1 Top Production Processes at Hadron Colliders By Paul Mellor.

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Top Production Processes at Hadron Colliders

By Paul Mellor

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Overview

• Why study the top quark?• Production of top quarks• Performing calculations• Numerical solutions• Analytic solutions• Diagram selections• Comparisons to an EFT• Future plans

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The Top Quark

• It is the SU(2)L partner of the bottom quark

• Its mass is obtained via EWSB

• Qt=+2/3 and is a colour triplet

• Discovered via top pair production by CDF and D0 collaboration in 1995

• Top is the only quark which decays before hadronizing

• Top quark interacts strongly with Higgs sector

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Why Study the Top?

• Top becomes an important background at LHC, understanding of its creation processes becomes vital

• Offers a method of measuring the tb element of the CKM matrix

• Sensitive to any non-standard tbW vertex effects

• Greater accuracy on parameters such as mt which are used in other calculations

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Top Production

Single top production Top pair production

• Good for probing EW interaction

• Can measure |Vtb|2

• Sensitivity to BSM

• Measure tbW coupling• Discovery channel

• Good for probing QCD interaction

• ~1 per second at LHC

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Single Top Production

S-channel Associated W T-channel

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Cro

ss S

ecti

on /p

b

T-channel Associated W S-channel

Production Channel

LHC

Tevatron*100

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T-channel Production

Choose an exact process by selecting

most abundant quark on incoming leg

Due to short lifetime of quark it makes

more sense to include its decay in diagram

Can then consider the various diagrams

which will contribute to the cross section

at NLO

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The Integralsp1

p2p3

p4

q

q + p2

q + p2 + p3

q + p2 + p3 + p4

Tensor Integrals

Scalar Integrals

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Method 1

• Reduce tensor integrals to scalar integrals via Davydychev reduction

• Factorise singularities using method of “Sector Decomposition”

• Solve integrals to obtain final analytic expression for the integral

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q + p3

p1 - pN

p2 - p1p3 - p2

pN - pN-1

q + p1

q + p2

q + pN

Davydychev Reduction

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Davydychev Reduction

Davydychev developed the following reduction formula:

where

We can rewrite a tensor integral as a sum of scalar integrals, for example, a rank 1 tensor triangle integral can be written as:

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Sector DecompositionRecall from lectures that we introduce Feynman parameters;

Extension to a larger number of denominator factors can introduce overlapping singularities:

Factorise singularities by dividing integration range into sectors and remapping integration range;

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The Problems

• Davydychev reduction introduces integrals in higher dimensions

• Sector decomposition is an iterative procedure leading to factorial increase in number of integrals

• Extraction of poles from sector decomposition introduces complicated logarithms

Expressions too complicated to be solved analytically

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Creating Something Useful

• Already had Mathematica programs to automate above processes

• Expressions can be solved numerically

• Numerical integration runs into problems at poles

• Deform path of integral to avoid these problems

• Program which solved tensor integrals numerically

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Back to the Drawing Board

• Reduce integrals as far as possible by rewriting kinematic factors where applicable

• Use Passarino-Veltman style reduction to complete reduction to scalar master integrals

• Obtain final expression by using known equations for scalar master integrals

Find an alternative way to reduce tensor integrals while trying to avoid the problems

encountered earlier

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Initial Reductionp1

p2p3

p4

q

q + p2

q + p2 + p3

q + p2 + p3 + p4

Rewrite the numerator as a combination of denominator factors:

This reduces the tensor integral to a sum of scalar triangles and a scalar box;

p1

p2p3

p4

q

q + p2

q + p2 + p3

q + p2 + p3 + p4p1

p2p3

p4

q

q + p2

q + p2 + p3

q + p2 + p3 + p4p1

p2p3

p4

q

q + p2

q + p2 + p3

q + p2 + p3 + p4

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Further Reduction

•Expand tensor integral in a sum of all possible Lorentz structures

•Contract with external momenta to obtain coefficients

Use v and w tensors as basis for Lorentz structures

where

such that

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Further ReductionWrite the tensor, q, as;

Under integration this becomes, using the properties of v and w;

This can the be rewritten in terms of the denominator factors;

where we have;

in the massless case. Modifying this to include masses we have;

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Reducing Pentagons

After full tensor reduction we can further simplify our result by rewriting the scalar pentagon integrals as a sum of scalar box integrals using

where

Using this along with the equation for the rank 1 pentagon integral it can be shown that;

Note: The factors containing Gram determinants do not appear in this expression!

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Using the Tools

Now we have the tools, we need the diagrams to apply them to

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Diagram Selection

Try to pick out the diagrams which give the largest contributions

Split diagrams into groups with similar properties

Resonant Non-Resonant Gluon Ex.

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Diagram SelectionResonant

Connected

Resonant Disconnected

Resonant Electroweak

Non-Resonant Connected

Non-Resonant Disconnected

Gluon Exchange 1

Gluon Exchange 2

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Diagram Selection

Squaring the tree level amplitudes and power counting in the small parameter;

The leading order contribution comes from squaring the

resonant diagram

The next highest order in the small parameter comes from the terms

However, due to colour factors;

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Required Diagrams

To order δ3/2 the diagrams required for the calculation are;

Resonant Tree

Triangle 1 Triangle 2

Triangle 3 Box 1

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Performing the Calculation

• Use FeynArts to create required diagrams

• Insert Feynman rules using FeynCalc

• Compute squared amplitude using FeynCalc

• Reduce tensor integrals

• Insert expressions for scalar integrals

• Obtain final expression

• Reduce tensor integrals

• Insert expressions for scalar integrals

• Compute squared amplitude using FeynCalc

• Obtain final expression

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Effective Field Theory

• Collaboration with Adrian Signer and Pietro Falgari

• Same diagrams as above calculated with an EFT

• EFT calculations give only the leading order expressions

• This method gives results to all orders and can be expanded if required

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The Next Steps?

• Perform calculations to next order in small parameter

• Include real corrections to tree level diagrams

• Perform phase space integration to calculate cross section and compare to current results

• Modify calculation to include decay of W-boson

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Recent Developments

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Long Term Goals

• Complete t-channel calculations

• Move on to study associated W and s-channel production processes

• Look at top pair production processes

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Summary

• Understanding of top production process vital for fully understanding LHC data

• Difficulty in calculating Feynman diagrams due to tensor integrals• Can reduce these tensor integrals via a number of methods to sums of

known scalar integrals• Have the tools to calculate all tensor integrals in diagrams for single top

production• Agreement with an results from effective field theory gives another possible

method of calculation

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Thanks for listening!

Any questions?

The End