Universality of the Collins-Soper-Sterman nonperturbative function in gauge boson production

0.2 MB·English

by  Anton V. Konychev

Universality of the Collins-Soper-Sterman
nonperturbative function
in vector boson production
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2 Anton V. Konycheva and Pavel M. Nadolskyb

n
a
J aDepartment of Physics, Indiana University, Bloomington, IN 47405-7105, U.S.A.

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1 bHigh Energy Physics Division, Argonne National Laboratory,

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Argonne, IL 60439-4815, U.S.A.
v
5
2
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6 Abstract
0
5
0
/ We revise the b∗ model for the Collins-Soper-Sterman resummed form factor to
h
p improve description of the leading-power contribution at nearly nonperturbative
-
p
e impact parameters. This revision leads to excellent agreement of the transverse
h
v: momentum resummation with the data in a global analysis of Drell-Yan lepton pair
i
X
and Z boson production. The nonperturbative contributions are found to follow
r
a
universal quasi-linear dependence on the logarithm of the heavy boson invariant
mass, which closely agrees with an estimate from the infrared renormalon analysis.
Transverse momentum distributions of heavy Drell-Yan lepton pairs, W, or Z
bosons produced in hadron-hadron collisions present an interesting example
of factorization for multi-scale observables. If the transverse momentum q of
T
the electroweak boson is much smaller than its invariant mass Q, dσ/dq at
T
an n-th order of perturbation theory includes large contributions of the type
αnlnm(q2/Q2)/q2 (m = 0,1...2n 1), which must be summed through all
s T T −
Preprint submitted to Elsevier Science 2 February 2008
orders of α to reliably predict the cross section [1]. The feasibility of all-order
s
resummation is proved by a factorization theorem, first formulated for e+e−
hadroproduction [2,3], stated by Collins, Soper, and Sterman (CSS) for the
Drell-Yan process [4], and recently proved by detailed investigation of gauge
transformations of k -dependent parton densities [5,6].
T
The heavy bosons acquire non-zero q mostly by recoiling against QCD radi-
T
ation. The CSS formalism accounts for both the short- and long-wavelength
QCD radiation by means of a Fourier-Bessel transform of a resummed form
factor W(b) introduced in impact parameter (b) space. The perturbative con-
tributiofn, characterized by b . 0.5 GeV−1, dominates in W and Z boson
production at all values of q . The nonperturbative contribution from b &
T
0.5 GeV−1 is not negligible at q < 20 GeV in the precision measurements
T
of the W boson mass M at the Tevatron and LHC [7]. The model for the
W
nonperturbative recoil is the major source of theoretical uncertainty in the
extraction of M from the experimental data. This uncertainty must be re-
W
duced inorder tomeasureM withaccuracyofabout30MeVintheTevatron
W
Run-2 and 15 MeV at the LHC. The nonperturbative model presented below
approaches the level of accuracy desired in these measurements.
The nonperturbative component [described by the function (b,Q) given
NP
F
in Eq.(4)] can be constrained in a few experiments by exploiting process-
independence, oruniversality, of (b,Q),just astheuniversal k -integrated
NP T
F
partondensities areconstrained with thehelp of inclusive scattering data.The
universality of (b,Q) in unpolarized Drell-Yan-like processes and semi-
NP
F
inclusive deep-inelastic scattering (SIDIS) follows from the CSS factorization
theorem [5]. In the study presented here, we carefully investigate agreement of
the universality assumption with the data in a global analysis of fixed-target
2
Drell-Yan pair and Tevatron Z boson production. We revise the nonperturba-
tive model used in the previous studies [8,9] and improve agreement with the
data without introducing additional free parameters. Renormalization-group
invariance requires (b,Q) to depend linearly on lnQ [3,4]. With our lat-
NP
F
est revisions put in place, the global q fit clearly prefers a simple function
T
(b,Q) with universal lnQ dependence. The new (b,Q) has reduced
NP NP
F F
dependence on the collision energy √S comparatively to the earlier fits. The
slope of the lnQ dependence found in the new fit agrees numerically with its
estimate made with methods of infrared renormalon analysis [10,11].
Thefunction (b,Q)primarilyparametrizesthe“power-suppressed” terms,
NP
F
i.e.,termsproportionaltopositivepowersofb.Whenassessedinafit, (b,Q)
NP
F
also contains admixture of the leading-power terms (logarithmic in b terms),
which were not properly included in the approximate leading-power function
W (b) [cf. Eq. (4)]. In contrast, estimates of (b,Q) made in the infrared
LP NP
F
f
renormalon analysis explicitly remove all leading-power contributions from
(b,Q) [11]. While the recent studies [9,10,11,12,13] point to an approx-
NP
F
imately Gaussian form of (b,Q) [ (b,Q) b2], they disagree on the
NP NP
F F ∝
magnitude of (b,Q) and its Q dependence. The source of these differences
NP
F
can be traced to the varying assumptions about the form of the leading-power
function W (b) atb < 2 GeV−1,which is correlatedin thefit with (b,Q).
LP NP
F
The exacft behavior of W(b) at b > 2 GeV−1 is of reduced importance, as
f
W(b) is strongly suppressed at such b. The new improvements described here
f
(excellent agreement of (b,Q) with the data and renormalon analysis)
NP
F
result from modifications in the model for W (b) at b < 2 GeV−1. The im-
LP
f
provements are preserved under variations of the large-b form of W (b) in a
LP
f
significant range of the model parameters.
3
Our paper follows the notations in Ref. [9]. The form factor W(b) factorizes
f
at all b as [2,3,4]
(0)
σ
W(b) = j e−S(b,Q) (x ,b) (x ,b), (1)
S Pj 1 P¯j 2
jX=q,q¯
f
where σ(0)/S is a constant prefactor [4], and x e±yQ/√S are the Born-
j 1,2 ≡
level momentum fractions, with y being the rapidity of the vector boson. The
b-dependent parton densities (x,b) and Sudakov function
j
P
Q2
dµ¯2 Q2
(b,Q) (α (µ¯))ln + (α (µ¯)) (2)
S ≡ Z µ¯2 (cid:20)A s (cid:18)µ¯2(cid:19) B s (cid:21)
b20/b2
are universal in Drell-Yan-like processes and SIDIS [5]. When the momentum
scales Q and b /b (where b 2e−γE 1.123 is a dimensionless constant) are
0 0
≡ ≈
much larger than 1 GeV, W(b) reduces to its perturbative part W (b), i.e.,
pert
f f
its leading-power (logarithmic in b) part evaluated at a finite order of α :
s
(0)
σ
W(b) W (b) j e−SP(b,Q)
pert
f (cid:12)(cid:12)Q,b0/b≫1GeV ≈ f ≡ jX=q,q¯ S
(cid:12)
[ f] (x ,b; µ )[ f] (x ,b; µ ). (3)
× C ⊗ j 1 F C ⊗ ¯j 2 F
Here (b,Q) and [ f] (x,b; µ ) 1dξ/ξ (x/ξ,µ b)f (ξ,µ ) are
SP C ⊗ j F ≡ a x Cja F a F
P R
the finite-order approximations to the leading-power parts of (b,Q) and
S
(x,b). f (x,µ ) is the k -integrated parton density, computed in our study
j a F T
P
by using the CTEQ6M parameterization [14]. (x,µ b) is the Wilson coef-
ja F
C
ficient function. We compute (b,Q) up to O(α2) and up to O(α ).
SP s Cja s
In Z boson production, the maximum of bW(b) is located at b 0.25 GeV−1,
≈
f
and W (b) dominates the Fourier-Bessel integral. In the examined low-Q
pert
regionf, the maximum of bW(b) is located at b 1 GeV−1, where higher-order
≈
f
4
corrections in powers of α and b must be considered. We reorganize Eq. (1) to
s
separate the leading-power (LP) term W (b), given by the model-dependent
LP
continuation of W (b) to b & 1 GeVf−1, and the nonperturbative exponent
pert
f
e−FNP(b,Q), which absorbs the power-suppressed terms:
W(b) = W (b)e−FNP(b,Q). (4)
LP
f f
At b 0, the perturbative approximation for W(b) is restored: W W ,
LP pert
→ →
f f f
0. The power-suppressed contributions are proportional to even pow-
NP
F →
ers of b [10]. Detailed expressions for some power-suppressed terms are given
in Ref. [11]. At impact parameters of order 1 GeV−1, we keep only the first
power-suppressed contribution proportional to b2:
b2(a +a ln(Q/Q )+a φ(x )+a φ(x ))+..., (5)
NP 1 2 0 3 1 3 2
F ≈
where a , a , and a are coefficients of magnitude less than 1 GeV2, and φ(x)
1 2 3
is a dimensionless function. The terms a ln(Q/Q ) and a φ(x ) arise from
2 0 3 j
(b,Q) and ln[ (x ,b)] in ln[W(b)], respectively. We neglect the flavor de-
j j
S P
f
pendence of φ(x) in the analyzed region dominated by scattering of light u
and d quarks. is consequently a universal function within this region. The
NP
F
dependence of on lnQ follows from renormalization-group invariance of
NP
F
the soft-gluon radiation [3]. The coefficient a of the lnQ term has been re-
2
lated to the vacuum average of the Wilson loop operator and estimated within
lattice QCD as 0.19+0.12 GeV2 [11].
−0.09
The preferred is correlated in the fit with the assumed large-b behavior of
NP
F
W . We examine this correlation in a modified version of the b model [3,4].
LP ∗
f
The shape of W is varied in the b model by adjusting a single parameter
LP ∗
f
5
b . Continuity of W and its derivatives, needed for the numerical stability
max
f
of the Fourier transform, is always preserved. We set W (b) W (b ),
LP pert ∗
≡
f f
with b (b,b ) b(1 + b2/b2 )−1/2. W (b) reduces to W (b) as b 0
∗ max ≡ max LP pert →
f f
and asymptotically approaches W (b ) as b . The b model with a
pert max ∗
→ ∞
relatively low b = 0.5 GeV−f1 was a choice of the previous q fits [8,9].
max T
However, it is natural to consider b above 1 GeV−1 in order to avoid ad
max
hoc modifications of W (b) in the region where perturbation theory is still
pert
applicable. To implemfent W (b ) for b > 1 GeV−1, we must choose the
pert ∗ max
f
factorization scale µ such that it stays, at any b and b , above the initial
F max
scaleQ = 1.3GeVoftheDGLAPevolutionfortheCTEQ6PDF’sf (x,µ ).
ini a F
We keep the usual choice µ = C /b (b,b ), where C b , for b
F 3 ∗ max 3 0 max
∼ ≤
b /Q 0.86 GeV−1. Such choice is not acceptable at b > b /Q , as it
0 ini max 0 ini
≈
would allow µ < Q . Instead, we choose µ = C /b (b,b /Q ) for b >
F ini F 3 ∗ 0 ini max
b /Q , i.e., we substitute b /Q for b in µ to satisfy µ Q at any
0 ini 0 ini max F F ini
≥
b. Aside from f (x,µ ), all terms in W (b) are known, at least formally, as
a F pert
f
explicit functions of α (1/b) at all b < 1/Λ . We show in Ref. [15] that
s QCD
this prescription preserves correct resummation of the large logarithms and is
numerically stable up to b 3 GeV−1.
max
∼
We perform a series of fits for several choices of b by closely following the
max
previous global q analysis [9]. We consider a total of 98 data points from
T
production of Drell-Yan pairs in E288, E605, and R209 fixed-target experi-
ments, as well as from observation of Z bosons with q < 10 GeV by CDF
T
and DØ detectors in the Tevatron Run-1. See Ref. [9] for the experimental
references. Overall normalizations for the experimental cross sections are var-
ied as free parameters. Our best-fit normalizations agree with the published
values within the systematical errors provided by the experiments, with the
6
1.2
E288
1 E605
CDFZ
0.8 D0Z
R209
D
2
V
e 0.6
G
@
a
0.4
a = 0.19 GeV2
2
0.2
b =1.5GeV-1
max
0
5 10 20 50 100 200
Q GeV
@ D
Fig. 1. The best-fit values of a(Q) obtained in independent scans of χ2 for the
contributing experiments. The vertical error bars correspond to the increase of χ2
by unity above its minimum in each Q bin. The slope of the line is equal to the
central-value prediction from the renormalon analysis [11].
exception of the CDF Run-1 normalization (rescaled down by 7%).
To test the universality of , we individually examine each bin of Q. We
NP
F
choose = a(Q)b2 and independently fit it to each of the 5 experimental
NP
F
data sets to determine the most plausible normalization in each experiment.
We then set the normalizations equal to their best-fit values and examine
χ2 at each Q as a function of a(Q). For b = 1 2 GeV−1, the best-fit
max
−
values of a(Q) follow a nearly linear dependence on lnQ [cf. Fig. 1]. The
slope a da(Q)/d(lnQ) is close to the renormalon analysis expectation
2
≡
of 0.19 GeV2 [11]. The agreement with the universal linear lnQ dependence
worsens if b is chosen outside the region 1-2 GeV−1. Since the best-fit a(Q)
max
are found independently in each Q bin, we conclude that the data support the
universality of , when b lies in the range 1 2 GeV−1. In another test,
NP max
F −
we find that each experimental data set individually prefers a nearly quadratic
dependence on b, = a(Q)b2−β, with β < 0.5 in all five experiments.
NP
F | |
7
150 0.3
C3=b0
140 C3=2b0 0.25
2VL
2χ130 Ge 0.2
H
a1
120 0.15
110 0.1
0
0.6
0.5 -0.05
22GeVGeVHLHL00..34 RaneanloyrsmisaHlToanfatL 2GeVHL-0.1
aa22 a3
0.2 -0.15
0.1
0 -0.2
0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5
bmaxHGeV−1L bmaxHGeV−1L
Fig. 2. The best-fit χ2 and coefficients a , a , and a in (b,Q) for different
1 2 3 NP
F
values of b , C = b (stars) and C = 2b (squares). The size of the symbols
max 3 0 3 0
approximately corresponds to 1σ errors for the shown parameters.
To further explore the issue, we simultaneously fit our model to all the data.
We parametrize a(Q) as a(Q) a +a ln[Q/(3.2 GeV)]+a ln[100x x ]. This
1 2 3 1 2
≡
parametrization coincides with the BLNY form [9], if the parameters are re-
named as g ,g ,g g (BLNY) a ,a ,a (here). It agrees with the generic
1 2 1 3 1 2 3
{ } → { }
form of (b,Q) in Eq. (5), if one identifies φ(x) = ln(x/0.1). We carry out
NP
F
two sequences of fits for C = b and C = 2b to investigate the stability of
3 0 3 0
our prescription for µ and sensitivity to (α2) corrections. The dependence
F O s
on C is relatively uniform across the whole range of b , indicating that our
3 max
choice of µ for b > b /Q is numerically stable.
F max 0 ini
Fig. 2 shows the dependence of the best-fit χ2, a , a , and a on b . As
1 2 3 max
b is increased above 0.5 GeV−1 assumed in the BLNY study, χ2 rapidly
max
decreases, becomes relatively flat at b = 1 2 GeV−1, and grows again
max
−
at b > 2 GeV−1. The global minimum of χ2 = 125(111) is reached at
max
b 1.5 GeV−1, where all data sets are described equally well, without
max
≈
major tensions among the five experiments. The improvement in χ2 mainly
8
Fig. 3. Differences between the measured (Data) and theoretical (Theory) cross
sections, divided by the experimental error δ in each (Q,q ) bin. The values of
exp T
χ2 for each experiment in the two fits are listed in the legend in the same order.
ensues from better agreement with the low-Q experiments (E288, E605, and
R209), while the quality of all fits to the Z data is about the same. This
is illustrated by Fig. 3, which shows the differences between the measured
and theoretical cross sections, divided by the experimental errors δ , as well
exp
as the values of χ2 in each experiment, in two representative fits for b =
max
0.5 GeV−1, C = b (squares) and b = 1.5 GeV−1, C = 2b (triangles).
3 0 max 3 0
The data are arranged in bins of Q (shown by gray background stripes) and
q , with both variables increasing from left to right. For b = 1.5 GeV−1,
T max
the (Data Theory) differences are reduced on average in the entire low-Q
−
sample, resulting in lower χ2 in three low-Q experiments. Two outlier points
in the E605 sample (the first point in the second Q bin and fifth point in the
fifth Q bin) disagree with the other E288 and E605 data in the same Q and x
region and contribute 15 25 units to χ2 at any b . If the two outliers were
max
−
removed, one would find χ2/d.o.f. 1 at the global minimum.
≈
The magnitudes of a , a , and a are reduced when b increases from 0.5
1 2 3 max
to 1.5 GeV−1. In the whole range 1 b 2 GeV−1, a agrees with the
max 2
≤ ≤
9
p+p-®Z0+X;(cid:143)!s!!=1.96TeV;Q=MZ;y=0 p+Cu®Μ+Μ-+X;(cid:143)!s!!=38.8GeV;Q=11GeV;y=0
0.6 40
bmax=1.5GeV-1,C3=b0
0.5 bmax=1.5GeV-1,C3=2b0
VD bmax=0.5GeV-1,C3=b0 GeVD30
Ge0.4 b(cid:144)
nb@(cid:144) xfBL@
,xABL0.3 Q,x,A20
ŽbWb,Q,xH00..12 Ž1-NbWb,fitH10 bbmmaaxx==11..55GGeeVV--11,,CC33==2bb00,,NNfifti=t=11..1095
bmax=0.5GeV-1,C3=b0,Nfit=1.09
Qiu-Zhang,bmax=0.3GeV-1,Nfit=1
0 0
0.2 0.4 0.6 0.8 1 1.2 1.4 0.25 0.5 0.75 1 1.25 1.5 1.75 2
b GeV-1 b GeV-1
@ D @ D
Fig. 4. The best-fit form factors bW(b) in (a) Tevatron Run-2 Z boson production;
(b)E605 experiment. IntheE605fcase, bW(b) aredividedby the best-fitnormaliza-
tions N for the E605 data, and the formffactor in the Qiu-Zhang parametrization
fit
[12] for bQZ = 0.3 GeV−1 is also shown.
max
renormalon analysis estimate. The coefficient a , which parametrizes devia-
3
tions from the linear lnQ dependence, is considerably smaller (< 0.05) than
both a and a ( 0.2). A reasonable quality of the fit is retained if a is set
1 2 3
∼
to zero by hand: χ2 increases by 5 in such a fit above its minimum in the
≈
fit with a free a . In contrast, χ2 increases by > 200 units if a = g g is set
3 3 1 3
to zero at b = 0.5 GeV−1, as it was noticed in the BLNY study.
max
The preference for the values of b between 1 and 2 GeV−1 indicates, first,
max
that the data do favor the extension of the b range where W (b) is approx-
LP
f
imated by the exact W (b). In Z boson production, this region extends up
pert
to 3 4 GeV−1 as a cfonsequence of the strong suppression of the large-b tail
−
by the Sudakov exponent. The fit to the Z data is actually independent of
b within the experimental uncertainties for b > 1 GeV−1. The best-fit
max max
form factors bW(b) for b = 0.5 and 1.5 GeV−1 in Z boson production are
max
f
shown in Fig. 4(a).
10