QCD and Event Generation for the Large Hadron Collider · Bryan Webber Burg Liebenzell, Sept 2014 PDF Uncertainties 10 M X 102 103 Gluon - Gluon Luminosity 0.8 0.85 0.9 0.95 1 1.05

Bryan Webber Burg Liebenzell, Sept 2014

QCD and Event Generation for the

Large Hadron Collider

1

Bryan Webber Cavendish Laboratory

University of Cambridge

Burg Liebenzell, Sept 2014


Outline

2

QCD BasicsFactorization, PDFs, running coupling

QCD and Higgs boson productionInclusive production cross sections

Differential cross sections, resummation

Monte Carlo event generation

QCD and Higgs boson decays

Quark masses

Uncertainties and prospects

Perturbative and non-perturbative components

Improvements: matching and merging

Survey of results


References

3

• R.K. Ellis, W.J. Stirling & B.R. Webber, “QCD and Collider Physics” (C.U.P. 1996)

• A. Buckley et al., “General-purpose event generators for LHC physics”, Phys.Rept. 504 (2011) 145 (MCNET-11-01, arXiv:1101.2599)

• M.H. Seymour & M. Marx, “Monte Carlo Event Generators”, MCNET-13-05, arXiv:1304.6677

• A. Siódmok, “LHC event generation with general-purpose Monte Carlo tools”, Acta Phys. Polon. B44 (2013)1587


QCD Basics

4


QCD Factorization

5

• Non-perturbative physics takes place over a much longer time scale, with unit probability

• Hence it cannot change the cross section

• Scale dependences of parton distribution functions and hard process cross section are perturbatively calculable, and cancel order by order

momentum fractions

parton distribution

functions at scale

hard process cross section, renormalised

at scale

( (

�pp!X(s) =X

i,j

Z 1

0dx1 dx2 fi(x1, µ

2F )fj(x2, µ

2F )�ij!X

�x1x2s,↵S(µ

2R), µ

2F , µ

2R

�

µF µR


PDF Evolution

6

• PDFs measured in various processes at various scales

• Global fits satisfying evolution equations give PDF sets

• Generally done at NNLO nowadays

µ

2 @

@µ

2f

i

(x, µ2) =X

j

Z 1

x

d⇠

⇠

P

ij

✓x

⇠

,↵S(µ2)

◆f

j

(⇠, µ2)


0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

HERAPDF1.5 NNLO (prel.)

exp. uncert.

model uncert. parametrization uncert.

x

xf 2 = 2 GeV2Q

vxu

vxd

xS

xg

HER

APD

F St

ruct

ure

Func

tion

Wor

king

Gro

up

Mar

ch 2

011H1 and ZEUS HERA I+II PDF Fit

0.2

0.4

0.6

0.8

1

Fraction of proton momentum

PDF Evolution


0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1


exp. uncert.


x

xf 2 = 10 GeV2Q

vxu

vxd

xS

xg

HER

APD

F St

ruct

ure

Func

tion

Wor

king

Gro

up

Mar

ch 2


0.2

0.4

0.6

0.8

1


PDF Evolution


0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1


exp. uncert.


x

xf 2 = 10000 GeV2Q

vxu

vxd

xS

xg

HER

APD

F St

ruct

ure

Func

tion

Wor

king

Gro

up

Mar

ch 2


0.2

0.4

0.6

0.8

1


PDF Evolution


PDF Uncertainties

10

XM210 310

Glu

on -

Glu

on L

umin

osity

0.80.850.9

0.951

1.051.1

1.151.2

1.251.3

= 0.118sαLHC 8 TeV - Ratio to NNPDF2.3 NNLO -

NNPDF2.3 NNLO

CT10 NNLO

MSTW2008 NNLO


XM210 310

Glu

on -

Glu

on L

umin

osity

0.80.850.9

0.951

1.051.1

1.151.2

1.251.3


NNPDF2.3 NNLO

ABM11 NNLO

HERAPDF1.5 NNLO


XM210 310

Qua

rk -

Glu

on L

umin

osity

0.80.850.9

0.951

1.051.1

1.151.2

1.251.3


NNPDF2.3 NNLO

CT10 NNLO

MSTW2008 NNLO


XM210 310

Qua

rk -

Glu

on L

umin

osity

0.80.850.9

0.951

1.051.1

1.151.2

1.251.3


NNPDF2.3 NNLO

ABM11 NNLO

HERAPDF1.5 NNLO


Figure 6: The gluon-gluon (upper plots) and quark-gluon (lower plots) luminosities, Eq. (2), forthe production of a final state of invariant mass MX (in GeV) at LHC 8 TeV. The left plots showthe comparison between NNPDF2.3, CT10 and MSTW08, while in the right plots we compareNNPDF2.3, HERAPDF1.5 and MSTW08. All luminosities are computed at a common value ofαs = 0.118.

11

Relevant PDFs (relatively) well known at x ~ MH/√s

Remains true at 13 TeV

Can be improved (in principle)

Parton luminosityZ

dx1dx2 fi(x1,M2X) fj(x2,M

2X) �

�x1x2s�M

2X

�Lij(M

2X , s) =

Ball et al., 1211.5142

Some disagreement with CT10 Lgg


QCD Running Coupling

11

24 2 Asymptotic freedom and confinement

However µ is an arbitrary parameter. The Lagrangian of QCD makesno mention of the scale µ, even though a choice of µ is required to definethe theory at the quantum level. Therefore, if we hold the bare couplingfixed, physical quantities such as R cannot depend on the choice made forµ. Since R is dimensionless, it can only depend on the ratio Q2/µ2 andthe renormalized coupling αS . Mathematically, the µ independence of Rmay be expressed by

µ2 d

dµ2R(Q2/µ2,αS) ≡

!

µ2 ∂

∂µ2+ µ2 ∂αS

∂µ2

∂

∂αS

"

R = 0 . (2.1)

To rewrite this equation in a more compact form we introduce the nota-tions

t = ln

#

Q2

µ2

$

, β(αS) = µ2∂αS

∂µ2. (2.2)

The derivative of the coupling in the definition of the β function is per-formed at fixed bare coupling. We rewrite Eq. (2.1) as

!

− ∂

∂t+ β(αS)

∂

∂αS

"

R(et,αS) = 0 . (2.3)

This first order partial differential equation is solved by implicitly defininga new function – the running coupling αS(Q2) – as follows:

t =% αS(Q2)

αS

dx

β(x), αS(µ2) ≡ αS . (2.4)

By differentiating Eq. (2.4) we see that

∂αS(Q2)

∂t= β(αS(Q2)),

∂αS(Q2)

∂αS=β(αS(Q2))

β(αS)(2.5)

and hence that R(1,αS(Q2)) is a solution of Eq. (2.3). The above analysisshows that all of the scale dependence in R enters through the running ofthe coupling constant αS(Q2). It follows that knowledge of the quantityR(1,αS), calculated in fixed-order perturbation theory, allows us to pre-dict the variation of R with Q if we can solve Eq. (2.4). In the next section,we shall show that QCD is an asymptotically free theory. This means thatαS(Q2) becomes smaller as the scale Q increases. For sufficiently largeQ, therefore, we can always solve Eq. (2.4) using perturbation theory.

2.2 The β function

The running of the coupling constant αS is determined by the renormal-ization group equation,

Q2 ∂αS

∂Q2= β(αS). (2.6)



µ2 d


!

µ2 ∂

∂µ2+ µ2 ∂αS

∂µ2

∂

∂αS

"

R = 0 . (2.1)


t = ln

#

Q2

µ2

$


∂µ2. (2.2)


!

− ∂

∂t+ β(αS)

∂

∂αS

"

R(et,αS) = 0 . (2.3)


t =% αS(Q2)

αS

dx

β(x), αS(µ2) ≡ αS . (2.4)


∂αS(Q2)

∂t= β(αS(Q2)),

∂αS(Q2)

∂αS=β(αS(Q2))

β(αS)(2.5)


2.2 The β function


Q2 ∂αS

∂Q2= β(αS). (2.6)



µ2 d


!

µ2 ∂

∂µ2+ µ2 ∂αS

∂µ2

∂

∂αS

"

R = 0 . (2.1)


t = ln

#

Q2

µ2

$


∂µ2. (2.2)


!

− ∂

∂t+ β(αS)

∂

∂αS

"

R(et,αS) = 0 . (2.3)


t =% αS(Q2)

αS

dx

β(x), αS(µ2) ≡ αS . (2.4)


∂αS(Q2)

∂t= β(αS(Q2)),

∂αS(Q2)

∂αS=β(αS(Q2))

β(αS)(2.5)


2.2 The β function


Q2 ∂αS

∂Q2= β(αS). (2.6)



µ2 d


!

µ2 ∂

∂µ2+ µ2 ∂αS

∂µ2

∂

∂αS

"

R = 0 . (2.1)


t = ln

#

Q2

µ2

$


∂µ2. (2.2)


!

− ∂

∂t+ β(αS)

∂

∂αS

"

R(et,αS) = 0 . (2.3)


t =% αS(Q2)

αS

dx

β(x), αS(µ2) ≡ αS . (2.4)


∂αS(Q2)

∂t= β(αS(Q2)),

∂αS(Q2)

∂αS=β(αS(Q2))

β(αS)(2.5)


2.2 The β function


Q2 ∂αS

∂Q2= β(αS). (2.6)



µ2 d


!

µ2 ∂

∂µ2+ µ2 ∂αS

∂µ2

∂

∂αS

"

R = 0 . (2.1)


t = ln

#

Q2

µ2

$


∂µ2. (2.2)


!

− ∂

∂t+ β(αS)

∂

∂αS

"

R(et,αS) = 0 . (2.3)


t =% αS(Q2)

αS

dx

β(x), αS(µ2) ≡ αS . (2.4)


∂αS(Q2)

∂t= β(αS(Q2)),

∂αS(Q2)

∂αS=β(αS(Q2))

β(αS)(2.5)


2.2 The β function


Q2 ∂αS

∂Q2= β(αS). (2.6)

Consider a dimensionless quantity R depending on a single hard scale QDependence on Q can only be via Q/mBut m is arbitrary, so overall dependence on it must vanish

Define

Introduce aS(Q2) such that

Then solution is

All scale dependence is absorbed in running coupling aS(Q2)



12

�(↵S) = �↵2S (�0 + �1↵S + . . .)

ln

✓Q2

µ2

◆=

1

�0

1

↵S(Q2)� 1

↵S(µ2)

�+ . . .

↵S(Q2) =

↵S(µ2)

1 + �0 ↵S(µ2) ln (Q2/µ2)+ . . .

,ln

✓Q2

µ2

◆=

Z ↵S(Q2)

↵S(µ2)

d↵S

�(↵S)

�0 = (33� 2nf )/12⇡

b-function known to 4 loops (b3)

b0>0 means asymptotic freedom



13

30 9. Quantum chromodynamics

Preliminary determinations of αs from CMS data on the ratio of inclusive 3-jet to2-jet cross sections [259], at NLO, and from the top-quark cross section [301], inNNLO, quote values of αs(M2

Z) = 0.1148± 0.0014(exp.)± 0.0018(PDF)+0.0050−0.0000(scale) and

αs(M2Z) = 0.1151+0.0033

−0.0032, respectively, indicating many new results to be expected forinclusion in upcoming reviews.

9.3.11. Electroweak precision fits :The N3LO calculation of the hadronic Z decay width was used in a revision of the globalfit to electroweak precision data [349], resulting in αs(M2

Z) = 0.1193± 0.0028, claiming anegligible theoretical uncertainty. For this Review the value obtained in Sec. Electroweakmodel and constraints on new physics from data at the Z-pole, αs(M2

Z) = 0.1197± 0.0028will be used instead, as it is based on a more constrained data set where QCD correctionsdirectly enter through the hadronic decay width of the Z. We note that all theseresults from electroweak precision data, however, strongly depend on the strict validityof Standard Model predictions and the existence of the minimal Higgs mechanism toimplement electroweak symmetry breaking. Any - even small - deviation of nature fromthis model could strongly influence this extraction of αs.

0.11 0.12 0.13αα ((ΜΜ ))s ΖΖ

LatticeDIS e+e- annihilation

τ-decays

Z pole fits

Figure 9.3: Summary of values of αs(M2Z) obtained for various sub-classes

of measurements (see Fig. 9.2 (a) to (d)). The new world average value ofαs(M2

Z) = 0.1185 ± 0.0006 is indicated by the dashed line and the shaded band.

9.3.12. Determination of the world average value of αs(M2Z) :

Obtaining a world average value for αs(M2Z) is a non-trivial exercise. A certain

arbitrariness and subjective component is inevitable because of the choice of measurementsto be included in the average, the treatment of (non-Gaussian) systematic uncertaintiesof mostly theoretical nature, as well as the treatment of correlations among the variousinputs, of theoretical as well as experimental origin.

We have chosen to determine pre-averages for classes of measurements which areconsidered to exhibit a maximum of independence between each other, consideringexperimental as well as theoretical issues. These pre-averages are then combined to thefinal world average value of αs(M2

Z), using the χ2 averaging method and error treatmentas described above. The five pre-averages are summarized in Fig. 9.3; we recall that these

December 18, 2013 12:00

9. Quantum chromodynamics 33

QCD αs(Mz) = 0.1185 ± 0.0006

Z pole fit

0.1

0.2

0.3

αs (Q)

1 10 100Q [GeV]

Heavy Quarkonia (NLO)

e+e– jets & shapes (res. NNLO)

DIS jets (NLO)

Sept. 2013

Lattice QCD (NNLO)

(N3LO)

τ decays (N3LO)

1000

pp –> jets (NLO)(–)

Figure 9.4: Summary of measurements of αs as a function of the energy scale Q.The respective degree of QCD perturbation theory used in the extraction of αs isindicated in brackets (NLO: next-to-leading order; NNLO: next-to-next-to leadingorder; res. NNLO: NNLO matched with resummed next-to-leading logs; N3LO:next-to-NNLO).

9.4. Acknowledgments

We are grateful to J.-F. Arguin, G. Altarelli, J. Butterworth, M. Cacciari, L. delDebbio, D. d’Enterria, P. Gambino, C. Glasman Kuguel, N. Glover, M. Grazzini, A.Kronfeld, K. Kousouris, M. Luscher, M. d’Onofrio, S. Sharpe, G. Sterman, D. Treille,N. Varelas, M. Wobisch, W.M. Yao, C.P. Yuan, and G. Zanderighi for discussions,suggestions and comments on this and earlier versions of this review.

References:1. R.K. Ellis, W.J. Stirling, and B.R. Webber, “QCD and collider physics,” Camb.

Monogr. Part. Phys. Nucl. Phys. Cosmol. 81 (1996).2. C. A. Baker et al., Phys. Rev. Lett. 97, 131801 (2006).3. H. -Y. Cheng, Phys. Reports 158, 1 (1988).4. G. Dissertori, I.G. Knowles, and M. Schmelling, “High energy experiments and

theory,” Oxford, UK: Clarendon (2003).5. R. Brock et al., [CTEQ Collab.], Rev. Mod. Phys. 67, 157 (1995), see also

http://www.phys.psu.edu/~cteq/handbook/v1.1/handbook.pdf.6. A. S. Kronfeld and C. Quigg, Am. J. Phys. 78, 1081 (2010).7. R. Stock (Ed.), Relativistic Heavy Ion Physics, Springer-Verlag Berlin, Heidelberg,

2010.8. Proceedings of the XXIII International Conference on Ultrarelativistic Nucleus–

Nucleus Collisions, Quark Matter 2012, Nucl. Phys. A, volumes 904–905.9. Special Issue: Physics of Hot and Dense QCD in High-Energy Nuclear

Collisions, Ed. C. Salgado. Int. J. Mod. Phys. A, volume 28, number 11.[http://www.worldscientific.com/toc/ijmpa/28/11].

December 18, 2013 12:00

Bethke, Dissertori, Salam, RPP 2013

Figure 26: α(5)

MS(MZ), the coupling constant in the MS scheme at the Z mass. The results

labeled Nf = 0, 2 use estimates for Nf = 3 obtained by first extrapolating in Nf fromNf = 0, 2 results. Since this is not a theoretically justified procedure, these are not includedin our final estimate and are thus given a red symbol. However, they are shown to indicatethe progress made since these early calculations. The PDG entry indicates the outcome oftheir analysis excluding lattice results (see section 9.9.4).

perturbative truncation errors, which are difficult to estimate. This concern also applies tomany non-lattice determinations. Further, all results except for those of sections 9.3, 9.6 arebased on extractions of αMS that are largely influenced by data with αeff ≥ 0.3. At smallerα the momentum scale µ quickly is at or above a−1. We have included computations usingaµ up to 1.5 and αeff up to 0.4, but one would ideally like to be significantly below that.Accordingly we wish at this stage to estimate the error ranges in a conservative manner, andnot simply perform weighted averages of the individual errors estimated by each group.

Many of the methods have thus far only been applied by a single collaboration, and withsimulation parameters that could still be improved. We therefore think that the followingaspects of the individual calculations are important to keep in mind, and look forward toadditional clarification and/or corroboration in the future.

• The potential computations Brambilla 10 [505], ETM 11C [504] and Bazavov 12 [503] giveevidence that they have reached distances where perturbation theory can be used. However,in addition to ΛQCD, a scale is introduced into the perturbative prediction by the processof subtracting the renormalon contribution. The extractions of Λ are dominated by data

170

Lattice (MZ)

FLAG WG: Aoki et al., 1310.8555

aS(MZ)=0.1183(12) [non-lattice]aS(MZ)=0.1185(5) [lattice]


Lattice QCD Coupling

14

Collaboration Ref. Nf publicationstatus

renorm

alisationscale

perturbative

behaviour

continuum

extrapolation

αMS(MZ) Method Table

ETM 13D [544] 2+1+1 A ◦ ◦ ! 0.1196(4)(8)(16) gluon-ghost vertex 37ETM 12C [545] 2+1+1 A ◦ ◦ ! 0.1200(14) gluon-ghost vertex 37ETM 11D [546] 2+1+1 A ◦ ◦ ! 0.1198(9)(5)(+0

−5) gluon-ghost vertex 37

Bazavov 12 [503] 2+1 A ◦ ◦ ◦ 0.1156(+21−22) Q-Q potential 33

HPQCD 10 [73] 2+1 A ◦ ◦ ◦ 0.1183(7) current two points 36HPQCD 10 [73] 2+1 A ◦ ⋆ ⋆ 0.1184(6) Wilson loops 35PACS-CS 09A [486] 2+1 A ⋆ ⋆ ◦ 0.118(3)# Schrodinger functional 32Maltman 08 [517] 2+1 A ◦ ◦ ◦ 0.1192(11) Wilson loops 35HPQCD 08B [85] 2+1 A ! ! ! 0.1174(12) current two points 36HPQCD 08A [514] 2+1 A ◦ ⋆ ⋆ 0.1183(8) Wilson loops 35HPQCD 05A [513] 2+1 A ◦ ◦ ◦ 0.1170(12) Wilson loops 35

QCDSF/UKQCD 05[518] 0, 2 → 3 A ⋆ ! ⋆ 0.112(1)(2) Wilson loops 35Boucaud 01B [539] 2 → 3 A ◦ ◦ ! 0.113(3)(4) gluon-ghost vertex 37SESAM 99 [519] 0, 2 → 3 A ⋆ ! ! 0.1118(17) Wilson loops 35Wingate 95 [520] 0, 2 → 3 A ⋆ ! ! 0.107(5) Wilson loops 35Davies 94 [521] 0, 2 → 3 A ⋆ ! ! 0.115(2) Wilson loops 35Aoki 94 [522] 2 → 3 A ⋆ ! ! 0.108(5)(4) Wilson loops 35El-Khadra 92 [523] 0 → 3 A ⋆ ◦ ◦ 0.106(4) Wilson loops 35

# Result with a linear continuum extrapolation in a.

Table 38: Results for αMS(MZ). Nf = 3 results are matched at the charm and bottomthresholds and scaled to MZ to obtain the Nf = 5 result. The arrows in the Nf columnindicates which Nf (Nf = 0, 2 or a combination of both) were used to first extrapolate toNf = 3 or estimate the Nf = 3 value through a model/assumption. The exact proceduresused vary and are given in the various papers.

Nf = 2 + 1 and Nf = 2 + 1 + 1 simulations. For comparison, we also include results fromNf = 0, 2 simulations, which are not relevant for phenomenology. For the Nf ≥ 3 simulations,the conversion from Nf = 3 to Nf = 5 is made by matching the coupling constant at thecharm and bottom quark thresholds and using the scale as determined or used by the authors.For Nf = 0, 2 the results for αMS in the summary table come from evaluations of αMS at alow scale and are extrapolated in Nf to Nf = 3.

As can be seen from the tables and figures, at present there are several computationssatisfying the quality criteria to be included in the FLAG average. We note that none of

those calculations of α(5)

MS(MZ) satisfy all of our more stringent criteria: a ⋆ for the renor-

malization scale, perturbative behaviour and continuum extrapolation. The results, however,

168

FLAG WG: Aoki et al., 1310.8555


QCD and the Higgs Boson

15


Higgs production cross sections

16 [TeV]s

7 8 9 10 11 12 13 14

H+X

) [pb

]A

(pp

m

-110

1

10

210

LHC

HIG

GS

XS W

G 2

013

H (NNLO+NNLL QCD + NLO EW)App

H (NNLO QCD + NLO EW)q qApp

WH (NNLO QCD + NLO EW)App

ZH (NNLO QCD + NLO EW)App

H (NLO QCD)t tApp

Georgi et al. PRL40(1978)692

Jones & Petcov PLB84(1979)440

Kunszt!NPB247(1984)339

Glashow et al. PRD18(1978)1724

The challenges: (taken from [R. Tanaka, talk at Aspen Higgs WS 03/13 ])

Sven Heinemeyer, Royal Society WS: Exploiting the Higgs Breakthrough, 22.01.2014 12

The challenges: (taken from [R. Tanaka, talk at Aspen Higgs WS 03/13 ])


& ET


LHC Higgs Cross Section Working Group

19

3. Success stories

LHCHXSWG work is documented in:

Appeared: 01/11, 01/12, 07/13

Authors: 64, 120, 157

Pages: 151, 275, 404

Citations: 621, 271, 70


arXiv:1101.0593, 1201.3084, 1307.1347


Higgs Production Cross Section

20


Inclusive (QCD) Cross Section

21

Table B.10: ggF cross sections at the LHC at 8 TeV and corresponding scale and PDF+αs uncertainties computedaccording to the PDF4LHC recommendation.

MH[GeV] σ [pb] QCD Scale [%] PDF+αs [%]124.4 19.45 +7.2 −7.9 +7.5 −6.9124.5 19.42 +7.2 −7.9 +7.5 −6.9124.6 19.39 +7.2 −7.9 +7.5 −6.9124.7 19.36 +7.2 −7.9 +7.5 −6.9124.8 19.33 +7.2 −7.8 +7.5 −6.9124.9 19.30 +7.2 −7.8 +7.5 −6.9125.0 19.27 +7.2 −7.8 +7.5 −6.9125.1 19.24 +7.2 −7.8 +7.5 −6.9125.2 19.21 +7.2 −7.8 +7.5 −6.9125.3 19.18 +7.2 −7.8 +7.5 −6.9125.4 19.15 +7.2 −7.8 +7.5 −6.9125.5 19.12 +7.2 −7.8 +7.5 −6.9125.6 19.09 +7.2 −7.8 +7.5 −6.9125.7 19.06 +7.2 −7.8 +7.5 −6.9125.8 19.03 +7.2 −7.8 +7.5 −6.9125.9 19.00 +7.2 −7.8 +7.5 −6.9126.0 18.97 +7.2 −7.8 +7.5 −6.9126.1 18.94 +7.2 −7.8 +7.5 −6.9126.2 18.91 +7.2 −7.8 +7.5 −6.9126.3 18.88 +7.2 −7.8 +7.5 −6.9126.4 18.85 +7.2 −7.8 +7.5 −6.9126.5 18.82 +7.2 −7.8 +7.5 −6.9126.6 18.80 +7.2 −7.8 +7.5 −6.9126.7 18.77 +7.2 −7.8 +7.5 −6.9126.8 18.74 +7.1 −7.8 +7.5 −6.9126.9 18.71 +7.1 −7.8 +7.5 −6.9127.0 18.68 +7.1 −7.8 +7.5 −6.9127.1 18.65 +7.1 −7.8 +7.5 −6.9127.2 18.62 +7.1 −7.8 +7.5 −6.9127.3 18.59 +7.1 −7.8 +7.5 −6.9127.4 18.57 +7.1 −7.8 +7.5 −6.9127.5 18.54 +7.1 −7.8 +7.5 −6.9127.6 18.51 +7.1 −7.8 +7.5 −6.9127.7 18.48 +7.1 −7.8 +7.5 −6.9127.8 18.45 +7.1 −7.8 +7.5 −6.9127.9 18.42 +7.1 −7.8 +7.5 −6.9128.0 18.40 +7.1 −7.8 +7.5 −6.9128.1 18.37 +7.1 −7.8 +7.5 −6.9128.2 18.34 +7.1 −7.8 +7.5 −6.9128.3 18.31 +7.1 −7.8 +7.5 −6.9128.4 18.28 +7.1 −7.8 +7.5 −6.9128.5 18.26 +7.1 −7.8 +7.5 −6.9128.6 18.23 +7.1 −7.8 +7.5 −6.9128.7 18.20 +7.1 −7.8 +7.5 −6.9128.8 18.17 +7.1 −7.8 +7.5 −6.9128.9 18.15 +7.1 −7.8 +7.5 −6.9129.0 18.12 +7.1 −7.8 +7.5 −6.9

289

HXSWG vol.3

sggF(8 TeV) = 19.1+ 2.0 pbNNLO

�H(s) =X

i,j

Zdx1dx2 fi(x1, µ

2F ) fj(x2, µ

2F ) �ij

�x1x2s,m

2H,↵S(µ

2R), µ

2F , µ

2R

�



22

�ggF ⇡ �LO

⇥1 + {�↵S + (�↵S)

2 + (�↵S)3 + · · · }

⇤

LO NLO NNLO N3LOXS order

σ[pb]

0

5

10

15

20

25

Fixed PDF LO

Fixed PDF NLO

Fixed PDF NNLO

Figure 1: The cross section for Higgs production in gluon fusion, computed varying theperturbative order of the matrix element. The label on the x-axis denotes the order ofthe matrix element, while in each case the three points from left to right are obtainedrespectively using LO, NLO, and NNLO PDFs. The uncertainties are obtained varyingthe renormalization scale by a factor two about µR = mH . The N3LO result is theapproximation of Ref. [4].

Gluon fusion, the dominant Higgs production channel at the LHC, has a slowly con-vergent expansion in perturbative QCD: the inclusive cross section is currently known upto next-to-next-to-leading order (NNLO) [1–3], and a recent approximate determinationof the N3LO result has been presented [4], while rapid progress on the exact computationhas been reported [5].

With N3LO results around the corner, it is natural to ask whether these will be ofany use, given that fully consistent N3LO parton distributions (PDFs) are not likely to beavailable any time soon, essentially because the determination of N3LO anomalous dimen-sions would require a fourth-order computation, for instance of deep-inelastic structurefunctions, or Wilson coefficients. Clearly, this question is related to the more general issueof theoretical uncertainties on PDFs: current PDF uncertainties [6] only reproduce theuncertainty in the underlying data, and of the procedure used to propagate it onto PDFs,but not that related to missing higher-order corrections in the theory used for PDF deter-mination. Henceforth in this paper we will call ‘theoretical uncertainty’ the uncertaintydue to the fixed-order truncation of the perturbative expansion, sometimes [7] also calledmissing higher-order uncertainty, or MHOU.

Here we address this set of issues in the specific context of Higgs production in gluonfusion. We use the dependence on the perturbative order of the prediction for this processas either the PDF or the matrix element are taken at different orders as an estimate thetheoretical uncertainty on either. We then address the more general issue of how one mayestimate theoretical uncertainties on PDFs and matrix elements, specifically by using theapproach of Cacciari and Houdeau [8].

We first compute the cross-section using the ggHiggs code [4,9], with default settings 1.

1We have checked that similar results are obtained using ihixs [10] version 1.3.3. Note that previous

2

⇡

pp @ 8 TeV

Forte, Isgro, Vita, 1312.6688

Higher-order effects are larger than x2 scale variation estimates

� ⇡ 5.6

Ball et al., 1303.3590



23

�ggF ⇡ �LO

⇥1 + {�↵S + (�↵S)

2 + (�↵S)3 + · · · }

⇤⇡ �LO

1� +

1� �↵S

�


σ[pb]

0

5

10

15

20

25

Fixed PDF LO

Fixed PDF NLO

Fixed PDF NNLO







2

⇡

pp @ 8 TeV



� ⇡ 5.6

Ball et al., 1303.3590



24

�ggF ⇡ �LO

⇥1 + {�↵S + (�↵S)

2 + (�↵S)3 + · · · }

⇤⇡ �LO

1� +

1� �↵S

�


σ[pb]

0

5

10

15

20

25

Fixed PDF LO

Fixed PDF NLO

Fixed PDF NNLO







2

⇡

pp @ 8 TeV



� ⇡ 5.6

Ball et al., 1303.3590

23.6 pb



25


σ[pb]

0

5

10

15

20

25

Fixed PDF LO

Fixed PDF NLO

Fixed PDF NNLO







2

⇡

pp @ 8 TeV

David & Passarino, 1307.1843: 22.5 + 2.6 pb

sggF(8 TeV) = 19.1+ 2.0 pbNNLO

Series extrapolation: 23.6 + ?? pb

Full N3LO coming soon (Anastasiou et al.)



26

ATLAS-CONF-2014-009 (Moriond EW)

ττ,ZZ*,WW*,γγ

ggF+ttHµ

-2 -1 0 1 2 3 4 5 6

ττ,Z

Z*,

WW

*,γγ VB

F+

VH

µ

-2

0

2

4

6

8

10Standard Model

Best fit

68% CL

95% CL

γγ →H

4l→ ZZ* →H

νlν l→ WW* →H

ττ →H

PreliminaryATLAS

-1Ldt = 4.6-4.8 fb∫ = 7 TeV s-1Ldt = 20.3 fb∫ = 8 TeV s

= 125.5 GeVHm

Figure 2: Likelihood contours in the (µ fggF+ttH , µ

fVBF+VH) plane for the channels f=H! ��,

H!ZZ⇤! 4`, H!WW⇤! `⌫`⌫, H ! ⌧⌧ and a Higgs boson mass mH = 125.5 GeV. The sharp loweredge of the H!ZZ⇤! 4` contours is due to the small number of events in this channel and the require-ment of a positive pdf. The best-fit values to the data (⇥) and the 68% (full) and 95% (dashed) CLcontours are indicated, as well as the SM expectations (+).

5 Coupling fits

In the previous section signal strength scale factors µ fi for given Higgs boson production or decay modes

are discussed. However, for a measurement of Higgs boson couplings, production and decay modescannot be treated independently. Scenarios with a consistent treatment of Higgs boson couplings inproduction and decay modes are studied in this section. All uncertainties on the best-fit values shown inthis Section take into account both experimental and theoretical systematic values.

5.1 Framework for coupling scale factor measurements

Following the leading order (LO) tree level motivated framework and benchmarks recommended inRef. [14], measurements of coupling scale factors are implemented for the combination of all analysesand channels summarised in Table 1. This framework is based on the following assumptions:

• The signals observed in the di↵erent search channels originate from a single narrow resonancewith a mass near 125.5 GeV. The case of several, possibly overlapping, resonances in this massregion is not considered.

• The width of the assumed Higgs boson near 125.5 GeV is neglected, i.e. the zero-width approxi-mation is used. Hence the product � ⇥ BR(i ! H ! f ) can be decomposed in the following wayfor all channels:

� ⇥ BR(i! H ! f ) =�i · �f

�H,

where �i is the production cross section through the initial state i, �f the partial decay width intothe final state f and �H the total width of the Higgs boson.

• Only modifications of couplings strengths, i.e. of absolute values of couplings, are taken intoaccount, while the tensor structure of the couplings is assumed to be the same as in the SM. This

7

ATLAS, but not CMS, find ggF excess in gg and ZZ* channels

CMS PAS HIG-13-005

16 4 Results

ggH,ttHµ

-1 0 1 2 3

VBF,

VHµ

0

2

4

6ττ →H

WW→H ZZ→H bb→H γγ →H

CMS Preliminary -1 19.6 fb≤ = 8 TeV, L s -1 5.1 fb≤ = 7 TeV, L s

Figure 4: The 68% (solid lines) CL region for the signal strength in the gluon-gluon-fusion-plus-ttH and in the VBF-plus-VH production mechanisms, µggH+ttH and µVBF+VH, respectively. Thedifferent colours show the results obtained by combining data from each of the five analyseddecay modes: gg (green), WW (blue), ZZ (red), tt (violet), bb (cyan). The crosses indicate thebest-fit values. The diamond at (1,1) indicates the expected values for the SM Higgs boson. Acombination of the different decay modes is not possible without making assumptions on therelative branching fractions.

VBFµ

0 1 2 3

ln L

∆- 2

0123456789

10Observed

Exp. for SM H


ttHµ, VHµ, qqHµ, ggHµ

VHµ

0 1 2 3

ln L

∆- 2

0123456789

10Observed

Exp. for SM H


ttHµ, VHµ, qqHµ, ggHµ

Figure 5: Likelihood scan versus µVBF (left) and µVH (right). The solid curve is the data. Thedashed line indicates the expected median results in the presence of the SM Higgs boson. Cross-ings with the horizontal thick and thin red lines denote the 68% CL and 95% CL intervals.


Cross Sections at 13 TeV

27

SM Higgs production cross sections at √s = 13-14 TeVSM Higgs production cross sections (calculated official numbers)

√s = 13.0 TeV√s = 13.5 TeV√s = 14.0 TeV

Higgs cross section estimation via parton-luminosity functionParton luminosity ratio

SM Higgs production cross sections (calculated official numbers)See 8 TeV XS TWiki page for more information on calculation conditions.All ggF and VBF numbers are based upon complex-pole-scheme (CPS), while WH/ZH and ttH numbers are with zero-width-approximation (ZWA).

√s = 13.0 TeV

gluon-gluon Fusion Process

All cross sections are in complex-pole-scheme from the dFG program. They are computed at NNLL QCD and NLO EW.

mH (GeV) Cross Section (pb) +QCD Scale % -QCD Scale % +(PDF+αs) % -(PDF+αs) %

125.0 43.92 +7.4 -7.9 +7.1 -6.0

125.5 43.62 +7.4 -7.9 +7.1 -6.0

126.0 43.31 +7.4 -7.9 +7.1 -6.0

VBF Process

At NNLO QCD and NLO EW. All cross sections are in complex-pole-scheme.


125.0 3.748 +0.7 -0.7 +3.2 -3.2

125.5 3.727 +1.0 -0.7 +3.4 -3.4

126.0 3.703 +1.3 -0.6 +3.1 -3.1

WH Process

The cross section are calculated at NNLO QCD and NLO EW.


125.0 1.380 +0.7 -1.5 +3.2 -3.2

125.5 1.362 +0.9 -1.5 +3.2 -3.2

126.0 1.345 +0.8 -1.5 +3.2 -3.2

ZH Process

The cross section are calculated at NNLO QCD and NLO EW.


125.0 0.8696 +3.8 -3.8 +3.5 -3.5

125.5 0.8594 +3.8 -3.8 +3.5 -3.5

126.0 0.8501 +3.8 -3.9 +3.5 -3.5

ttH Process

The cross section are calculated at NLO QCD.


125.0 0.5085 +5.7 -9.3 +8.8 -8.8

125.5 0.5027 +5.7 -9.3 +8.8 -8.8

126.0 0.4966 +5.7 -9.3 +8.8 -8.8

bbH Process

The cross sections are the Santander matched numbers with 5FS (NNLO) and 4FS (NLO) (see Section 12.2.2 in CERN Report 2).

TWiki > LHCPhysics Web > CrossSections > CERNYellowReportPageAt1314TeV(05 Apr 2014, ReiTanaka)

HXSWG 05/04/2014


Cross Section vs Energy

28

sggF(13 TeV) = 43.3 + 4.3 pbNNLO

sggF(13 TeV) = 55.9 + ?? pbSeries extrapolation:

sggF(13 TeV) = 53 + 8 pbMy best guess:

http://theory.fi.infn.it/grazzini/hcalculators.html

http://theory.fi.infn.it/grazzini/hcalculators.html


Higgs Differential Cross Sections

29


Higgs qT & ET

30

pTi

Higgs transverse momentum

qT = �X

pTi

Radiated transverse energy

ET =X

|pTi|

Bozzi et al. 0705.3887

Catani & Grazzini, 1011.3918Mantry & Petriello, 0911.4135

de Florian et al. 1109.2109

Papaefstathiou, Smillie, BW, 1002.4375+Grazzini, 1403.3394


Higgs qT (fixed order)

31

Large logs of mH2/qT2 need resummation

(N)LO �!qT!0

(�)1


Higgs qT & ET (fixed order)

32

Large logs of mH2/ET2 need resummation

(N)LO �!ET!0

(�)1


Resummation of Higgs qT

33

1

�gg

d2�gg

dq2T

⇠ �2(qT) + ↵S

Zd2pT

Ag

p2T

lnm2

H

p2T

+Bg

p2T

�

+

�2(qT + pT) + . . .

d� =

Zdx1dx2 fa(x1, µ)fb(x2, µ) d�ab(x1x2s, µ, . . .)



34

⇠Z

d2b

(2⇡)2eib·qT

⇢1 + ↵S

Zd2pT

Ag

p2T

lnm2

H

p2T

+Bg

p2T

� �eib·pT � 1

�+ . . .

�

1

�gg

d2�gg

dq2T

⇠ �2(qT) + ↵S

Zd2pT

Ag

p2T

lnm2

H

p2T

+Bg

p2T

�

+

�2(qT + pT) + . . .

d� =




35

⇠Z

d2b

(2⇡)2eib·qT

⇢1 + ↵S

Zd2pT

Ag

p2T

lnm2

H

p2T

+Bg

p2T

� �eib·pT � 1

�+ . . .

�

1

�gg

d2�gg

dq2T

⇠ �2(qT) + ↵S

Zd2pT

Ag

p2T

lnm2

H

p2T

+Bg

p2T

�

+

�2(qT + pT) + . . .

d� =


⇠Z

d

2b

(2⇡)2e

ib·qTexp

⇢↵S

Zd

2pT

Ag

p2T

ln

m2H

p2T

+

Bg

p2T

� �e

ib·pT � 1

��


Resummation & matching of Higgs qT

36

⇠Z

d2b

(2⇡)2eib·qT

⇢1 + ↵S

Zd2pT

Ag

p2T

lnm2

H

p2T

+Bg

p2T

� �eib·pT � 1

�+ . . .

�

1

�gg

d2�gg

dq2T

⇠ �2(qT) + ↵S

Zd2pT

Ag

p2T

lnm2

H

p2T

+Bg

p2T

�

+

�2(qT + pT) + . . .

d� =


⇠Z

d

2b

(2⇡)2e

ib·qTexp

⇢↵S

Zd

2pT

Ag

p2T

ln

m2H

p2T

+

Bg

p2T

� �e

ib·pT � 1

��

d�

dqT=

d�

dqT

�

resum

�d�

dqT

�

resum,NLO

+

d�

dqT

�

NLO


Higgs transverse momentum: 8 TeV

37

Resummation affects spectrum out to larger qT

Peak at ~10 GeV: log(mH2/qT2)~5.1


Higgs transverse momentum: 14 TeV

Peak at ~10 GeV: log(mH2/qT2)~5.1

Resummation affects spectrum out to larger qT



39

⇠Z

d2b

(2⇡)2eib·qT

⇢1 + ↵S

Zd2pT

Ag

p2T

lnm2

H

p2T

+Bg

p2T

� �eib·pT � 1

�+ . . .

�

1

�gg

d2�gg

dq2T

⇠ �2(qT) + ↵S

Zd2pT

Ag

p2T

lnm2

H

p2T

+Bg

p2T

�

+

�2(qT + pT) + . . .

d� =


⇠Z

d

2b

(2⇡)2e

ib·qTexp

⇢↵S

Zd

2pT

Ag

p2T

ln

m2H

p2T

+

Bg

p2T

� �e

ib·pT � 1

��


Resummation of Higgs ET

40

⇠Z

d2b

(2⇡)2eib·qT

⇢1 + ↵S

Zd2pT

Ag

p2T

lnm2

H

p2T

+Bg

p2T

� �eib·pT � 1

�+ . . .

�

1

�gg

d2�gg

dq2T

⇠ �2(qT) + ↵S

Zd2pT

Ag

p2T

lnm2

H

p2T

+Bg

p2T

�

+

�2(qT + pT) + . . .

d� =


⇠Z

d

2b

(2⇡)2e

ib·qTexp

⇢↵S

Zd

2pT

Ag

p2T

ln

m2H

p2T

+

Bg

p2T

� �e

ib·pT � 1

��

⇠Z

d⌧

2⇡e

i⌧ETexp

⇢↵S

Zd

2pT

Ag

p2T

ln

m2H

p2T

+

Bg

p2T

�⇣e

�i⌧ |pT| � 1

⌘�

1

�gg

d�gg

dET⇠ �(ET ) + ↵S

Zd2pT

Ag

p2T

lnm2

H

p2T

+Bg

p2T

�

+

�(ET � |pT|) + . . .



41

Defined for ET 0

1

�gg

d�gg

dET⇠

Z +1

�1

d⌧

2⇡e

i⌧ETexp

⇢↵S

Zd

2pT

Ag

p2T

ln

m2H

p2T

+

Bg

p2T

�⇣e

�i⌧ |pT| � 1

⌘�

><



42

Defined for ET 0

1

�gg

d�gg

dET⇠

Z +1

�1

d⌧

2⇡e

i⌧ETexp

⇢↵S

Zd

2pT

Ag

p2T

ln

m2H

p2T

+

Bg

p2T

�⇣e

�i⌧ |pT| � 1

⌘�

><

For ET < 0, can close t-contour in lower half-plane



43

Defined for ET 0

1

�gg

d�gg

dET⇠

Z +1

�1

d⌧

2⇡e

i⌧ETexp

⇢↵S

Zd

2pT

Ag

p2T

ln

m2H

p2T

+

Bg

p2T

�⇣e

�i⌧ |pT| � 1

⌘�

><


No singularities in lower half-plane



44

Defined for ET 0

1

�gg

d�gg

dET⇠

Z +1

�1

d⌧

2⇡e

i⌧ETexp

⇢↵S

Zd

2pT

Ag

p2T

ln

m2H

p2T

+

Bg

p2T

�⇣e

�i⌧ |pT| � 1

⌘�

><


No singularities in lower half-plane


Resummation & matching of Higgs ET

45

to all orders in ↵s. The resummation applies to leading, next-to-leading and next-to-next-

to-leading logarithms of ET /Q ((N)NLL) where Q is the hard process scale, taken to be the

Higgs mass mH . Thus it improves the treatment of the small ET region, where the fixed-

order prediction diverges whereas the actual distribution must tend to zero as ET ! 0.

By matching the resummed prediction to the NLO one we improve the treatment of both

small and large ET .

Our approach follows on from Ref. [3], based in turn on the early work on ET resumma-

tion in vector boson production [4–6] and closely related to the resummation of transverse

momentum in vector boson [7, 8, 11] and Higgs production [10, 12]. We make a number of

improvements relative to Ref. [3], including:

• Predictions for the experimentally relevant Higgs mass of 126 GeV, at centre-of-mass

energies 8 and 14 TeV;

• Fixed-order predictions to NLO;

• Expansion of the ET resummation formula to NLO, and demonstration that to this

order all the logarithms are consistent with the fixed-order prediction;

• Matching of the resummed and NLO predictions across the whole range of ET ;

• A constraint on the perturbative unitarity of the prediction, using the method of

Ref. [9], which reduces the impact of logarithmic terms in the large-ET region;

• Monte Carlo studies of the e↵ects of rapidity cuts and preclustering.

The paper is organized as follows. In Section 2, we review the resummation calcula-

tion and then describe the necessary modifications to implement the unitarity condition

mentioned above. This involves new formalism and the evaluation of new integrals in this

prescription. In Section 3, we expand our resummed result to next-to-leading order in

order to match our results to the fixed-order prediction at this accuracy. This renders

our predictions positive throughout the ET -range. In Section 4, we investigate the ET

distribution further through Monte Carlo studies. We first reweight Monte Carlo results to

our analytic distribution and then investigate the impact of hadronisation and underlying

event. We end the main text in Section 5 with conclusions and discussion. A number of

appendices then contain supplementary results.

2. Resummation of logarithms

Here we summarize the results of refs. [3,5] as applied to Higgs production. The resummed

component of the transverse energy distribution in the process h1h2 ! HX at scale Q has

the form

d�HdQ2 dET

�

res.

=1

2⇡

X

a,b

Z 1

0dx1

Z 1

0dx2

Z +1

�1d⌧ e�i⌧ET fa/h1

(x1, µ) fb/h2(x2, µ)

· WHab (x1x2s;Q, ⌧, µ) (2.1)

– 2 –

to all orders in ↵s. The resummation applies to leading, next-to-leading and next-to-next-

to-leading logarithms of ET /Q ((N)NLL) where Q is the hard process scale, taken to be the

Higgs mass mH . Thus it improves the treatment of the small ET region, where the fixed-

order prediction diverges whereas the actual distribution must tend to zero as ET ! 0.

By matching the resummed prediction to the NLO one we improve the treatment of both

small and large ET .

Our approach follows on from Ref. [3], based in turn on the early work on ET resumma-

tion in vector boson production [4–6] and closely related to the resummation of transverse

momentum in vector boson [7, 8, 11] and Higgs production [10, 12]. We make a number of

improvements relative to Ref. [3], including:

• Predictions for the experimentally relevant Higgs mass of 126 GeV, at centre-of-mass

energies 8 and 14 TeV;

• Fixed-order predictions to NLO;

• Expansion of the ET resummation formula to NLO, and demonstration that to this

order all the logarithms are consistent with the fixed-order prediction;

• Matching of the resummed and NLO predictions across the whole range of ET ;

• A constraint on the perturbative unitarity of the prediction, using the method of

Ref. [9], which reduces the impact of logarithmic terms in the large-ET region;

• Monte Carlo studies of the e↵ects of rapidity cuts and preclustering.

The paper is organized as follows. In Section 2, we review the resummation calcula-

tion and then describe the necessary modifications to implement the unitarity condition

mentioned above. This involves new formalism and the evaluation of new integrals in this

prescription. In Section 3, we expand our resummed result to next-to-leading order in

order to match our results to the fixed-order prediction at this accuracy. This renders

our predictions positive throughout the ET -range. In Section 4, we investigate the ET

distribution further through Monte Carlo studies. We first reweight Monte Carlo results to

our analytic distribution and then investigate the impact of hadronisation and underlying

event. We end the main text in Section 5 with conclusions and discussion. A number of

appendices then contain supplementary results.

2. Resummation of logarithms

Here we summarize the results of refs. [3,5] as applied to Higgs production. The resummed

component of the transverse energy distribution in the process h1h2 ! HX at scale Q has

the form

d�HdQ2 dET

�

res.

=1

2⇡

X

a,b

Z 1

0dx1

Z 1

0dx2

Z +1

�1d⌧ e�i⌧ET fa/h1

(x1, µ) fb/h2(x2, µ)

· WHab (x1x2s;Q, ⌧, µ) (2.1)

– 2 –

where fa/h(x, µ) is the parton distribution function (PDF) of parton a in hadron h at

factorization scale µ, taken to be the same as the renormalization scale here (we illustrate

the impact of varying this scale in section 3). In what follows we use the MS renormalization

scheme. To take into account the constraint that the transverse energies of emitted partons

should sum to ET , the resummation procedure is carried out in the domain that is Fourier

conjugate to ET . The transverse energy distribution (2.1) is thus obtained by performing

the inverse Fourier transformation with respect to the “transverse time”, ⌧ . The factor

WHab is the perturbative and process-dependent partonic cross section that embodies the

all-order resummation of the large logarithms ln(Q⌧). Since ⌧ is conjugate to ET , the limit

ET ⌧ Q corresponds to Q⌧ � 1.

As in the case of transverse momentum resummation [15], the resummed partonic cross

section can be written in the following form:

WHab (s;Q, ⌧, µ) =

Z 1

0dz1

Z 1

0dz2 Cga(↵S(µ), z1; ⌧, µ) Cgb(↵S(µ), z2; ⌧, µ) �(Q

2 � z1z2s)

· �Hgg(Q,↵S(Q)) Sg(Q, ⌧) . (2.2)

Here �Hgg is the cross section for the partonic subprocess of gluon fusion, gg ! H, through

a massive-quark loop:

�Hgg(Q,↵S(Q)) = �(Q2 �m2

H)�H0 , (2.3)

where in the limit of infinite quark mass

�H0 =

↵2S(mH)GFm

2H

288⇡p2

. (2.4)

Sg(Q, ⌧) is the appropriate gluon form factor, which in the case of ET resummation takes

the form [5,6]

Sg(Q, ⌧) = exp

⇢�2

Z Q

0

dq

q

2Ag(↵S(q)) ln

Q

q+Bg(↵S(q))

� �1� eiq⌧

��. (2.5)

The functions Ag(↵S), Bg(↵S), as well as the coe�cient functions Cga in Eq. (2.2), contain

no ln(Q⌧) terms and are perturbatively computable as power expansions with constant

coe�cients:

Ag(↵S) =1X

n=1

⇣↵S

⇡

⌘nA(n)

g , (2.6)

Bg(↵S) =1X

n=1

⇣↵S

⇡

⌘nB(n)

g , (2.7)

Cga(↵S, z) = �ga �(1� z) +1X

n=1

⇣↵S

⇡

⌘nC(n)ga (z) . (2.8)

Thus a calculation to NLO in ↵S involves the coe�cients A(1)g , A(2)

g , B(1)g , B(2)

g and C(1)ga .

Where logarithmically enhanced terms are concerned, knowledge of A(1)g leads to the resum-

mation of the leading logarithmic (LL) contributions at small ET , which in the di↵erential

– 3 –













WHab (s;Q, ⌧, µ) =

Z 1

0dz1

Z 1


2 � z1z2s)

· �Hgg(Q,↵S(Q)) Sg(Q, ⌧) . (2.2)



�Hgg(Q,↵S(Q)) = �(Q2 �m2

H)�H0 , (2.3)


�H0 =

↵2S(mH)GFm

2H

288⇡p2

. (2.4)


the form [5,6]

Sg(Q, ⌧) = exp

⇢�2

Z Q

0

dq

q

2Ag(↵S(q)) ln

Q

q+Bg(↵S(q))

� �1� eiq⌧

��. (2.5)



coe�cients:

Ag(↵S) =1X

n=1

⇣↵S

⇡

⌘nA(n)

g , (2.6)

Bg(↵S) =1X

n=1

⇣↵S

⇡

⌘nB(n)

g , (2.7)

Cga(↵S, z) = �ga �(1� z) +1X

n=1

⇣↵S

⇡

⌘nC(n)ga (z) . (2.8)


g , B(1)g , B(2)

g and C(1)ga .



– 3 –













WHab (s;Q, ⌧, µ) =

Z 1

0dz1

Z 1


2 � z1z2s)

· �Hgg(Q,↵S(Q)) Sg(Q, ⌧) . (2.2)



�Hgg(Q,↵S(Q)) = �(Q2 �m2

H)�H0 , (2.3)


�H0 =

↵2S(mH)GFm

2H

288⇡p2

. (2.4)


the form [5,6]

Sg(Q, ⌧) = exp

⇢�2

Z Q

0

dq

q

2Ag(↵S(q)) ln

Q

q+Bg(↵S(q))

� �1� eiq⌧

��. (2.5)



coe�cients:

Ag(↵S) =1X

n=1

⇣↵S

⇡

⌘nA(n)

g , (2.6)

Bg(↵S) =1X

n=1

⇣↵S

⇡

⌘nB(n)

g , (2.7)

Cga(↵S, z) = �ga �(1� z) +1X

n=1

⇣↵S

⇡

⌘nC(n)ga (z) . (2.8)


g , B(1)g , B(2)

g and C(1)ga .



– 3 –













WHab (s;Q, ⌧, µ) =

Z 1

0dz1

Z 1


2 � z1z2s)

· �Hgg(Q,↵S(Q)) Sg(Q, ⌧) . (2.2)



�Hgg(Q,↵S(Q)) = �(Q2 �m2

H)�H0 , (2.3)


�H0 =

↵2S(mH)GFm

2H

288⇡p2

. (2.4)


the form [5,6]

Sg(Q, ⌧) = exp

⇢�2

Z Q

0

dq

q

2Ag(↵S(q)) ln

Q

q+Bg(↵S(q))

� �1� eiq⌧

��. (2.5)



coe�cients:

Ag(↵S) =1X

n=1

⇣↵S

⇡

⌘nA(n)

g , (2.6)

Bg(↵S) =1X

n=1

⇣↵S

⇡

⌘nB(n)

g , (2.7)

Cga(↵S, z) = �ga �(1� z) +1X

n=1

⇣↵S

⇡

⌘nC(n)ga (z) . (2.8)


g , B(1)g , B(2)

g and C(1)ga .



– 3 –

Figure 3: Transverse-energy distribution in Higgs boson production at the LHC at 8 and 14 TeV.Blue: resummed only. Red: resummed and matched to NLO. Green: NLO only. The solid curvescorrespond to renormalization scale mH , the dashed to 2mH and mH/2.

Away from the small-ET region, the NLO data are then well described by a parametriza-

tion of the form"d�HdET

� d�HdqT

��qT=ET

#

NLO

= Logarithmic terms +a1ET

mH(mH + a2ET ) + a3E2T

, (3.16)

as shown by the red curves in Fig. 2, with the parameter values shown.

To match the resummed and NLO ET distributions, we have to subtract the NLO

logarithmic terms (3.14), which are already included in the resummation, and replace

them by the full NLO result:

d�HdET

=

d�HdET

�

resum

�d�HdET

�

resum,NLO

+

d�HdET

�

NLO

. (3.17)

3.3 Results

The resulting resummed and matched ET distributions at the LHC at 8 and 14 TeV are

shown in Fig. 3. For all these predictions we use the best-fit value B(2)g = �3.0 found

from the NLO data. The distribution peaks at around ET = 35 GeV at both centre-of-

mass energies, considerably above the peak in the Higgs transverse-momentum distribution,

around qT = 12 GeV [11]. Thus in the peak region of ET the resummed logarithms are

not so dominant as in the corresponding region of qT . On the other hand, the fixed-order

NLO prediction is rising rapidly and unphysically towards smaller values of ET .2

The purely resummed distribution becomes slightly negative at small and large ET ,

which is also unphysical. The e↵ect of matching is to raise the distribution to positive

values, close to the fixed-order prediction at high ET . The matched prediction is still

somewhat unstable at small ET , owing to the delicate cancellation of diverging logarithmic

2At very small ET it turns over and tends to �1, as seen in Fig. 1.

– 12 –


Transverse energy distribution

Peak at ~35 GeV: log(mH2/ET2)~2.6

Resummation affects spectrum out to much larger ET

Unlike qT, the Underlying Event also contributes…


Summary

47

Scale dependence is a (rough) guide to precision

Higgs ggF cross section at 13 TeV is still very uncertain

My estimate: sggF(13 TeV) = 53 + 11 pb

Higgs transverse momentum resummed to NNLL+NLO

Peak qT ~10 GeV, independent of energy

Radiated transverse energy resummed to (N)NLL+NLO

Peak ET ~35 GeV in associated transverse energy

Contribution from Underlying Event to be considered …

QCD factorization allows precise predictions for LHC

Documents

QCD and Event Generation for the Large Hadron Collider · Bryan Webber Burg Liebenzell, Sept 2014 PDF Uncertainties 10 M X 102 103 Gluon - Gluon Luminosity 0.8 0.85 0.9 0.95 1 1.05