Q2

34

s5 = (p1 + p2 − p3 − p4)2 − m2

5

ˆ s = ˆ s + ˜ t13 + ˜ t14 + ˜ t23 + ˜ t24 + s34 − m2

5

ˆ s

Threshold resummation for t¯ tH 11

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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 αs and L: ✞ ✝ ☎ ✆ ˜ σ ∼ ˜ σLO × C(αs) exp [Lg1(αsL) + g2(αsL) + αsg3(αsL) + · · · ] ⇓ ⇓ ⇓ With orders of precision: LL NLL NNLL ⇓ ⇓ ⇓ αn

s logn+1(N)

αn

s logn(N)

αn+1

s

logn(N)

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¯ tH 12

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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: q¯ q 1 : δa2a1δa3a4 8 : T D

a2a1T D a3a4

gg 1 : δA1A2δa3a4 8S : T D

a3a4dDA1A2

8A : iT D

a3a4f DA1A2

Basis for diagonalization of soft anomalous dimension in absolute threshold limit. Same as for t¯ t production

Threshold resummation for t¯ tH 13

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Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results

Soft wide-angle

★ ✧ ✥ ✦

[Kidonakis et al., ’97-’01]

˜ Sij→kl Q µN

  • =

¯ P exp Q/N

µ

dq q Γ†

ij→kl

  • αs
  • q2
  • ˜

Sij→kl ×P exp Q/N

µ

dq q Γij→kl

  • αs
  • q2
  • Threshold resummation for t¯

tH 14

  • V. Theeuwes
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Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results

Soft anomalous dimension matrix

Γ(1) q¯ q→Q ¯ QX,11 = − αs 2π 2CF

  • Lβ34 + 1
  • Γ(1)

q¯ q→Q ¯ QX,12 = αs 2π 2CF NC Ω3 = CF 2NC Γ(1) q¯ q→Q ¯ QX,21 Γ(1) q¯ q→Q ¯ QX,22 = αs 2π

  • NC − 2CF

Lβ34 + 1

  • + NC Λ3 +
  • 8CF − 3NC
  • Ω3
  • Ω3

= (T13 + T24 − T14 − T23) /2 Λ3 = (T13 + T24 + T14 + T23) /2 Tij = log   m2 j − tij mj √s   + iπ − 1 2

with tij = (pi − pj)2 and β2

34 = 1 − (m3 + m4)2/s34

Threshold resummation for t¯ tH 15

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Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results

Soft anomalous dimension matrix

Γ(1) q¯ q→Q ¯ QX,11 = − αs 2π 2CF

  • Lβ34 + 1
  • Agrees with q¯

q → Q ¯ Q

Γ(1) q¯ q→Q ¯ QX,12 = αs 2π 2CF NC Ω3 = CF 2NC Γ(1) q¯ q→Q ¯ QX,21

for p5 → 0 and m5 → 0

Γ(1) q¯ q→Q ¯ QX,22 = αs 2π

  • NC − 2CF

Lβ34 + 1

  • + NC Λ3 +
  • 8CF − 3NC
  • Ω3
  • Ω3

= (T13 + T24 − T14 − T23) /2 Λ3 = (T13 + T24 + T14 + T23) /2 Tij = log   m2 j − tij mj √s   + iπ − 1 2

with tij = (pi − pj)2 and β2

34 = 1 − (m3 + m4)2/s34

⇓ ⇓ For 2 → 2: t13 = t24 s34 = ˆ s t23 = t14 β34 = β

Threshold resummation for t¯ tH 15

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Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results

Soft wide-angle emission absolute threshold

Absolute threshold limit for soft anomalous dimension: 2ReΓ(1)

q¯ q→Q ¯ QB,IJ,thr.

= αs π diag (0, −NC) 2ReΓ(1)

gg→Q ¯ QB,IJ,thr.

= αs π diag (0, −NC, −NC) results in soft wide-angle contribution: ∆NLL

I

= exp [h2(αsL, −CI)] With CI the quadratic Casimir invariant

Threshold resummation for t¯ tH 16

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Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results

Hard Matching Coefficient

C(αs) = 1 + αs π C(1) + . . .

  • Massive final state dipoles [Catani,Dittmaier,Seymour,Trócsányi,’02]
  • Virtual contribution from PowHeg-Box [Hartanto,Jäger,Reina,Wackeroth,’15]

confirmed by aMC@NLO [Hirschi,Frederix,Frixione,Garzelli,Maltoni,Pittau,’11]

  • Include Coulomb correction

1 β34

Threshold resummation for t¯ tH 17

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Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results

Matching to Fixed Order

Resummed Cross Section

σ(NLO+NLL)(τ) = σ(NLO)(τ) +

  • CT

dN 2πiτ −N ˜ fg/p(N + 1) ˜ fg/p(N + 1) ×

  • ˜

ˆ σ(NLL)(N) − ˜ ˆ σ(NLL)(N)|(NLO)

  • Matching to fixed order required to avoid double counting.

Threshold resummation for t¯ tH 18

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Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results

Results

[Kulesza, Motyka, Stebel, VT, ’15 and in preperation]

PDFs used: MMHT2014NLO

0.00 200.00 400.00 600.00 800.00 1000.00 0.2 0.5 1 2 5 µ/µ0 σ(pp → Ht¯ t + X)[fb] √ S = 14 TeV µ0 = mt + mH/2 mH = 125 GeV

LO NLO NLO+NLL (Q) NLO+NLL (Q w C)

0.00 200.00 400.00 600.00 800.00 1000.00 0.2 0.5 1 2 5 µ/µ0 σ(pp → Ht¯ t + X)[fb] √ S = 14 TeV µ0 = mt + mH/2 mH = 125 GeV

LO NLO NLO+NLL (M w C) NLO+NLL (Q w C)

preliminary

Threshold resummation for t¯ tH 19

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Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results

Results

[Kulesza, Motyka, Stebel, VT, in preperation]

PDFs used: MMHT2014NLO

0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 σ(pp → Ht¯ t + X)[fb per bin] √ S = 14 TeV µ0 = mt + mH/2 mH = 125 GeV 0.80 1.00 1.20 500 600 700 800 900 1000 1100 1200 Q[GeV]

σX/σNLO NLO NLO+NLL (Q w C)

preliminary

Threshold resummation for t¯ tH 20

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Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results

Results

[Kulesza, Motyka, Stebel, VT, ’15 and in preperation]

PDFs used: MMHT2014NLO

√ S [TeV] NLO [fb] NLO+NLL M with C NLO+NLL Q with C Value [fb] K-factor Value [fb] K-factor 13 506+5.9%

−9.4%

537+8.2%

−5.5%

1.06 512+5.1%

−6.2%

1.01 14 613+6.2%

−9.4%

650+7.9%

−5.7%

1.06 619+5.2%

−6.4%

1.01 Using 7-point method: (µF /µ0, µR/µ0) = {(0.5, 0.5), (0.5, 1), (1, 0.5), (1, 1), (1, 2), (2, 1), (2, 2)}

preliminary

Threshold resummation for t¯ tH 21

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Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results

Results

[Kulesza, Motyka, Stebel, VT, ’15 and in preperation]

PDFs used: MMHT2014NLO

√ S [TeV] NLO [fb] NLO+NLL M with C NLO+NLL Q with C Value [fb] K-factor Value [fb] K-factor 13 506+5.9%

−9.4%

537+8.2%

−5.5%

1.06 512+5.1%

−6.2%

1.01 14 613+6.2%

−9.4%

650+7.9%

−5.7%

1.06 619+5.2%

−6.4%

1.01 Using 7-point method: (µF /µ0, µR/µ0) = {(0.5, 0.5), (0.5, 1), (1, 0.5), (1, 1), (1, 2), (2, 1), (2, 2)}

preliminary

Thank you for your attention

Threshold resummation for t¯ tH 21

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Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results

Backup

[Kulesza, Motyka, Stebel, VT, ’in preperation]

PDFs used: MMHT2014NLO

0.00 200.00 400.00 600.00 800.00 1000.00 0.2 0.5 1 2 5 µF /µ0 σ(pp → Ht¯ t + X)[fb] √ S = 14 TeV µ0 = mt + mH/2 mH = 125 GeV

LO NLO NLO+NLL (M) NLO+NLL (M w C) NLO+NLL (Q w C)

0.00 200.00 400.00 600.00 800.00 1000.00 0.2 0.5 1 2 5 µR/µ0 σ(pp → Ht¯ t + X)[fb] √ S = 14 TeV µ0 = mt + mH/2 mH = 125 GeV

LO NLO NLO+NLL (M) NLO+NLL (M w C) NLO+NLL (Q w C)

preliminary

Threshold resummation for t¯ tH 22

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Motivation Threshold Resummation Application for 2 → 3 Hard matching coefficient Numerical Results

Backup

[Kulesza, Motyka, Stebel, VT, in preperation]

PDFs used: MSTW2008NNLO

350.00 400.00 450.00 500.00 550.00 1 2 3 4 5 µ/µ0 σ(pp → Ht¯ t + X)[fb] √ S = 13 TeV µ0 = mt + mH/2 mH = 125 GeV

NLO (no qg 1510.01914) NLL|NLO (Q w C)

preliminary

Threshold resummation for t¯ tH 23

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