Higgs phenomenology with antenna subtraction Juan M Cruz Martinez - - PowerPoint PPT Presentation

Higgs phenomenology with antenna subtraction

Juan M Cruz Martinez

IPPP (Durham University) Supervisor: E.W.N. Glover NNLOJET Collaboration

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Outline

1

Introduction

2

Phenomenological challenges of NNLO calculations

3

Testing grounds

4

Vector Boson Fusion Higgs Production

5

Summary

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HEP Introduction

The Standard Model (SM) is one of the most robust theories known to phyisicists. As it stands, the SM fails to explain many aspects of High Energy Physics (neutrino masses, gravity, baryion assymetry, etc). Despite its many flaws, we are uncapable of finding any significant deviations from the SM in our current experiments at collideres.

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The Higgs Boson

Just five years ago...

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The Higgs Boson

Why is the Higgs that important? The High Energy Physics community have been looking for hints of new physics for many years now. As it stands, the newly discovered boson (which could be just one of many Higgs, a composite particle or the standard model elemental scalar proposed fifty years ago) could provide some clues about where to find new physics. Sadly it looks very standard model-like...

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Some motivation: Higgs boson couplings

From hep-ex/1507.04548 Experimental efforts are currently focused on finding deviations from the Standard Model in all the different channels that can be measured at the LHC. Confirming that the Higgs boson is not the one predicted by the Standard Model will

• pen the door for many theories

that would be ruled out otherwise.

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Some motivation: Higgs boson

Even assuming all couplings were to be fully compatible with the SM, the Higgs would still have some problems of its own. The Higgs parameters require some fine tunning that could be avoided (thus restoring naturalness) by supersymmetry (SUSY) or by being a composite scalar with an inverse length of the order of the TeV scale. In the light of these issues, we expect (we hope) to find new physics at the TeV scale associated with EWSB and the Higgs.

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We need precision

These are just some of the reasons we need more precision, both in the theoretical and experimental side. However, even when we have the most precise measurement and calculations, the comparison between theory and experiment with the minimum loss of information is still highly non trivial.

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The Big Picture

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The Big Picture

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The experimental side

After a very sucessful run I at 7 and 8 TeV, the LHC is currently delivering 13 TeV data. Five years ago, during Run I, it was able to confirm the existence of a boson which was compatible with the highly anticipated Higgs Boson. And only one year ago the HEP community was turned upside down by a misterious resonance

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The experimental side

After a very sucessful run I at 7 and 8 TeV, the LHC is currently delivering 13 TeV data. Five years ago, during Run I, it was able to confirm the existence of a boson which was compatible with the highly anticipated Higgs Boson. And only one year ago the HEP community was turned upside down by a misterious resonance Sadly the world is not always that beautiful

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The experimental side

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An example: φ∗

There are other ways of improving the precision of a theory-experiment comparison without the need to get more data or better calculations. By selecting the right observable for the right process we can compare results that are bound by different (better behaved) errors.

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An example: φ∗

One simple example of this is the low pt regime of the Z boson. In this case the pt of the Z is computed using the energy of its decay products, which could introduce big relative errors due to the energy resolution of the detector. The observable φ∗, sensible to the same physics as the pt in the low energy regime, is computed using the direction of the decay products of the Z, which are measured with far better resolution.

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The theoretical side

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The theoretical side

Even with that knowledge at hand, there are still many known unknowns and unknown unknowns: Strong CP problem Dark Matter Neutrino masses The Higgs Boson Gravity ... ... and the list goes on

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The theoretical side

Even with that knowledge at hand, there are still many known unknowns and unknown unknowns: Strong CP problem Dark Matter Neutrino masses The Higgs Boson Gravity ... ... and the list goes on

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The theoretical side

And, even knowing there exists a Higgs, there are still many open questions: Is it a fundamental particle? Is it composite? Is it just part of a spectrum, as SUSY predicts? Is it the Higgs of the Standard Model, as simple as that? Improving our theoretical knowledge of the Higgs boson could help find a deviation from the Standard Model: maybe the couplings are not quite right? Maybe we are seeing more (or less) Higgses than we should?

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And how do we improve precision:

There are many things that can be improved in the theory side of particle physics: Parton Distribution Fuctions

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And how do we improve precision:

There are many things that can be improved in the theory side of particle physics: Parton Distribution Fuctions Parton shower

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And how do we improve precision:

There are many things that can be improved in the theory side of particle physics: Parton Distribution Fuctions Parton shower Fixed order calculations

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And how do we improve precision:

Better precision in fixed order calculations means, basically, two things p1 p2

More legs

p

1

p

2

More loops

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State of the art

Many legs calculations can be obtained with tools currently on the market in a mostly automatised manner. The only actual limitation is the computational power. Ie, Leading Order (LO) calculation are, up to a certain point, a solved problem. The challenges that arise for Next to Leading Oredr (NLO) calculations (one extra loop with respect to LO) reduce the number

• f processes that can be currently automatised. Nonetheless, many

processes have been already computed with many legs (and one loop) in the final state (W+5j, Z+4j, 5j, WW+3j). Many techniques and tools have been developed to address the challenges of NLO calculations.

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State of the art

Next-to-Next-to Leading Order (NNLO) calculations (two extra loops with respecto to LO) are currently the state of the art and they will continue be as long as most two loop matrix element are still unknown. Two loops amplitude are known for some processes as H (+ one extra jet), Weak Vector Boson (+ one extra jet) and jet inclusive cross section. Some N3LO calculations have been performed for some processes on very particular approximations (inclusive VBF, Higgs production on EFT)

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State of the art

At NLO, new channels for production of a given final state can

• appear. This means that, for NLO calculations we might be taking

into account certain channels with Leading Order precision. At two loops or NNLO the theoretical results are much more reliable as any new channels will be expected to be small in comparison with LO and NLO and the scale dependence stabilises

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State of the art

At NLO, new channels for production of a given final state can

• appear. This means that, for NLO calculations we might be taking

into account certain channels with Leading Order precision. At two loops or NNLO the theoretical results are much more reliable as any new channels will be expected to be small in comparison with LO and NLO and the scale dependence stabilises Although this is not always necessaryly true.

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Higgs at Next-to-Next-to-Next-to Leading Order

A very well known case for this is the production of Higgs boson through gluon fusion. The scale uncertainty is still quite large at NNLO and a N3LO calculation has been performed in hep-ph/1504.06056 by C. Anastasiou, C. Duhr, F. Dulat, F. Herzog and B. Mistlberger

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Phenomenological challenges: NLO

In order to introduce the challenges that will arise at NNLO let us review NLO calculations first: Leading Order is equivalent to selecting only those processes that produce exactly the final state we are looking for at the minimum power on (if we are doing QCD) αs. At Next to Leading Order we need to consider any process, with an extra power of αs that will produce the same final state. This means we need to consider loop diagrams and diagrams with one extra leg.

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Phenomenological challenges: NLO

Real radiation Virtual diagram

As the extra particle was not on the base process, it is necessary to integrate it over the entire phase space, this will introduce singularities at high momentum (Ultraviolet, UV) and low momentum (Infrared, IR)

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Phenomenological challenges: NLO

Energy conservation will stop UV divergences from appearing in the Real radiation diagram, whereas renormalisation will remove UV divergences appearing in loop diagrams. Infrared divergences can appear in both loop and radiation diagrams. KLN Theorem: dσNLO = dσR + dσV = finite The cross section is finite, so such a cancelation will occur. However, Virtual and Real diagrams have different multiplicities: integrals with different dimensionality Real divergences show up as implicit divergences in the phase space integral Dimensional regularisation make loop divergences appear as poles on ǫ where D = 4 − 2ǫ.

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Subtraction schemes

We need a way of controlling these divergences: dσR

finite = dΦN+1

• M0

N+1−F 0 3 M0 2

• ,

dσV

finite = dΦN

• M1

N+F0 3M0 2

• ,

where Mloop

legs stands for the physical matrix element where F is some

known function we will use to remove the divergences. Without changing the physical result: F0

3 =

• dΦ3F 0

3 ,

where dΦ3 is the phase space of the unresolved parton.

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Subtraction schemes

Various subtraction schemes have been extended to NNLO and have been used for some NNLO calculations. Sector improved subtraction (arxiv: 1210.6832, 1207.0236) N-jettiness (arvix: 1505.03893, 1504.02131, 1512.01291) Projection to born (arxiv: 1506.02660) Antenna subtraction

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Antenna functions implementations

At the moment they have been implemented at NNLO in a number of processes e+e− → 3 jets (0806.4601, 0710.0346) t¯ t production (1404.6493) Higgs plus jet production (1607.01749, 1408.5325) Z plus jet production (1610.01843, 1605.04295) Jet inclusive production (1611.01460, 1407.5558) Deep Inelastic Scattering (1606.03991)

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Antenna functions theory

In any unresolved limit, a colour ordered matrix element (MN) with N particles in its final state will factorise such that: lim

x→0 MN = PxMN−1,

Where Px are universal functions that depend only on the partons involved in the limit. We can thus define a set of functions Ax =

MN MN−1 so that:

lim

x→0 Ax = Px

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Phenomenological challenges

The infrared structure is much more complicated than it was at NLO, and so the structure of the subtraction function is much more complicated. At two loops not all master integrals are known: not all two loops amplitudes are available. It is very hard to make the control of divergences numerically stable even when they cancel. The computational effort in order to achieve the desired accuracy can be orders of magnitude larger than it was at NLO.

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Phenomenological challenges

The infrared structure is much more complicated than it was at NLO, and so the structure of the subtraction function is much more complicated. At two loops not all master integrals are known: not all two loops amplitudes are available. It is very hard to make the control of divergences numerically stable even when they cancel. The computational effort in order to achieve the desired accuracy can be orders of magnitude larger than it was at NLO. Once everything is in place, testing the complete framework can be quite challenging

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NNLO ingredients

A numerical integration method that allow us to produce differential distributions Physical matrix elements derived from the theory Subtraction scheme: some method to analytically remove explicit divergences and numerically (at integration time) remove the implicit divergences. A Parton Distribution Function provider A phase space generator A framework to glue them all together A way of proving your results

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Monte Carlo Integration

In order to integrate the cross section for this kind of processes, Monte Carlo methods are usually used

• fdV ≃ V f ± V
• f 2 − f 2

N , with f = 1 N

N

• i=1

f (xi) f 2 = 1 N

N

• i=1

f 2(xi). Disadvantages: The error goes as

1 √ N

Advantages: This is true for any number of dimensions

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Importance and stratified sampling: Vegas

Change the probability density from 1

V to a function g that behaves as f :

f g gdV ≃ f g ±

• f

g 2 − f g 2

N . It is impossible to know the form of f before integrating f , so simply go through a few “warmup” iterations to generate a function g such that: g ∝ |f |

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Importance and stratified sampling: Vegas

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 10 grid iteration Dimension 14

Probabily density plot for a variable that maps to the momentum fraction

• f one of the initial partons

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NNLO tests

Many ingredients of NNLO calculations are actually used during NLO calculations thus providing a powerful testing ground for both inclusive quantities and differential results. The numerical algorithm we use focuses on reducing the statistical error associated with the integral by importance sampling. If we are not careful, this could lead to some potentially important regions of the phase space left undersampled. Scale uncertainty: the scale evolution of any observable depend only

• f lower order quantities.

Note: Our calculation must go through all these test to be correct, however, going through all tests doesn’t necessaryly mean it is correct.

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Can the importance sampling go wrong?

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 10 grid iteration Dimension 11

If some regions of the integration greatly dominate the integral, some

• ther regions could go severely undersampled

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Divergence cancellation

Cancelling the divergences will be done at integration time by a function that behaves as the physical matrix element in the limit in which one of the parton goes unresolved. Computers don’t deal with infinite quantities very well and in order to stabilise the results we need to define some technical cut x on our invariants such that: dσ =

• sij ≤ xˆ

s dσ

• therwise

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Divergence cancellation

For ease of mind we also wrote a program to drive both the subtraction term and the physical matrix element onto the unresolved regions. Ie, we can check the cancellation by taking both the cross section (dσR) and the subtraction term (dσS) extremely close to the divergence. If the subtraction term is behaving correctly, as we reduce the technical cut (x) we should find dσS → dσR. Ie, lim

x→0

dσS dσR

• (x) −

→ 1

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Divergence cancellation

Limit g1||qf , gf → 0 Limit g1||Qf , qi||gf

Limits for a process gq → gqQ ¯ Q

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Divergence cancellation

Limit g||qf Limit g → 0

Limits for a process Qq → qQg with a loop

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The running of αS(µ0) as a testing ground

Loop calculations introduce both Infrared (IR) and Ultraviolet (UV) divergences. The former are to be removed upon integration of the unresolved

• partons. This is achieved in our framework through the antenna

subtraction method. UV divergences, however, are removed during renormalisation. This process introduces an unphysical scale (µR) in our calculations through dimensional regularisation.

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Scale dependence

When we look at a QCD cross section, each term of the series depends on an unphysical scale (µR), even though the full calculation does not depend on it. σij = αS(µ0)n σ(0)

ij (µ0) + αS(µ0)σ(1) ij (µ0) + αS(µ0)2σ(2) ij (µ0) + ...

• When we truncate the series (at leading order, next to leading order,

next-to-next-to leading order...) we keep some dependence on this unphysical scale which will introduce a theoretical error in our prediction. One of the main motivations for moving further on the series is to reduce said uncertainty and make our predictions more accurate.

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The running of αS(µ0)

The running of the strong coupling between two scales µR and µ0 can be derived from the renormalisation equation of αb

S.

αb

Sµ2ǫ 0 = ZααSµ2ǫ

β(α) = µ2 dαs dµ2 At two loops it reads: αS(µ0) = αS(µR)

• 1 + β0 log

µ2

R

µ2 αS(µR) 2π

• +
• β2

0 log2

µ2

R

µ2

• + β1 log

µ2

R

µ2 αS(µR) 2π 2 + O

• α4

S(µR)

• Juan M Cruz Martinez

(IPPP) Higgs Pheno 40 / 63

The running of αS(µ0)

σij(µR) = αS(µR) 2π n σ(0)

ij (µ0)

+ αS(µR) 2π n+1 σ(1)

ij (µ0) + n log

µ2

R

µ2

• β0σ(0)

ij (µ0)

• +

αS(µR) 2π n+2 σ(2)

ij (µ0)

+ log µ2

R

µ2 (n + 1)β0σ(1)

ij (µ0) + nβ1σ(0) ij (µ0)

• + log2

µ2

R

µ2 n(n + 1) 2 β2

0σ(0) ij (µ0)

• + O

αS(µR) 2π n+3 Note: only the Nx−1LO cross section contribute to the running of a NxLO calculation!

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The running of σij(µ0)

A powerful check of a NNLO calculation is testing whether it depends

• n µ as the previous slide predicts.

In particular, it is a powerful self-consistency check for the subtraction scheme. Many spurious terms that depend on the value of µ are introduced during the integration. They all must cancel internally, the remaining scale dependence must fully agree with the formula shown. It can also be used to test the implementation of effective theories. It will also tell us how sizeable the scale variation at the next level will be.

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The running of σij(µ0)

A powerful check of a NNLO calculation is testing whether it depends

• n µ as the previous slide predicts.

In particular, it is a powerful self-consistency check for the subtraction scheme. Many spurious terms that depend on the value of µ are introduced during the integration. They all must cancel internally, the remaining scale dependence must fully agree with the formula shown. It can also be used to test the implementation of effective theories. It will also tell us how sizeable the scale variation at the next level will be. However: this test does not give us information on the actual size of the NNLO correction since a central reference value σ(2)

ij (µ0) needs to be

computed.

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The running of σij(µ0)

Figure: Comparison between the expected running (+) and the numerical results (x) for Higgs production. Reference scale µR = µF = mH.

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Introduction to Vector Boson fusion

Vector boson fusion is a very important channel for Higgs phenomenology: At the LHC its production rate is second only to gluon fusion. Its very clean signature make this channel very easy to isolate

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Vector Boson Fusion: VBF Cuts

In order to enhance the signal, we impose the following cuts to our results: One outgoing jet in each hemisphere (y1y2 < 0, where yi = rapidity

• f the ith-hardest jest).

Furthermore, we ask the two hardest jets to have a ∆y > 4.5. We apply a cut in the dijet invariant mass of 600 GeV. Finally we ask the two hardest jets pT to be above 25 GeV.

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Vector Boson Fusion amplitudes

First of all, we need to define what are we calling “Vector Boson Fusion” Diagrams in which the vector boson is exchanged in the t channel Not including exchange of gluons between upper and lower legs Not including same flavour quark annihilation These contributions are estimated to be negligible when VBF cuts are applied.

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VBF: Leading Order

At Leading Order is straightforward to see we only include squared matrix elements that look like H

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VBF: Leading Order

By taking this approximation we are already neglecting contributions at Leading order where identical quark diagrams would allow for contributions

• f the form:

H These are colour suppressed by a factor of 1

N and its phase space is very

suppressed when taking usual VBF cuts.

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VBF : Next to Leading Order

At NLO are not including the interference between gluons radiated from different legs (and its corresponding loop diagrams): 1 H 2 1 2 These are colour forbidden due to Tr (ta) = 0, but they can be used to estimate contributions neglected at higher orders.

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Antenna Subtraction on VBF at NLO

Our Antenna Subtraction prescription means for every real contribution we generate a counterterm of the form: 1 i H 2 k j

−X 0

3 (1, j, i)

¯ 1 ˜ ij H 2 k

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Antenna Subtraction on VBF at NLO

And similarly for virtual contributions (X 0

2 =

• X 0

3 )

1 i H 2 k

−X 0

2 (1, i)

1 i H 2 k

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VBF Results at NLO

Figure: Comparison between NNLOJET and MCFM at NLO. Transverse momentum of the Higgs

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VBF Results at NLO

Figure: Comparison between NNLOJET and MCFM at NLO. Transverse momentum of the hardest jet

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VBF at NNLO

Inclusive cross section known for a few years up to NNLO in the structure function approach. NNLO corrections believed to be small (around 1% from NLO at 13 TeV) (hep-ph/1003.4451, P. Bolzonoi,

• F. Maltoni, S. Moch, M. Zaro, 2010)

Recent study suggest a NNLO correction for the total cross section greater than 5% when typical VBF cuts are applied (and even greater for differential distributions) (hep-ph/1506.02660, M. Cacciari, F. Dreyer, A. Karlberg, G. Salam, G. Zanderighi, 2015) With the antenna prescription we can obtain differential distributions without relaying on approximations: fully differential result. Other processes included in the program are Z+J (hep-ph/1507.02850), H+J (hep-ph/1408.5325), DY, H, or Dijet

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VBF at NNLO: structure function approach

The total cross section can be written as a product of a VV → H Matrix Element and two DIS tensors1: dσ ∝ WµνMµρMνσWρσ (1) This method implicitly integrates over the NLO and NNLO kinematics of the process: the jet algorithm is not sensitive to the radiated jets.

1arXiv:hep-ph/9206246, T. Han, G. Valencia, S. Willenbrock, 1992 Juan M Cruz Martinez (IPPP) Higgs Pheno 55 / 63

VBF at NNLO: structure function approach

The total cross section can be written as a product of a VV → H Matrix Element and two DIS tensors1: dσ ∝ WµνMµρMνσWρσ (1) Where the DIS tensor W is defined by means of the standard DIS structure functions. These functions are known at NNLO. We lose information on the radiation pattern of the process → we cannot apply cuts since we only have access to the Born kinematics of the process This method allows us to compute the inclusive cross section at NNLO.

1arXiv:hep-ph/9206246, T. Han, G. Valencia, S. Willenbrock, 1992 Juan M Cruz Martinez (IPPP) Higgs Pheno 55 / 63

VBF at NNLO: projection to Born

Based on the structure function approach, this method2 relies on a NLO calculation of H + 3j process to obtain differential results. They recover the dependence of the cross section on the two hardest jet → it allows for cuts.

2hep-ph/1506.02660, M. Cacciari, F. Dreyer, A. Karlberg, G. Salam, G. Zanderighi,

2015

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VBF at NNLO: our implementation

We have implemented this process up to NNLO in the program we are

• developing. This allows us to achieve fully differential results:

Access information on all four jets. Allows for different cuts and differential distributions. We apply the same factorisation assumptions that are implicit within the Structure Function approach.

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VBF at NNLO: RR

We apply a factorisation similar to what we used at NLO. The neglected contribution have been estimated to be small3 1 H 2

M1

1 H 2

M2

1 H 2

M3

These are some representative Double Real diagrams. We only include contributions MiM∗

j for i = j.

3hep-ph/1109.3717, P. Bolzonoi, F. Maltoni, S. Moch, M. Zaro, 2011 Juan M Cruz Martinez (IPPP) Higgs Pheno 58 / 63

VBF at NNLO: RV

Most diagrams with crosstalk between the upper and lower legs are still forbidden at NNLO and we don’t include diagrams that are colour and kinematically suppressed. 1 H 2

Colour (1/N2) and kinematically suppressed

The neglected contribution is estimated to be, at most, a 10% of the NNLO correction.

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VBF at NNLO: RV

Most diagrams with crosstalk between the upper and lower legs are still forbidden at NNLO and we don’t include diagrams that are colour and kinematically suppressed. 1 H 2 H 1 2

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VBF at NNLO: VV

And similarly at the double virtual level 1 H 2

Colour (1/N2) and kin. suppressed

H

Contributes at leading colour

H

Contributes at leading colour

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VBF at NNLO: µR dependence

We have also checked the different channels present the correct RGE behaviour for different values of µR.

Quarks initiated channel for different values of µR Quark-gluon initiated channel for different values of µR

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Summary

High precision calculations are necessary to probe the Standard Model. NNLO calculations present many theoretical and practical challenges that need to be addressed. The renormalisation scale dependence of the cross section can be a powerful check for a NNLO implementation. We have made the first steps towards a fully differential VBF NNLO calculation.

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Thanks!

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