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,

2010-07-20, 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.103:092002,2009

D0,Phys.Rev.Lett.103:092001,2009

5 σ discovery!

Motivation

Combined CDF and D0 results (arXiv:0908.2171, The Tevatron Electroweak working Group for the CDF and D0 Collaborations)

Motivation

Time like W Space like W Real 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 (1006.0681)

2s

Generic Cross Section in pQCDTake 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)

0

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

1 42

4

1 lnn n

t

sLs m

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

4~ ~hard t softt

Q Q smm

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.

H

After full theory match onto SCET at the hard scale , hard function (H) can be obtained

h

• Factorization: step 2-integrated out the jet function

JH

Usually final state jet has larger invariant mass: need to be integrated out.

After integrate out the final state collinear gluons at the scale j

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.

H

• Factorization: step 3-integrated out the soft function

JH

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.

Qs CD

s

JH

• Factorization: step 4, match onto PDFs

JH

JH

Finally, the remaining initial state collinear effects are absorbed into the PDFs:

baf f

Match at scale f

ba ff JHS

• Description of the factorization

baf f JHS

Hard function

Jet function

Soft function

PDFs

h

j

s

f

Large logarithms of scale ratio: , ln , ln , lnjh s

f f f

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 resummationUtilizing 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

2 22

2 2~ exp ln lnh hcusp h

Can be evaluated at some scale and the run to the lower scale by RGE to match on jet and soft scale.

h

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( )

( )

( ), )( )

(s

s

SSa d

Soft anomalous dimension

Laplace transformed soft function

It can be solved directly in momentum space (Becher, Neubert, 2006):

• The quark jet functionDefinition:

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:

ImJ [ ]RGE:

Solved with the similar techniques as soft function:

is the Laplace transformation of the jet function.j

• 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 formlog( / )h log( / )j log( / )s

• 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.

80s GeV 50j GeV

• Numerical results

We present the factorization scale dependence of the resummed cross section in terms of R ratio:

( )( 200 )

F

F

RGeV

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 1.03-1.05 , when comparing with NLO results.

• Our formalism can be extended to other massless or massive colored parton production process at hadron colliders.