Resummation of Large Logs in DIS at x->1

Xiangdong JiUniversity of Maryland

SCET workshop, University of Arizona, March 2-4, 2006

Outline

Introduction to DIS at large x and resummation of large logarithms

Resummation to N3LL in the standard and EFT approaches

Puzzles in SCET factorization Cancellation of the spurious scale Summary

Inclusive DIS

Consider the text-book example of inclusive DIS on a proton target

As Q->∞, at a fixed Bjorken x, the process can be factorized, as shown by the reduced diagram on the right.

QCD factorization

Standard QCD factorization for DIS

f is the parton distribution function, nonperturbativeC is the coefficient function, a power series in coupling αs

)1(

...)1(ln)1(ln)(

),()(22

212

1

x

xaxaxC

xCxCn

nn

nn

nnsn

As Bjorken x->1, the pQCD series converges slowly

A resummation is needed to get reliable predictions

Physical origin

As x-> 1, the hadron final state has an invariant mass Q2(1-x), which becomes an independent scale.

Thus, the hadron final state is restricted to a hadronic jet plus arbitrary number of soft gluons radiations.

Soft gluons contribution is usually large due to infrared enhancement near the edge of phase-space.

One must sum these soft gluons, just like in the case of QED where one must sum over soft photon contributions when the detector resolution is high (large logarithms).

In moment space

In moment space, the factorization becomes

)()/,()( NsNN qQCQF

...])lnln

lnlnlnln(

)lnlnlnln(

)lnln(1[

30312

32

332

434

535

636

3

20212

223

234

242

10112

120

cNcNc

NcNcNcNc

cNcNcNcNc

cNcNcCC

s

s

sNN

The expansion parameter is αsln2N!

Exponentiation

The large logarithms exponentiate! A property obvious easily seen in QED. In QCD, it requires some additional study of color factors,

...])lnln

lnln(

)lnlnln(

)lnln(

30312

32

332

434

3

20212

223

232

10112

12

gNgNg

NgNg

gNgNgNg

gNgNgG

s

s

sN

The expansion parameter is now αslnN!

Resummation

Consider αslnN is of order 1, sum over all terms of same order in αs such αslnN, (αslnN)2, (αslnN)3, etc

where = 0αslnN. The expansion is now in αs

We need to find what gn() are g1(): Leading Logarithms (LL)

g2(): Next-to-Leading Logarithm (NLL)

g3(): Next-to-Next-to-Leading Logarithm (N2LL)

Sterman’s approach

Re-factorization of the DIS structure function at new scale Q2(1-x). Introducing new ingredients such as jet functions, soft factor, and real hard contribution

Write done (complicated) differential equations for jets and soft factor at large x, which when solved yield exponentiated x-dependence.

Result

A is the anomalous dimension of a Wilson-line cusp A= αsn An

B is a perturbation series B= αsn Bn which can be extracted

from fixed order calculation

LL: A1 NLL: A1,A2,B1

N2LL: A1-A3,B1,B2 N3LL: A1-A4,B1-B3

Resummed functions

Up to N3LL, all are known except A4

An EFT Approach

A. Manohar, Phys. Rev. 68, 114019 (2003) Based on SCET, conceptually simple and readily

generalizable to other processes. Result obtained to NLL, agrees with old approach

Improvements and to N3LL (Idilbi, Ji, Ma and Yuan, hep-ph/0509294) Take Q-> first and (1-x) is small but not correlated

with Q. An actual formulation of effective field theory, such as

SCET is entirely unnecessary. Result agrees with the old one to all orders in principle,

and to N3LL explicitly.

EFT Approach in a nutshell

Main idea: integrating out physics at different scales stepwise and connecting different scales using renormalization group running.

Main steps: Integrating out physics at scale Q2 by matching to

effective current Taking care of physics between Q2 and

Q2 (1-x) by RG running of the effective current Integrating out physics at scale Q2 by matching to parton

distribution function RG running of PDF through DGLAP

Matching at Q2

At scale Q, one can integrate out perturbative physics from virtual gluons in the vertex type of diagrams,

Running from Q2 to Q2(1-x)

The physics between scale Q2 to Q2(1-x) can be taken care of by solve the renormalization group equation for the scale evolution of the effective current

Where B is the related to the coefficient of the delta function in the anomalous dimension

Matching at Q2(1-x)

At this scale, one must consider soft gluon radiations. Integrating out these radiations matches the theory to parton distributions. The calculation is exactly the same as in the full QCD, therefore, one can take the full QCD result in the soft-collinear limit,

where the logarithms of type lnQ/N has been set to zero

Final Result in EFT

Put all factors together

some additional manipulation shows the full equivalence with the traditional approach.

Comments No actual EFT is needed! Only new scale Q2(1-x) appears, which is assumed to

be perturbative. Power counting in 1-x. Resummation is entirely accomplished. Conceptually

much simpler than original approach.

How does one connect the EFT approach to Sterman’s approach?

Need an actual formulation of EFT

Is it SCET? Maybe: Expansion parameter (1-x) is can be

identified as SCET expansion parameter 2 = (1-x) «1

Maybe Not: In the usual resummation, (1-x)αQ » ΛQCD, for any α>0. In SCET, Q is usually ~ ΛQCD .Thus SCET is defined in a very small kinematic region, whereas

the usual resummation works in a much wider region. In this limit kinematic region, SCET may or may not

generate the correct resummation, because the scale Q is generally non-perturbative.

Questions over SCETFactorization

B. D. Pecjak, JHEP10 (2005) 040. Non-factorizable contribution to DIS at large x

In principle, this is not a problem because there is no proof that the DIS in this region is factorizable.

J. Chay & C. Kim, hep-ph/0511066. There is a non-perturbative soft contribution in

additional to the usual parton distribution.

Soft contribution is at scale Q(1-x) and is non-perturbative.

A different factorization and hence the resumed perturbative part is different from the usual coefficient function.

SCET factorization

In the second stage matching, one can obtain a SCET factorization by matching the DIS process in SCETI on to a product of jet function, soft factor and parton distribution

Chay & Kim

Puzzles

Jet functions reproduces entirely the matching at Q2(1-x)

There is no room for the soft contribution

Role of soft function?

Explicit calculation shows that the soft factor has no infrared divergence and lives in the scale Q(1-x) which is on the order of ΛQCD Only in that sense the soft factor is non-perturbative!

New factorization beyond the usual pQCD factorization?

Scale cancellation?

Thus, SCET factorization is in principle outside of the usual pQCD factorization range.

Since the coefficient function is at the scale Q2(1-x), thus the physics in the soft factor must be cancelled by that in the jet function and parton distributions.

Therefore the non-pert. scale Q(1-x) in SCET is spurious: although it is non-perturbative, but its dependence cancels.

Similar scale cancellation may happen for the calculation of Pecjak, in a way more subtle than that suggested by A. Manohar.

Summary

Using EFT concepts, resummation of large logs in DIS at large x can be done very simply using the renormalization group approach. (Now to N3LL)

SCET factorization of DIS at large x introduces a new small scale Q(1-x). However, this scale cancels in the product. Thus, the DIS resummation works even when Q(1-x) is on the

order of ΛQCD SCET factorization is not the most efficient way to

characterize the important regions of momentum flow.

Generating large-x partons

Large x-partons are generated through soft-gluon radiation

One can write done a differential equation for large-x parton distribution

Knowing the kernal, the solution can be written formally as

Large-x jet function

In the large-x region, the jet function satisfy the following equation

Solution of the equation

Explicit form of factorization

In moment space

Large double logs

Large double logs