From Factorization to Resummation for Single Top Production at the LHC with Effective Theory Chong Sheng Li ITP, Peking Unversity Based on the work with Hua Xing Zhu, Jian Wang, Jia jun Zhang, , 乌兰浩特

Motivation Single top already discovered at the Tevatron! Relay on precise theoretical predictions of signal and background CDF, Phys.Rev.Lett.103: ,2009 D0, Phys.Rev.Lett.103: , σ discovery!

Motivation Combined CDF and D0 results (arXiv: , The Tevatron Electroweak working Group for the CDF and D0 Collaborations)

Motivation Time like WSpace like WReal W Allow measurement of V_tb per channel Easier to check the chiral structure of Wtb vertex than t-tbar production t-channel can be used to measurement the b quark density ( Campbell, Frederix, Maltoni, Tramontano ) Sensitive to FCNC (t-channel), or W’ resonances (s-channel) Single top production is important because:

Motivation Precise prediction for single top quark production cross section is need! NLO QCD corrections: Bordes, van Eijk, Stelzer, Willenbrock, Smith, S. Zhu, Sullivan, Q.-H. Cao, Yuan, et.al. NNLL-NNLO threshold logarithms (expanded up to ) Kidonakis, 2010 SM NLO QCD predicitons: We present a complete and up to date NNLL resummation for single top production, using SCET (soft-collinear-effective-theory) Hua Xing Zhu, Chong Sheng Li, Jian Wang, Jia Jun Zhang ( )

Generic Cross Section in pQCD Take s-channel as an example, other channels are similar. Extremely complicated at higher orders: initial state splitting, final state splitting, soft gluon emission, etc. hard particles soft particles collinear particles Hard and soft, collinear particles are entangled! Large logarithms appear in partonic cross section: ω: final state energy configure : energy flow operator (C. W. Bauer,et. al., PRD, 2009)

Near threshold fixed order expansion of the cross section In general, the singular part (near threshold) of the partonic cross section C can be written as Here we denote, and treat delta function as 1. Large Logarithms spoil convergence, must sum them up to all orders.

LL NLL NNLL Resummation as a reorganization We could rewrite the perturbative series as: Resummation is a reorganization of the perturbative series in order to improve the convergence.

Resummation from factorization SCET: [Bauer, Fleming, Pirjol, Stewart, Rothstein, Beneke, Chapovsky, Diehl, Feldmann, Yang Gao, C. S. Li, Idilbi, Ji ]

SCET approach vs. Traditional approach Resummation in SCET has advantage over traditional approach of resummation as following reasons: Completely separates the effects associated with different scales in the problem. Avoid the Landau-pole ambiguities inherent in the traditional approach. Using conventional RG equations to resum logarithms of scale ratios. Especially, momentum space resummation in SCET(Neubert, 2005) is simpler comparing with the Mellin moment space approach.

Some Recent Important Progress in SCET approach to Resummation Momentum space resummation in SCET (Becher and Neubert, 2006) Direct photon production with effective theory (Becher and Schwartz, 2009) Top quark pair production at NNLL ( Beneke etc.; Neubert etc., 2009,2010) Drell-Yan production away from threshold via beam function ( Stewart etc., 2009,2010) Threshold resummation of boosted multijet processes ( Bauer etc., 2010) Jet function with Realistic jet algorithms in SCET ( Jouttenus; S. D. Ellis, etc, 2009)

Basics of SCET SCET is an effective theory describing collinear and soft interaction (Bauer,Fleming, Pirjol, Rothstein, Stewart) Collinear quark and gluon field Soft and collinear interaction decoupled by field redefinition SCET Lagrangian factorized:

Factorization in SCET For N jets production, factorization form can be written as (Bauer, Hornig, Tackmann, 2008) However, actual application in resummation is non-trivial. For example, for s and t-channel, there are three light-like direction and a time-like direction. (Never considered before) Initial state light- like direction Final state light- like direction Top quark, final state time-like direction

Factorization for process of single top production In the following slides, we show our recent work, which present for the first time the general idea of factorization for process of single top production with SCET and its numerical results. We mainly concentrate on the s- channel case, and give some comment on the other two channel. Born diagram for s-channel single top production:

Factorization: step 1-integrated out the hard function Idea: Match the full theory cross section onto SCET by integrating out hard momentum mode. SCET is constructed to reproduce the long distance physics, thus short distance physics is not described by the dynamics of the effective field theory. The short distance information carried by hard mode (H) is absorbed into Wilson coefficient, the hard function. After full theory match onto SCET at the hard scale, hard function (H) can be obtained

Factorization: step 2-integrated out the jet function Usually final state jet has larger invariant mass: need to be integrated out. After integrate out the final state collinear gluons at the scale The jet function J is the final state analog of the parton distribution functions. Jet function describe how the final partons from the hard interaction evolve into the observed jets, and contain all dependence on the actual jet algorithm.

Factorization: step 3-integrated out the soft function The remaining parton interact with soft gluon through eikonal interaction, and can be absorbed into soft Wilson line by field redefinition. Match at The soft function S describes the emission of soft partrons from the soft Wilson line. If we define the soft scale Then one can perturbatively calculate it and avoid the Landau pole problem.

Factorization: step 4, match onto PDFs Finally, the remaining initial state collinear effects are absorbed into the PDFs: Match at scale

Description of the factorization Hard function Jet function Soft function PDFs Large logarithms of scale ratio:,Resummed by RGE! Note: Above figure do not mean that factorization scale is lower than both the soft scale and the jet scale!

From factorization to resummation Full theory SCET Initial collinear field operator Final collinear and heavy quark field operator Soft operator: product of 4 Wilson line singlet octect

From factorization to resummation Utilizing the properties that the different collinear sectors decouple and soft interactions factorized into Wilson line, one can derive a factorized expression: Hard function Jet function Soft function Short distance Wilson coefficient, can be obtain from NLO QCD corrections. Describes final collinear emission from light parton, sensitive to algorithms for jet definition. Describes soft radition between energetic jet and heavy particles.

The hard function The hard function is In dimensional regularization, the IR divergences and UV divergences in the SCET 1-loop diagrams calculations cancel It is a 2x2 matrix in color space Hard function=full theory virtual corrections – SCET virtual corrections RGE: Solution of this equation sum double log and single log of the form Can be evaluated at some scale and the run to the lower scale by RGE to match on jet and soft scale. Anomalous dimension matrix known to 2-loop

The soft function The soft function is a time ordered product of Wilson line. It describes the effects of soft gluon emission. It can be obtained from the following eikonal diagrams Initial state corrections (Becher, Schawrtz, 2009): Final state corrections: Initial state real gluon emission diagram, similar to Drell-Yan production Final state real gluon emission diagram, similar to b-quark shape function calculation

RG evolution of the soft function Sudakov double logarithms Single logarithms Soft anomalous dimension Laplace transformed soft function It can be solved directly in momentum space (Becher, Neubert, 2006):

The quark jet function Definition: Matrix elements of the collinear fields associated with the jet. Here the momentum p is referred to collinear momentum with zero bin subtraction. (Manohar, Stewart, 2006), which can be obtained from calculating the following diagrams in SCET: RGE: Solved with the similar techniques as soft function: is the Laplace transformation of the jet function.

Final expression for the resummed cross section Hard function, summing logarithms of the form Soft function, summing logarithms of the form Jet function, summing logarithms of the form If desired, it can be used to generate higher order expansion of threshold singular terms. For example, the 2-loop singular terms are given by:

Numerical results In contrast to fixed order calculation, threshold resummation with effective theory require the determination of 4 scale: the hard scale, the soft scale the jet scale and the factorization scale. We choose: hard and factorization scale: 200 GeV The soft scale: We choose the soft scale to minimize the 1-loop soft corrections (Becher, Neubert, Xu), thus the large logarithms will appear in the evolution factor. The jet scale: Similar to the soft scale, we choose the jet scale to minimize the 1-loop jet corrections.

Numerical results We present the factorization scale dependence of the resummed cross section in terms of R ratio: At the Tevatron, the resummation effects reduce the factorization scale dependence. At the LHC, the resummation effects do not improve the factorization scale dependence: CM energy is so large that threshold approximation may not be a good approximation.

Numerical results The cross section in blue color is our best prediction. We can see that the resummation effects enhance the NLO cross section by about 3%-5%. The total uncertainties is obtained from varying the hard, soft, jet and factorization scale separately by a factor ½ and 2, and then adding up the individual variations in quadrature.

Comment on the other two channels The same factorization formalism applied to the t-channel and tW associated production. The only differences would be the hard function and soft function. Born diagram Soft function

Summery Single top production is very important at the Tevatron and LHC, and precise theoretical predictions are needed. We present a calculation of NNLL resummation effects in single top production with soft-collinear effective theory, which are different with Kidonakis’s NNLO results. We find mild enhancement of the total cross section for s-channel single top production. The K factors are about , when comparing with NLO results. Our formalism can be extended to other massless or massive colored parton production process at hadron colliders.