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1 Top Production Processes at Hadron Colliders By Paul Mellor

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2 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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3 The Top Quark It is the SU(2) L partner of the bottom quark Its mass is obtained via EWSB Q t =+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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4 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 m t which are used in other calculations

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5 Top Production Single top production Top pair production Good for probing EW interaction Can measure |V tb | 2 Sensitivity to BSM Measure tbW coupling Discovery channel Good for probing QCD interaction ~1 per second at LHC

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6 Single Top Production S-channel Associated W T-channel

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7 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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8 The Integrals p1p1 p2p2 p3p3 p4p4 q q + p 2 q + p 2 + p 3 q + p 2 + p 3 + p 4 Tensor Integrals Scalar Integrals

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9 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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10 q + p 3 p 1 - p N p 2 - p 1 p 3 - p 2 p N - p N-1 q + p 1 q + p 2 q + p N Davydychev Reduction

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11 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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12 Sector Decomposition Recall 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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13 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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14 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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15 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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16 Initial Reduction p1p1 p2p2 p3p3 p4p4 q q + p 2 q + p 2 + p 3 q + p 2 + p 3 + p 4 Rewrite the numerator as a combination of denominator factors: This reduces the tensor integral to a sum of scalar triangles and a scalar box; p1p1 p2p2 p3p3 p4p4 q q + p 2 q + p 2 + p 3 q + p 2 + p 3 + p 4 p1p1 p2p2 p3p3 p4p4 q q + p 2 q + p 2 + p 3 q + p 2 + p 3 + p 4 p1p1 p2p2 p3p3 p4p4 q q + p 2 q + p 2 + p 3 q + p 2 + p 3 + p 4

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17 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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18 Further Reduction Write 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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19 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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20 Using the Tools Now we have the tools, we need the diagrams to apply them to

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23 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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24 Diagram Selection Resonant Connected Resonant Disconnected Resonant Electroweak Non-Resonant Connected Non-Resonant Disconnected Gluon Exchange 1 Gluon Exchange 2

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25 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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26 Required Diagrams To order δ 3/2 the diagrams required for the calculation are; Resonant Tree Triangle 1 Triangle 2 Triangle 3 Box 1

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27 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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28 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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29 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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30 Recent Developments

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31 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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32 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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33 Thanks for listening! Any questions? The End

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