Geometric IR Subtraction for Real Radiation Universita degli studi - - PowerPoint PPT Presentation

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Geometric IR Subtraction for Real Radiation Universita degli studi - - PowerPoint PPT Presentation

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Geometric IR Subtraction for Real Radiation Universita degli studi di Milano & INFN 3 rd of May 2018 Franz Herzog The problem of IR divergences in differential calculations Individually both contributions diverge Final state IR

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Geometric IR Subtraction for Real Radiation

Universita degli studi di Milano & INFN

3rd of May 2018

Franz Herzog

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The problem of IR divergences in differential calculations

  • Individually both contributions diverge
  • Final state IR Divergences cancel in the sum
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A solution is to subtract

The real is rendered fjnite by the counterterm - which is subsequently added back to the virtual in integrated form

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[slide shamelessly stolen from Gavin Salam]

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A very brief history of Subtraction

  • Subtraction at NLO:

–

1991: phase space slicing [Glover, Giele,Kosower]

–

1995: FKS (residue subtraction) [Firxione, Kunszt, Signer]

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1996: Dipole subtraction [Catani, Seymour]

  • Subtraction at NNLO:(fully differential)

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2003: Sector Decomposition [Binoth, Heinrich;Anastasiou, Melnikov,Petriello] (limited in fjnal states, numerical CT, subtraction)

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2004: Subtraction for NNLO [Grazzini, Frixione]

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2005: Antennas [Glover, Gehrmann, Gehrmann-De Ridder] (subtraction, general?; analytic CT, phasespace generation complicated)

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2006: colorful subtraction scheme for jets [Somogyi, Trocsanyi, Del Duca] (subtraction, fjnal states; inital?, numerical CT)

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2007: Kt-subtraction [Catani, Grazzini] (slicing; analytic CT, color singlets; and limited number of fjnal states)

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2010: General subtraction with sector decomposition [Czakon; Boughezal, Melnikov, Petriello]

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2010: Non-linear Mappings [Anastasiou, FH, Lazopoulos] (limited massive colored fjnal states and color-singlets, numerical CT)

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2015: N-jettiness Subtraction [Boughezal, Focke, Giele, Liu, Petriello; Gaunt, Stahlhofen, Tackmann, Walsh] (general?, complicated soft function, numerical CT)

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2015: Projection to Born [Cacciari, Dreyer, Karlberg, Salam, Zanderigh] (limited in applications)

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Conventions

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Phase Space Measures and Volumes

Shorthand for massless particles: Shorthand for massive sums of momenta: The integrated volume: The familiar Lorentz invariant on-shell phase space measure:

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Phase Space Factorisation

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A simple Example

Collinear singularities: 1||2 and 2||3 Soft singularity: 2 →

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Singularities in Invariant Space

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Singularities evaluate to Poles Dimensional Regularisation

It is impractical to have to evaluate phase space integrals in D- dimensions! How can we subtract singularities before integration in a minimal way?

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A simple Slicing Scheme

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A simple Slicing Scheme

Partition of unity:

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Collinear Region

The collinear limit can be parameterised choosing as a normal coordinate:

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Collinear Phase Space

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Soft Phase Space

is parameterised by the normal coordinate since and

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Soft Phase Space

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Soft Collinear Phase Space

Order limits such that

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Singular Phase Spaces and Integrals

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Sum of Singular Regions

Counter terms reproduce correct poles and simple fjnite parts

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Evaluation of fjnite part

  • Use two different approaches:

i) Slicing ii) Subtraction

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Slicing Subtraction

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The Slicing scheme can be promoted to a fully local subtraction scheme. The Slicing scheme allows to defjne simple (to integrate) counter terms. Subtraction method easily outperforms its parent slicing method numerically. A slicing scheme can be defjned based on the phase space factorisation property.

What we learned from this simple example?

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The BIG Questions: Can we generalise to multi particle amplitudes? To NNLO? beyond?

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General Formalism

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Overlap contributions

Using normal coordinates to defjne regions we partition the phase space into a singular and a fjnite region The fjnite region can expressed as Where is the set of all singular regions. Such that for our simple example:

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Overlap contributions II

Combining and multiplying out we obtain: where the sum goes over all non empty subsets of . So for our simple example we just get: Which agrees with our previous expression if we further demand the geometric cancellation identity:

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Overlap contributions III

Introduce the measurement-function which allows for no more than unresolved partons. We then obtain: is the set of soft and/or collinear singularities which: i) pass the criteria of the the measurement function and ii) survive the region cancellations We will refer to the set as the set of IR forests.

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Normal coordinates and

  • rdering of regions

Regions are defjned by: We then impose the following strict order:

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Region Cancellations I

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Region Cancellations II

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Region Cancellations III

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The set of IR forests factorises

as a consequence of region cancellations/ordering

  • Conjecture:

– is a set of soft forests – is a set of collinear forests

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Counter terms for final states in Yang Mills

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Defjne an observable

In the following wish to compute for l=1,2 ; the integrated counterterm:

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Key idea

Independent sums/classes of Feynman Diagrams Insert different volumes in different sets of Feynman diagrams

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N-particle final state at NLO

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Poles of single real are isolated by singular volume contribution:

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Soft Region: Collinear Region:

It is suffjcient to defjne insertion in the limit

(almost any decomposition, which satisfjes these will do)

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Integrated counter-terms are simple!

Convenient to defjne a soft subtracted collinear counterterm:

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The integrated NLO counterterm for n emissions:

agrees with usual 1-loop Catani operator

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N-particles final state at NNLO

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Normal Coordinates and Measures at NNLO

Limits Normal coordinate Phase Space Measure bound

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The Double Soft Measure

Unlike the single the double soft measure has further support:

  • Double soft integrals are not (completely) trivial.
  • Evaluation can be simplifjed by IBPs.
  • The corresponding 2 double soft Master integrals known [1208.3130]
  • In fact even tripple soft masters (hard!, which enter at N3LO) are already

known from Higgs soft expansion at N3LO

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Triple Collinear Masters

  • Slightly harder than double soft but

same as N-jettiness beam function

  • 4 Master Integrals
  • Evaluated by Ritzmann and Waalewijn for initial and fjnal states

(to all orders in eps in terms of 4F3 and 3F2) [1407.3272 ]

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Double Soft - Triple Collinear Overlap

Asymptotic measure:

Double soft triple collinear Master integrals can be extracted from the double soft Masters!

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Singular double real contribution

Task is to fjnd a suitable insertion of volumes:

  • NLO limits are inserted as before!
  • NNLO limits require a new prescription
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Collinear phase spaces factorise (in limit)

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What to do with the double soft?

Soft momenta factorised but color kinematic correlations with up to 4 Wilson lines Double soft momenta correlated, but only 2 Wilson lines

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Let the kinematics follow the color!

This fjxes all the overlaps at NNLO!

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Iterated double soft limits: 3 different eikonals in iterated limit contribute to each

non-abelian double soft factor

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Caveat: although vanishes. survives, due to single soft Phase space

Rescale invariance of eikonal factor is not satisfjed by the soft volume bound

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IR forests at NNLO

Reality is slightly better since some terms can be combined into

  • ne term..
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Primitive Measures

All limits of phase space measures at NNLO are expressable using Other overlapping regions are all iterated or factorising integrals of the NLO ones and evaluate to Gamma-functions.

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Convenient to combine sets regions:

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Leads to following basic integrated counterterm building blocks:

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The integrated NNLO counterterm

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Check for H gg double real emission →

Analytic result is easy to obtain:

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Poles check out! Finite terms remain to be checked.

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Outlook & Conclusion

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  • Application of the differential cross section calculations still requires

adequate mappings

  • They should exist, but not completely trivial
  • Generalisation to initial states and real-virtual is not much work
  • Required tripple collinear integrals already known
  • Generalisation to massive colored states (tops)
  • possible, but requires eikonal factors with massive Wilson lines (more

challenging; integrals may not be known?)

  • N3LO should be possible
  • tripple soft known; double real-virtual: double soft known also; ….

Outlook

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Conclusions

  • Presented a new subtraction scheme based on different slicing
  • bservables for different sets of Feynman diagrams
  • Integrated counterterms are simple and can be recycled from higgs soft

expansion and n-jettiness jet function

  • Scheme is useless as a slicing scheme!
  • Numerically unstable
  • Proposition: Scheme can be promoted to a fully local subtraction scheme,

after including proper mappings.. (remains to be shown!)

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Scalar integral Checks

Checked that sum of integrated counterterms reproduces poles of the following to integrals:

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