Resummation of Large Logs in DIS at x->1
description
Transcript of Resummation of Large Logs in DIS at x->1
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