Towards N 3 LO QCD Higgs production cross section Takahiro Ueda TTP - - PowerPoint PPT Presentation

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Towards N 3 LO QCD Higgs production cross section Takahiro Ueda TTP KIT Karlsruhe, Germany Based on work in collaboration with: Chihaya Anzai, Maik Hschele, Jens Hoff, Matthias Steinhauser [Some results are from a sub-project paper,

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Towards N3LO QCD Higgs production cross section

Takahiro Ueda TTP KIT Karlsruhe, Germany

HP2.5 3-5 Sep 2014, GGI, Florence

Based on work in collaboration with: Chihaya Anzai, Maik Höschele, Jens Hoff, Matthias Steinhauser

[Some results are from a sub-project paper, arXiv:1407.4049 Maik Höschele, Jens Hoff, TU]

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Introduction

  • The discovery of a Higgs boson SM? BSM?

Needs for studying its production and decay processes

  • Higgs production @ LHC
  • Dominated by the

gluon fusion channel

  • Mainly QCD corrections
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Introduction

  • NNLO QCD corrections to the partonic cross sections in the

gluon fusion channel are known

  • Converge slowly The scale uncertainty is the main

source of theoretical uncertainty

  • In total: large theoretical uncertainty 10-15 %

[Harlander, Kilgore ’02; Anastasiou, Melnikov ’02; Ravindran, Smith, van Neerven ’03] [Figs. from arXiv:hep-ph/0201206]

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Improve TH prediction

  • To improve the theoretical prediction:
  • Resummations
  • Approximated N3LO
  • PDFs or
  • Other contributions (finite top mass, bottom, EW etc.)
  • Our ultimate goal:

Improve the the theoretical prediction by computing N3LO QCD corrections in the gluon fusion channel

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Heavy-Top Effective Theory

  • Top-loop induced ggh vertex
  • Wilson coefficient is known up to 3-loop
  • Small power corrections in at NNLO
  • Good approximation to the result in the full theory

[Chetyrkin, Kniehl, Steinhauser ’98; Krämer, Laenen, Spira ’98] [Harlander, Ozeren ’09; Harlander, Mantler, Marzani, Ozeren ’10; Pak, Rogal, Steinhauser ’09-’11]

: VEV

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Overview

  • Dimensional regularization
  • Consider all possible cuts in forward scattering diagrams
  • One kinematic variable
  • (Unrenormalized) partonic cross sections have expansions like

LO NLO NNLO N3LO

: Higgs mass : partonic center-of-mass energy : Higgs threshold / soft partons

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Status of N3LO Calculations

  • Triple-virtual (form factor)
  • Double-virtual real
  • Real virtual-squared
  • Triple-virtual
  • Double-real virtual
  • IR subtraction terms

known Soft limit known known known First two terms in soft expansion known

[Baikov, Chetyrkin, Smirnov2, Steinhauser ’09; Gehrmann, Glover, Huber, Ikizlerli, Studerus ’10] [Duhr, Gehrmann ’13; Li, Zhu ’13] [Anastasiou, Duhr, Dulat, Herzog, Mistlberger ’13; Kilgore ’13]

Soft limit known

[Anastasiou, Duhr, Dulat, Furlan, Gehrmann, Herzog, Mistlberger ’14; Li, Manteuffel, Schabinger, Zhu ’14] [Anastasiou, Duhr, Dulat, Mistlberger ’13] [Höschele, Hoff, Pak, Steinhauser, TU ’13; Buehler, Lazopoulos ’13]

  • Hadronic Higgs production cross section at threshold up to N3LO

[Anastasiou, Duhr, Dulat, Furlan, Gehrmann, Herzog, Mistlberger ’14]

Mistlberger's talk Dulat's talk

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Reverse Unitarity Technique

  • In the same way as loop integrals, cut integrals can be

reduced using integration-by-parts (IBP) relations Reduction to master integrals (MIs)

  • We used
  • an in-house implementation of Laporta algorithm

with TopoID for automatization and symmetries

  • FIRE

[Anastasiou, Melnikov ’02] [Smirnov]

[1-loop example]

[Hoff, Pak]

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Differential Equation Method

  • Consider the derivatives of MIs with respect to
  • Dependence on comes only from the Higgs propagator
  • Equivalent to increasing the index of the Higgs propagator

(put a dot) up to a constant factor

  • Resultant integrals can be reduced to the MIs

System of linear differential equations (DEs) for MIs

  • Can be quite complicated for higher loop orders

[Kotikov ’91; Bern, Dixon, Kosower ’94; Remiddi ’97; Gehrmann, Remiddi ’00]

[1-loop example]

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Change of Basis

  • The choice of MIs are not unique. We can choose another

basis of MIs:

  • The system of DEs for the new MIs are:

i.e., is determined by

  • : the system of DEs in the old basis
  • : how the new basis integrals are reduced into the old

basis integrals

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Canonical Basis

  • Conjecture: MIs for QCD integrals can be chosen as pure

functions of uniform weight, which makes it strikingly simple to solve the system of DEs

  • Choose a basis satisfying the following canonical form
  • The expansion of the system of DEs in is triangular; easy

to solve if the boundary condition is fixed at some We use the values at the soft limit

  • The solutions are expressed in iterated integrals of uniform

weight; in this case, harmonic polylogarithms

[Henn ’13] [Remiddi, Vermaseren ’00]

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Canonical Basis

[1-loop example]

cf.

  • r
  • r
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Find Canonical MIs

  • No general recipe, but some tips exist to find candidates that

may give the canonical form

  • Hints from the lower loop orders results
  • Massless bubble with a dot
  • is a good candidate at NNLO

Canonical MI at NLO Integrate out

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Find Canonical MIs

  • is one of 2-coupled MIs. Need another MI
  • One may try a similar integral but not canonical
  • It turns out a linear combination is better
  • Feynman parameter representation tells us
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Find Canonical MIs

  • Need a normalization factor

gives gives a canonical basis

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Add Another MI

  • Need a normalization factor
  • Adding a MI to a system is relatively easy:

Trial & error for each candidate, adjusting normalization gives gives a canonical basis

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Add Coupled MIs

  • Can one of the added coupled MIs be a canonical MI if the
  • ther is adequately chosen?
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Characteristic Form of Higher Order DE

  • Point: from a set of DEs of and , we can eliminate

and get a higher order DE of

  • The coefficients , and are independent on the

choice of

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Characteristic Form of Higher Order DE

  • For and , we have
  • If is a canonical MI (there exists a proper choice of , and

and gives in the canonical form), the coefficients have the following specific form because is linear in

  • Equating the C's both from and , we get a set of

(differential) equations for even though is unknown (at this point)

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Characteristic Form of Higher Order DE

  • Once all the elements in are obtained, one can get the

transformation from to :

  • is expressed as a linear combination of

Give linear equations (not DEs) cf.

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Algorithm for Coupled MIs

  • Input: a basis (and )
  • Assume is a canonical MI. This leads , where

is expressed as a linear combination of

  • Any contradiction means cannot be a canonical MI
  • C's do not have the specific form
  • Obtained is not of the canonical form
  • No consistent solution for
  • Answer to the previous question: is canonical
  • In principle, this algorithm can be extended to 3 or more

coupled MIs

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Canonical MIs at NNLO

3-particle cut diagrams (17 MIs) coupled

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coupled

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Canonical MIs at NNLO

3-particle cut diagrams (17 MIs)

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Canonical MIs at NNLO

2-particle cut diagrams (6 MIs)

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Canonical MIs at NNLO

2-particle cut diagrams (6 MIs)

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Canonical MIs at NNLO

  • The canonical basis and the reduction basis:
  • In the reduction basis, the values of the soft limit are obtained

from the corresponding phase space integrals

  • Translate the soft limits in reduction basis into those in the

canonical basis (Avoid direct evaluation of soft limits of diagrams with dotted cut propagators)

  • Solve MIs in the canonical basis with the soft limits as the

boundary condition

  • Checked with known results
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Example at N3LO

  • Sea-snake topology (11 MIs)
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Example at N3LO

coupled

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coupled diagrammatically similar to MIs of off-shell K4

[Henn, Smirnov2 ’14]

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Example at N3LO

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Example at N3LO

  • As the boundary condition, compute values of the soft limit in

the reduction basis, translated into those in the canonical basis

  • For this purpose, the method of soft MIs are used, where

the coefficients in soft expansions are expressed in a small number of soft MIs

  • The soft MI for this topology is only the phase-space
  • Without the reduction to the soft MI, the leading terms in

the soft expansions of the MIs can be computed with the standard MB techniques

  • Solved up to weight 6, enough for N3LO

[Anastasiou, Duhr, Dulat, Mistlberger ’13]

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Work in Progress

  • Other topologies: example: Tennis coat topology
  • As a preliminary result, we have obtained a canonical basis
  • The last 2 MIs are coupled. Applying aforementioned

algorithm to a candidate for the 35th MI succeeded

  • For the boundary condition, the standard MB technique

works??

  • 3- and 4-particle cuts (36 MIs)

1 2 1 2

Canonical MI at NNLO

MB Tools (MB.m and Mbasymptotics.m by Czakon; barnesroutines.m by Kosower)

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Summary

  • We studied MIs needed for the Higgs production via the DE

method with canonical bases

  • Recomputed MIs up to NNLO
  • Application to MIs at N3LO
  • We developed an algorithm to see whether or not an integral

in a coupled system of DEs can be a canonical MI

  • Work in progress for other topologies at N3LO