QED x QCD Resummation and Shower/ME Matching for LHC Physics
QED⊗QCD Resummation and Shower/ME
Matching for LHC Physics∗
B.F.L. Ward and S.A. Yost
Department of Physics, Baylor University, Waco, TX, USA
BU-HEPP-07/02
November 4, 2018
Abstract
We present the theory of QED⊗QCD resummation and its inter-
play with shower/matrix element matching in precision LHC physics
scenarios. We illustrate the theory using single heavy gauge boson
production at hadron colliders.
PACS numbers: 12.38.Cy, 12.15.Lk, 11.25.Db
∗Presented by S.A.Y. at the Cracow Epiphany Conference on Precision Physics and
Monte Carlos for LHC, 4 – 6 January, 2007.
http://arxiv.org/abs/0704.0294v1
1 Introduction
In the imminent LHC environment, where one expects to have an experi-
mental luminosity precision tag at the level of 2%, [1] the requirement for
the theoretical precision tag on the corresponding luminosity processes, such
as single W,Z production with the subsequent decay into light lepton pairs,
should be at the 0.67% level in order not to compromise, unnecessarily, the
over-all precision of the respective LHC luminosity determinations. This dic-
tates that multiple gluon and photon radiative effects must be controlled at
the stated precision. The theory of QED ⊗ QCD exponentiation [2] allows
for the simultaneous resummation of multiple gluon and multiple photon ra-
diative effects in LHC physics processes, to be realized ultimately by MC
methods on an event-by-event basis in the presence of parton showers, in
a framework which allows us to systematically improve the accuracy of the
calculations without double-counting of effects, in principle to all orders in
both αs and α. Such a theoretical framework opens the way to the desired
theoretical precision tag on the LHC luminosity processes.
Our starting point for the new QED⊗QCD resummation theory [2] is
the QCD resummation theory presented in Ref. [3]. This resummation is an
exact rearrangement of the QCD perturbative series based on the N = 1 term
in the exponent in the formal proof of exponentiation in non-Abelian gauge
theories in the eikonal approximation, as given in Ref. [4]. This exponential
is augmented with a sum of residuals which take into account the remaining
contributions to the perturbative series exactly to all orders in αs.
therefore have an exact result whereas the resummation theory in Ref. [4] and
those in Refs. [5–7] are approximate. Recently, an alternative resummation
theory, the soft-collinear effective theory(SCET) [8], has been developed to
treat double resummation of soft and collinear effects. Since we have an exact
re-arrangement of the perturbative series, we could introduce the results from
Refs. [5–8] into our representation as well. Such introductions will appear
elsewhere.
The need for the extension of the QCD resummation theory to QED⊗
QCD resummation was already suggested by the results in Refs. [9–14], where
1If desired, our overall expoential factor can be made to include all of the terms in the
exponent in Ref. [4], in principle.
it was shown that in the evolution of the structure functions the inclusion of
the QED contributions leads to effects at the level of ∼ 0.3%, already almost
half of the error budget discussed above. We will find similar size effects from
the threshold region of heavy gauge boson production. All of these must be
taken into account if one wants ∼ 1.0% for the theoretical precision tag.
The discussion is organized as follows. In Section 2, we review the exten-
sion of the YFS theory to an exact resummation theory for QCD. Section
3 presents the further extension to QED⊗QCD. Section 4 contains the ap-
plication to heavy gauge boson production with the attendant discussion of
shower/ME matching. Section 5 contains some concluding remarks.
2 Extension of YFS Theory to QCD
We consider a parton-level single heavy boson production process such as
q + q̄′ → V + n(g) + X → ℓ̄ℓ′ + n(g) + X , where V = W±, Z, and ℓ =
e, µ, ℓ′ = νe, νµ(e, µ) respectively for V = W
+(Z), and ℓ = νe, νµ, ℓ
′ = e, µ
respectively for V = W−. It has been established [3] that the cross section
may be expressed as
dσ̂exp =
dσ̂n = eSUMIR(QCD)
(2π)4
eiy·(p1+p2−q1−q2−
kj)+DQCD
× ˜̄βn(k1, . . . , kn)
where gluon residuals ˜̄βn(k1, . . . , kn), defined by Ref. [3], are free of all in-
frared divergences to all orders in αs(Q). The functions SUMIR(QCD) and
DQCD, together with the basic infrared functions B
QCD, B̃
QCD, and S̃
QCD are
specified in Ref. [3]. We call attention to the essential compensation between
the left over genuine non-Abelian IR virtual and real singularities between
the phase space integrals
dPh β̄n and
dPh β̄n+1 that really allows us to
isolate ˜̄βj and distinguishes QCD from QED, where no such compensation
occurs. The result in (1) has been realized by Monte Carlo methods [3]. See
also Refs. [15–17] for exact O(α2s) and Refs. [18–20] for exact O(α) results
on the heavy gauge boson production processes which we discuss here.
Apparently, we can not emphasize too much the exactness of (1). Some
confusion seems to exist because it does not show explicitly an ordered ex-
ponential operator for an appropriate ordering prescription, path-ordered,
time-ordered, etc. The essential point is that, in (1), we have evaluated
the matrix elements of these operators and written the result in terms of
the over-all exponent shown therein and the residuals ˜̄βj. This allows us to
maintain exactness to all orders in αs.
3 QED⊗QCD Resummation Theory
The new QED⊗QCD theory is obtained by simultaneously exponentiating
the large IR terms in QCD and the exact IR divergent terms in QED, so that
we arrive at the new result
dσ̂exp = e
SUMIR(QCED)
n,m=0
d3kj1
d3k′j2
(2π)4
eiy·(p1+q1−p2−q2−
k′j2 )+DQCED
× ˜̄βn,m(k1, . . . , kn; k
1, . . . , k
where the new YFS [21,22] residuals, ˜̄βn,m(k1, . . . , kn;k
1, . . . , k
m), with n hard
gluons and m hard photons, defined in Ref. [2], represent the successive
application of the YFS expansion first for QCD and subsequently for QED.
The functions SUMIR(QCED), DQCED are determined from their QCD
analogs SUMIR(QCD), DQCD via the substitutions
BnlsQCD → B
QCD + B
QED ≡ B
QCED,
B̃nlsQCD → B̃
QCD + B̃
QED ≡ B̃
QCED, (3)
S̃nlsQCD → S̃
QCD + S̃
QED ≡ S̃
everywhere in expressions for the latter functions given in Ref. [3]. We stress
that if desired the exponent corresponding the N th Gatherall exponent for
N > 1 can be systematically included in the QCD exponents SUMIR(QCD),
DQCD if desired, with a corresponding change in the respective residuals
˜̄βn,m(k1, . . . , kn; k
1, . . . , k
m). The residuals
˜̄βn,m(k1, . . . , kn; k
1, . . . , k
m) are
free of all infrared singularities, and the result in (2) is a representation that
is exact and that can therefore be used to make contact with parton shower
MC’s without double counting or the unnecessary averaging of effects such as
the gluon azimuthal angular distribution relative to its parent’s momentum
direction.
In the respective infrared algebra (QCED) in (2), the average Bjorken x
values
xavg(QED) ∼= γ(QED)/(1 + γ(QED)),
xavg(QCD) ∼= γ(QCD)/(1 + γ(QCD)),
where γ(A) = 2αACA
(Ls − 1), A = QED, QCD, with CA = Q
f , CF , respec-
tively, for A = QED, QCD and the big log Ls, imply that QCD dominant
corrections happen an order of magnitude earlier than those for QED. This
means that the leading ˜̄β0,0-level gives already a good estimate of the size of
the interplay between the higher order QED and QCD effects which we will
use to illustrate (2) here.
4 QED⊗ QCD Threshold Corrections and
Shower/ME Matching at the LHC
The cross section for the processes pp → V +n(γ)+m(g)+X → ℓ̄ℓ′+n′(γ)+
m(g) +X , where V, ℓ, ℓ′ are the vector-boson / lepton combinations defined
in Section 3, may be constructed from the parton-level cross section via the
usual formula (we use the standard notation here [2])
dσexp =
dxidxjFi(xi)Fj(xj)dσ̂exp(xixjs), (4)
In this section, we will use the result in (2) here with semi-analytical methods
and structure functions from Ref. [23] to examine the size of QED⊗QCD
threshold corrections. A Monte Carlo realization will appear elsewhere [24].
First, we wish to make contact with the existing literature and stan-
dard practice for QCD parton showers as realized by HERWIG [25] and/or
PYTHIA [26]. Eventually, we will also make contact with the new parton
distribution function evolution MC algorithm in Ref. [27]. We intend to
combine our exact YFS-style resummation calculus with HERWIG and/or
PYTHIA by using the latter to generate a parton shower starting from the
initial (x1, x2) point at factorization scale µ, after this point is provided by
the {Fi}. This combination of theoretical constructs can be systematically
improved with exact fully exclusive results order-by-order in αs, where cur-
rently the state of the art in such a calculation is the work in Ref. [28] which
accomplishes the combination of an exact O(αs) correction with HERWIG,
where the gluon azimuthal angle is averaged in the combination.
The issue of this being an exact rearrangement of the QCD and QED
perturbative series requires some comment. Unlike the threshold resumma-
tion techniques in Refs. [5–7], we have a resummation which is valid over the
entire phase space. Thus, it is readily applicable to an exact treatment of
the respective phase space in its implementation via MC methods.
We may illustrate how the combination with PYTHIA/HERWIG may
proceed as follows. We note that, for example, if we use a quark mass
mq as our collinear limit regulator, DGLAP [29] evolution of the structure
functions allows us to factorize all the terms that involve powers of the big log
Lc = lnµ
2/m2q −1 in such a way that the evolved structure function contains
the effects of summing the leading big logs L = lnµ2/µ20 where the evolution
involves initial data at the scale µ0. This gives us a result independent of mq
for mq ↓ 0. In the DGLAP theory, the factorization scale µ represents the
largest pT of the gluon emission included in the structure function.
In practice, when we use these structure functions with an exact result
for the residuals in (2), it means that we must in the residuals omit the
contributions from gluon radiation at scales below µ. This can be shown to
amount in most cases to replacing Ls = ln ŝ/m
q − 1 → Lnls = ln ŝ/µ
2 but
in any case it is immediate how to limit the pT in the gluon emission
that we do not double count effects. In other words, we apply the standard
QCD factorization of mass singularities to the cross section in (2) in the
standard way. We may do it with either the mass regulator for the collinear
singularities or with dimensional regularization of such singularities. The
final result should be independent of this regulator and this is something
that we may use as a cross-check on the results.
This would in practice mean the following: We first make an event with
the formula in (4) which would produce an initial beam state at (x1, x2) for
the two hard interacting partons at the factorization scale µ from the struc-
ture functions {Fj} and a corresponding final state X from the exponenti-
ated cross section in dσ̂exp(xixjs), where we stress that the latter has had all
collinear singularities factorized so that it is much more convergent then its
analog in LEP physics for the electroweak theory for example. The standard
Les Houches procedure [30] of showering this event (x1, x2, X) would then be
used, employing backward evolution of the initial partons. If we restrict the
pT as we have indicated above, there would be no double counting of effects.
Let us call this pT matching of the shower from the backward evolution and
the matrix elements in the QCED exponentiated cross section.
It is possible, however, to be more accurate in the use of the exact result
in (2). Just as the residuals ˜̄βn,m(k1, . . . , kn; k
1, . . . , k
m) are computed order
by order in perturbation theory from the corresponding exact perturbative
results by expanding the exponents in (2) and comparing the appropriate
corresponding coefficients of the respective powers of αnαms , so too can the
shower formula which is used to generate the backward evolution be expanded
so that the product of the shower formula’s perturbative expansion, the per-
turbative expansion of the exponents in (2), and the perturbative expansions
of the residuals can be written as an over-all expansion in powers of αnαms
and required to match the respective calculated exact result for given order.
In this way, new shower subtracted residuals, {
βn,m(k1, . . . , kn; k
1, . . . , k
are calculated that can be used for the entire gluon pT phase space with an
accuracy of the cross section that should in principle be improved compared
with the first procedure for shower matching presented above. Both ap-
proaches are under investigation, where we note that the shower subtracted
2 Here, we refer to both on-shell and off-shell emitted gluons.
residuals have been realized for the exact O(α) luminosity Bhabha process
at DAPHNE energies by the authors in Ref. [31].
Returning to the general discussion, we compute, with and without QED,
the ratio rexp = σexp/σBorn, where we do not use the narrow resonance ap-
proximation, for we wish to set a paradigm for precision heavy vector boson
studies. The formula which we use for σBorn is obtained from that in (4) by
substituting dσ̂Born for dσ̂exp therein, where dσ̂Born is the respective parton-
level Born cross section. Specifically, we have from (1) the ˜̄β0,0-level result
σ̂exp(x1x2s) =
∫ vmax
dv γQCED v
γQCED−1FYFS(γQCED)
eδYFS σ̂Born((1− v)x1x2s)
where we intend the well-known results for the respective parton-level Born
cross sections and the value of vmax implied by the experimental cuts under
study.
What is new here is the value for the QED⊗QCD exponent
γQCED =
+ 2CF
Lnls (6)
where Lnls = ln x1x2s/µ
2 when µ is the factorization scale. The functions
FYFS(γQCED) and δYFS(γQCED) are well-known [22] as well:
FY FS(γQCED) =
e−γQCEDγE
Γ(1 + γQCED)
δY FS(γQCED) =
γQCED +
2ζ(2)−
where ζ(2) is Riemann’s zeta function of argument 2, i.e., π2/6, and γE is
Euler’s constant, i.e., 0.5772. . . .
Using these formulas in (4) allows us to get the results
rexp =
1.1901 , QCED ≡ QCD+QED, LHC
1.1872 , QCD, LHC
1.1911 , QCED ≡ QCD+QED, Tevatron
1.1879 , QCD, Tevatron.
We see that QED is at the level of .3% at both LHC and FNAL. This is stable
under scale variations [2]. We agree with the results in Refs. [15,16,18–20] on
both of the respective sizes of the QED and QCD effects. Furthermore, the
QED effect is similar in size to structure function results found in Refs. [9–13].
5 Conclusions
We have shown that YFS theory (EEX and CEEX), when extended to non-
Abelian gauge theory, allows simultaneous exponentiation of QED and QCD,
QED⊗QCD exponentiation. For QED⊗QCD we find that full MC event
generator realization is possible in a way that combines our calculus with
HERWIG and PYTHIA in principle. Semi-analytical results for QED (and
QCD) threshold effects agree with literature on Z production. As QED is
at the .3% level, it is needed for LHC theory predictions at . 1%. The cor-
responding analysis of the W production is in progress. We have illustrated
a firm theoretical basis for the realization of the complete O(α2s , ααs, α
2) re-
sults needed for the FNAL/LHC/RHIC/ILC physics and all of the latter are
in progress.
Acknowledgments
Work partly supported by US DOE grant DE-FG02-05ER41399 and by
NATO grant PST.CLG.980342. S.A.Y. thanks the organizers of the 2007
Cracow Epiphany Conference for hospitality. B.F.L.W. thanks Prof. W.
Hollik for the support and kind hospitality of the MPI, Munich, while a
part of this work was completed. We also thank Prof. S. Jadach for useful
discussions.
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http://arxiv.org/abs/hep-ph/0109068
	Introduction
	Extension of YFS Theory to QCD
	QEDQCD Resummation Theory
	QED QCD Threshold Corrections and Shower/ME Matching at the LHC
	Conclusions
