Joint resummation for slepton pair production at hadron colliders
ABSTRACT We present a precision calculation of the transversemomentum and invariantmass distributions for supersymmetric particle pair production at hadron colliders, focusing on Drell–Yan like slepton pair and slepton–sneutrino associated production at the CERN Large Hadron Collider. We implement the joint resummation formalism at the nexttoleading logarithmic accuracy with a processindependent Sudakov form factor, thus ensuring a universal description of softgluon emission, and consistently match the obtained result with the pure perturbative result at the first order in the strong coupling constant, i.e. at O(αs). We also implement three different recent parameterizations of nonperturbative effects. Numerically, we give predictions for production and compare the resummed cross section with the perturbative result. The dependence on unphysical scales is found to be reduced, and nonperturbative contributions remain small.

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ABSTRACT: If all strongly interacting sparticles (the squarks and the gluinos) in an unconstrained minimal supersymmetric standard model (MSSM) are heavier than the corresponding mass lower limits in the minimal supergravity (mSUGRA) model, obtained by the current LHC experiments, then the existing data allow a variety of electroweak (EW) sectors with light sparticles yielding dark matter (DM) relic density allowed by the WMAP data. Some of the sparticles may lie just above the existing lower bounds from LEP and lead to many novel DM producing mechanisms not common in mSUGRA. This is illustrated by revisiting the above squarkgluino mass limits obtained by the ATLAS Collaboration, with an unconstrained EW sector with masses not correlated with the strong sector. Using their selection criteria and the corresponding cross section limits, we find at the generator level using Pythia, that the changes in the mass limits, if any, are by at most 1012% in most scenarios. In some cases, however, the relaxation of the gluino mass limits are larger ($\approx 20%$). If a subset of the strongly interacting sparticles in an unconstrained MSSM are within the reach of the LHC, then signals sensitive to the EW sector may be obtained. This is illustrated by simulating the $blj$$\etslash$, $l= e and \mu $, and $b\tau j$$\etslash$ signals in i) the light stop scenario and ii) the light stopgluino scenario with various light EW sectors allowed by the WMAP data. Some of the more general models may be realized with nonuniversal scalar and gaugino masses.Journal of High Energy Physics 03/2012; 2012(6). · 5.62 Impact Factor  SourceAvailable from: ArXiv[Show abstract] [Hide abstract]
ABSTRACT: We present an exploratory study of gauginopair production in polarized and unpolarized hadron collisions, focusing on the correlation of beam polarization and gaugino/Higgsino mixing in the general Minimal Supersymmetric Standard Model. Helicitydependent cross sections induced by neutral and charged electroweak currents and squark exchanges are computed analytically in terms of generalized charges, defined similarly for charginopair, neutralinochargino associated, and neutralinopair production. Our results confirm and extend those obtained previously for negligible Yukawa couplings and nonmixing squarks. Assuming that the lightest chargino mass is known, we show numerically that measurements of the longitudinal singlespin asymmetry at the existing polarized pp collider RHIC and at possible polarization upgrades of the Tevatron or the LHC would allow for a determination of the gaugino/Higgsino fractions of charginos and neutralinos. The theoretical uncertainty coming from factorization scale and squark mass variations and the expected experimental error on the lightest chargino mass is generally smaller than the one induced by the polarized parton densities, so that more information on the latter would considerably improve on the analysis.Physical review D: Particles and fields 05/2008;  SourceAvailable from: export.arxiv.org[Show abstract] [Hide abstract]
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Page 1
arXiv:0709.3057v1 [hepph] 19 Sep 2007
Joint resummation for slepton pair production at hadron colliders
Giuseppe Bozzi
Insitut f¨ ur Theoretische Physik, Universit¨ at Karlsruhe, Postfach 6980, D76128 Karlsruhe, Germany
Benjamin Fuks and Michael Klasen∗
Laboratoire de Physique Subatomique et de Cosmologie,
Universit´ e Joseph Fourier/CNRSIN2P3, 53 Avenue des Martyrs, F38026 Grenoble, France
(Dated: February 5, 2008)
We present a precision calculation of the transversemomentum and invariantmass distributions
for supersymmetric particle pair production at hadron colliders, focusing on DrellYan like slepton
pair and sleptonsneutrino associated production at the CERN Large Hadron Collider. We imple
ment the joint resummation formalism at the nexttoleading logarithmic accuracy with a process
independent Sudakov form factor, thus ensuring a universal description of softgluon emission, and
consistently match the obtained result with the pure perturbative result at the first order in the
strong coupling constant, i.e. at O(αs). We also implement three different recent parameterizations
of nonperturbative effects. Numerically, we give predictions for ˜ eR˜ e∗
resummed cross section with the perturbative result. The dependence on unphysical scales is found
to be reduced, and nonperturbative contributions remain small.
Rproduction and compare the
PACS numbers: 12.60.Jv,13.85.Ni,14.80.Ly
I.INTRODUCTION
KATP152007
LPSC 0772
SFBCPP0726
One of the main tasks in the experimental programme of the CERN Large Hadron Collider (LHC) is to perform an
extensive and conclusive search of the supersymmetric (SUSY) partners of the Standard Model (SM) particles pre
dicted by the Minimal Supersymmetric Standard Model [1, 2]. Scalar leptons are among the lightest supersymmetric
particles in many SUSYbreaking scenarios [3, 4]. Presently, the experimental (lower) limits on electron, muon, and
tau slepton masses are 73 GeV, 94 GeV, and 81.9 GeV, respectively [5]. Since sleptons often decay into the corre
sponding SM partner and the lightest stable SUSY particle, the distinctive signature at hadron colliders will consist
in a highly energetic lepton pair and associated missing energy.
The leading order (LO) cross section for the production of nonmixing slepton pairs has been calculated in [6, 7, 8, 9],
while the mixing between the interaction eigenstates was included in [10]. The nexttoleading order (NLO) QCD
corrections have been calculated in [11], and the full SUSYQCD corrections with nonmixing squarks in the loops
have been added in [12]. Recently, an accurate calculation of the transversemomentum (qT) spectrum including
softgluon resummation at the nexttoleading logarithmic (NLL) accuracy has been performed [13], allowing for the
reconstruction of the mass and the determination of the spin of the produced particles by means of the Cambridge
(s)transverse mass variable [14, 15] and for distinguishing thus the SUSY signal from the SM background, mainly
due to WW and t¯t production [16, 17]. Very recently, the mixing effects relevant for the squarks appearing in the
loops have been investigated at NLO, and the thresholdenhanced contributions have been computed at NLL [18].
The numerical results show a stabilization of the perturbative results through a considerable reduction of the scale
dependence and a modest increase with respect to the NLO cross section.
Since the dynamical origin of the enhanced contributions is the same both in transversemomentum and threshold
resummations, i.e. the softgluon emission by the initial state, it would be desirable to have a formalism capable to
handle at the same time the softgluon contributions in both the delicate kinematical regions, qT ≪ M and M2∼ s,
M being the slepton pair invariantmass and s the partonic centreofmass energy. This joint resummation formalism
has been developed in the last eight years [19, 20]. The exponentiation of the singular terms in the Mellin (N) and
impactparameter (b) space has been proven, and a consistent method to perform the inverse transforms, avoiding the
Landau pole and the singularities of the parton distribution functions, has been introduced. Applications to prompt
photon [21], electroweak boson [22], Higgs boson [23], and heavyquark pair [24] production at hadron colliders have
exhibited substantial effects of joint resummation on the differential cross sections.
∗klasen@lpsc.in2p3.fr
Page 2
2
In this paper we apply the joint resummation formalism at the NLL level to the hadroproduction of slepton pairs
at the LHC, thus completing our programme (started in Ref. [13] and continued in Ref. [18]) of providing the first
precision calculations including softgluon resummation for slepton pair production at hadron colliders. In Sec. II,
we briefly review the theoretical formalism of joint resummation following Refs. [20, 22]. We reorganize the terms of
the resummed formula in a similar way as it was done for transversemomentum resummation in [25]. The inverse
transforms from the Mellin and impactparameter spaces and the matching of the resummed result with the fixed
order perturbative results are discussed in Sec. III. Sec. IV is devoted to phenomenological predictions for the LHC,
together with a comparison of the three types of resummation (transversemomentum, threshold, and joint), showing
their impact on the qTspectrum and on the invariantmass distribution. Our results are summarized in Sec. V.
II.JOINT RESUMMATION AT THE NEXTTOLEADING LOGARITHMIC ORDER
We consider the hard scattering process
ha(pa)hb(pb) → F(M,qT) + X, (1)
where F is a generic system of colourless particles, such as a Higgs boson or a DrellYan (s)lepton pair, M is the
invariant mass of the final state F, and qT is its transverse momentum. Thanks to the QCD factorization theorem,
the unpolarized hadronic cross section
d2σ
dM2dq2
T
=
?
a,b
?1
τ
dxa
?1
τ/xa
dxbfa/ha(xa;µF)fb/hb(xb;µF)
d2ˆ σab
dM2dq2
T
(z;αs,µR,µF)(2)
can be written as the convolution of the relevant partonic cross section ˆ σabwith the universal distribution functions
fa,b/ha,bof partons a,b inside the hadrons ha,b, which depend on the longitudinal momentum fractions of the two
partons xa,band on the unphysical factorization scale µF. The partonic scattering cross section depends on the strong
coupling constant αs, the unphysical renormalization and factorization scales µRand µF, and on the scaling variable
z = M2/s, where s = xaxbS and S = (pa+ pb)2are the partonic and hadronic centreofmass energies, respectively.
The lower limits for the integration over the longitudinal momentum fractions contain the quantity τ = M2/S, which
approaches the value τ = 1 when the process is close to the hadronic threshold M2∼ S. In Mellin Nspace, the
hadronic cross section naturally factorizes
d2σ
dM2dq2
T
=
?
a,b
?
C
dN
2πiτ−Nfa/ha(N + 1;µF)fb/hb(N + 1;µF)
d2ˆ σab
dM2dq2
T
(N;αs,µR,µF),(3)
where the contour C in the complex Nspace will be specified in Sec. III and the Nmoments of the various quantities
are defined according to the Mellin transform
F(N) =
?1
0
dxxN−1F(x)(4)
for x = xa,b,z,τ and F = fa/ha,b/hb, ˆ σ,σ, respectively. The jointly resummed hadronic cross section in Nspace can
be written at NLL accuracy as [20, 22, 23]
d2σ(res)
dM2dq2
T
(N;αs,µR,µF) =
?
c
ˆ σ(0)
c¯ cHc¯ c(αs,µR)
?
d2b
4πeib·qTCc/ha(N,b;αs,µR,µF)
× exp
?
E(PT)
c
(N,b;αs,µR)
?
C¯ c/hb(N,b;αs,µR,µF). (5)
The indices c and ¯ c refer to the initial state of the lowestorder cross section ˆ σ(0)
the final state F is assumed to be colourless.
For slepton pair and sleptonsneutrino associated production at hadron colliders,
c¯ c and can then only be q¯ q or gg, since
ha(pa)hb(pb) →˜li(p1)˜l(′)∗
j
(p2) + X,(6)
Page 3
3
we have M2= (p1+ p2)2, q2
T= (p1T− p2T)2, and
α2π β3
9M2
ˆ σ(0)
q¯ q
=
?
e2
qe2
lδij+
eqelδij(LqqZ+ RqqZ)Re(L˜li˜ljZ+ R˜li˜ljZ)
4xW(1 − xW)(1 − m2
???L˜li˜ljZ+ R˜li˜ljZ
???Lqq′WL˜li˜ νlW
Z/M2)
+
(L2
qqZ+ R2
32x2
qqZ)
???
???
2
W(1 − xW)2(1 − m2
Z/M2)2
?
,(7)
ˆ σ(0)
q¯ q′ =
α2π β3
9M2
?
2
32x2
W(1 − xW)2(1 − m2
W/M2)2
?
,(8)
where i,j denote slepton/sneutrino mass eigenstates with masses mi,j, mZand mW are the masses of the electroweak
gauge bosons, α is the electromagnetic fine structure constant, xW = sin2θW is the squared sine of the electroweak
mixing angle, and the velocity β is defined as
β =
?
1 + m4
i/M4+ m4
j/M4− 2(m2
i/M2+ m2
j/M2+ m2
im2
j/M4). (9)
The coupling strengths of the left and righthanded (s)fermions to the electroweak vector bosons are given by
{Lff′Z,Rff′Z} = (2T3
{L˜fi˜f′
{Lqq′W,Rqq′W} = {√2cWVqq′,0},
{L˜li˜ νlW,R˜li˜ νlW} = {√2cWS˜l∗
{L˜ qi˜ q′
f− 2efxW) × δff′,
˜ f
j1S
jZ,R˜fi˜f′
jZ} = {Lff′ZS
˜ f′∗
i1,Rff′ZS
˜ f
j2S
˜ f′∗
i2},
i1, 0},
i1S˜ q′
jW,R˜ qi˜ q′
lW} = {Lqq′WS˜ q∗
j1, 0},(10)
where the weak isospin quantum numbers are T3
cW is the cosine of the electroweak mixing angle, and Vff′ are the CKMmatrix elements. The unitary matrices S˜f
diagonalize the sfermion mass matrices, since in general the sfermion interaction eigenstates are not identical to the
sfermion mass eigenstates (see App. A).
The function Hc¯ cin Eq. (5) contains the hard virtual contributions and can be expanded perturbatively in powers
of αs,
f= ±1/2 for lefthanded and T3
f= 0 for righthanded (s)fermions,
Hc¯ c(αs,µR) = 1 +
∞
?
n=1
?αs(µR)
π
?n
H(n)
c¯ c(µR).(11)
The coefficients
Cc/ha(N,b;αs,µR,µF) =
?
a,b
Cc/b(N;αs(M/χ))Ub/a(N;M/χ,µF)fa/ha(N + 1;µF)(12)
and C¯ c/hb, defined analogously, allow to evolve the parton distribution functions fa,b/ha,bfrom the unphysical factor
ization scale µF to the physical scale M/χ with the help of the QCD evolution operator
Ub/a(N;µ,µ0) = exp
??µ2
µ2
0
dq2
q2γb/a(N;αs(q))
?
(13)
and to include, at this scale, the fixedorder contributions
Cc/b(N;αs) = δcb+
∞
?
n=1
?αs
π
?n
C(n)
c/b(N),(14)
that become singular when qT→ 0 (but not when z → 1). The QCD evolution operator fulfils the differential equation
dUb/a(N;µ,µ0)
dlnµ2
c
=
?
Ub/c(N;µ,µ0)γc/a(N;αs(µ)),(15)
Page 4
4
where the anomalous dimensions γc/a(N;αs) are the Nmoments of the AltarelliParisi splitting functions. The
function
χ(¯b,¯ N) =¯b +
¯ N
1 + η¯b/¯ N
with
¯b ≡ bM eγE/2 and
¯ N ≡ NeγE
(16)
organizes the logarithms of b and N in joint resummation. Its exact form is constrained by the requirement that
the leading and nexttoleading logarithms in¯b and¯ N are correctly reproduced in the limits¯b → ∞ and¯ N → ∞,
respectively. The choice of Eq. (16) with η = 1/4 avoids the introduction of sizeable subleading terms into perturbative
expansions of the resummed cross section at a given order in αs, which are not present in fixedorder calculations [22].
The perturbative (PT) eikonal exponent
E(PT)
c
(N,b;αs,µR) = −
?M2
M2/χ2
dµ2
µ2
?
Ac(αs(µ))lnM2
µ2+ Bc(αs(µ))
?
(17)
allows to resum soft radiation in the Aterm, while the Bterm accounts for the difference between the eikonal approx
imation and the full partonic cross section in the threshold region, i.e. the flavourconserving collinear contributions.
In the largeN limit, these coefficients are directly connected to the leading terms in the oneloop diagonal anomalous
dimension calculated in the MS factorization scheme [26]
γc/c(N;αs) = −Ac(αs)ln¯ N −Bc(αs)
2
+ O(1/N).(18)
They can thus also be expressed as perturbative series in αs,
Ac(αs) =
∞
?
n=1
?αs
π
?n
A(n)
c
andBc(αs) =
∞
?
n=1
?αs
π
?n
B(n)
c . (19)
Performing the integration in Eq. (17), we obtain the form factor up to NLL,
E(PT)
c
(N,b;αs,µR) = g(1)
c(λ) lnχ + g(2)
c(λ;µR)(20)
with
g(1)
c(λ) =
A(1)
c
β0
A(1)
c β1
β3
2λ + ln?1 − 2λ?
λ
?
A(1)
c
β0
µ2
R
,
g(2)
c (λ;µR) =
0
1
2ln2?1 − 2λ?+2λ + ln?1 − 2λ?
lnM2
β2
0
1 − 2λ
?
+
?
−A(2)
c
??
2λ
1 − 2λ+ ln?1 − 2λ??
+B(1)
c
β0
ln?1 − 2λ?
(21)
and λ = β0/παs(µR)lnχ. The first two coefficients of the QCD βfunction are
β0=
1
12(11CA− 4TRNf) and β1=
1
24(17C2
A− 10TRCANf− 6CFTRNf),(22)
Nf being the number of effectively massless quark flavours and CF = 4/3, CA= 3, and TR= 1/2 the usual QCD
colour factors.
In order to explicitly factorize the dependence on the parameter χ, it is possible to reorganize the resummation of
the logarithms in analogy to the case of transversemomentum resummation [25, 27]. The hadronic resummed cross
section can then be written as
d2σ(res)
dM2dq2
T
=
?
?
a,b
?
C
dN
2πiτ−Nfa/ha(N + 1;µF)fb/hb(N + 1;µF)
?∞
0
bdb
2
J0(bqT)
×
c
Hab→c¯ c
?
N;αs,µR,µF
?
exp[Gc(lnχ;αs,µR)].(23)
Page 5
5
The function Hab→c¯ cdoes not depend on the parameter χ and contains all the terms that are constant in the limits
b → ∞ or N → ∞,
?
n=1
Hab→c¯ c
?
N;αs,µR,µF
?
= ˆ σ(0)
c¯ c
δcaδ¯ cb+
∞
?
?αs(µR)
π
?n
H(n)
ab→c¯ c
?
N;µR,µF
??
. (24)
At O(αs), the coefficient H(1)
ab→c¯ cis given by
H(1)
ab→c¯ c
?
N;µR,µF
?
= δcaδ¯ cbH(1)
c¯ c(µR) + δcaC(1)
¯ c/b(N) + δ¯ cbC(1)
c/a(N) +
?
δcaγ(1)
¯ c/b(N) + δ¯ cbγ(1)
c/a(N)
?
lnM2
µ2
F
.(25)
The χdependence appearing in the Ccoefficient and in the evolution operator U of Eq. (12) is factorized into the
exponent Gc, which has the same form as E(PT)
Bc(αs) →˜Bc(N;αs) = Bc(αs) + 2β(αs)dlnCc/c(N;αs)
c
defined in Eq. (17) except for the substitution
dlnαs
+ 2γc/c(N;αs).(26)
At NLL accuracy, Eq. (20) remains almost unchanged, since only the coefficient g(2)
modified by
c
of Eq. (21) has to be slightly
B(1)
c
→˜B(1)
c (N) = B(1)
c
+ 2γ(1)
c/c(N). (27)
Although the firstorder coefficients C(1)
dependence cancels in the perturbative expression of Hab→c¯ c [25]. In the numerical code we developed for slepton
pair production, we implement the DrellYan resummation scheme and take Hq¯ q(αs,µR) ≡ 1. The Ccoefficients are
then given by
a/b(N) and H(1)
c¯ c(µR) are in principle resummationscheme dependent [27], this
C(1)
q/q(N) =
2
3N (N + 1)+π2− 8
3
and C(1)
q/g(N) =
1
2(N + 1)(N + 2).(28)
III. INVERSE TRANSFORM AND MATCHING WITH THE PERTURBATIVE RESULT
Once resummation has been achieved in N and bspace, inverse transforms have to be performed in order to get
back to the physical spaces. Special attention has to be paid to the singularities in the resummed exponent, related
to the divergent behaviour near χ = exp[π/(2β0αs)], i.e. the Landau pole of the running strong coupling, and near
¯b = −2¯ N and¯b = −4¯ N, where χ = 0 and infinity, respectively. The integration contours of the inverse transforms in
the Mellin and impact parameter spaces must therefore avoid hitting any of these poles.
The b−integration is performed by deforming the integration contour with a diversion into the complex bspace
[21], defining two integration branches
b = (cosϕ ± isinϕ)t with 0 ≤ t ≤ ∞,(29)
valid under the condition that the integrand decreases sufficiently rapidly for large values of b. The Bessel function
J0is replaced by two auxiliary functions h1,2(z,v) related to the Hankel functions
h1(z,v) ≡ −1
π
?−π+ivπ
?−ivπ
−ivπ
dθe−iz sinθ,
h2(z,v) ≡ −1
π
π+ivπ
dθe−iz sinθ.(30)
Their sum is always h1(z,v) + h2(z,v) = 2J0(z), but they distinguish positive and negative phases of the bcontour,
being then associated with only one of the two branches defined in Eq. (29).
The inverse Mellin transform is performed following a contour inspired by the Minimal Prescription [28] and the
Principal Value Resummation [29], where one again defines two branches
N = C + z e±iφwith 0 ≤ z ≤ ∞, π > φ >π
2.(31)
Page 6
6
The parameter C is chosen in such a way that all the singularities related to the Nmoments of the parton densities
are to the left of the integration contour. It has to lie within the range 0 < C < exp[π/(2β0αs) − γE] in order to
obtain convergent inverse transform integrals for any choice of φ and ϕ.
A matching procedure of the NLL resummed cross section to the NLO result has to be performed in order to keep the
full information contained in the fixedorder calculation and to avoid possible doublecounting of the logarithmically
enhanced contributions. A correct matching is achieved through the formula
d2σ
dM2dq2
T
=d2σ(F.O.)
dM2dq2
T
(αs) +
?
CN
dN
2πiτ−N
?
bdb
2
J0(bqT)
?d2σ(res)
dM2dq2
T
(N,b;αs) −d2σ(exp)
dM2dq2
T
(N,b;αs)
?
,(32)
where d2σ(F.O.)is the fixedorder perturbative result, d2σ(res)is the resummed cross section discussed above, and
d2σ(exp)is the truncation of the resummed cross section to the same perturbative order as d2σ(F.O.). Here, we have
removed the scale dependences for brevity.
At NLO, the doubledifferential partonic cross section
dˆ σ(F.O.)
ab
dM2dq2
T
(z;αs,µR) = δ(q2
T)δ(1 − z) ˆ σ(0)
ab+αs(µR)
π
ˆ σ(1)
ab(z) + O(α2
s)(33)
receives contributions from the emission of an extra gluon jet and from processes with an initial gluon splitting into
a q¯ q pair,
TR
2sAqg(s,t,u)σ(0)
ˆ σ(1)
g¯ q(z) =
2sAqg(s,u,t)σ(0)
ˆ σ(1)
q¯ q(′)(z) =
ˆ σ(1)
qg(z) =
q¯ q(′),(34)
TR
q¯ q(′),(35)
CF
2sAqq(s,t,u)σ(0)
q¯ q(′)(M) (36)
with [30]
Aqg(s,t,u) = −
Aqq(s,t,u) = −Aqg(u,t,s).
?s
t+t
s+2uM2
st
?
,(37)
(38)
The Mandelstam variables s, t, and u refer to the 2 → 2 scattering process ab → γ, Z0, W±+ X and are related to
the invariant mass M (or scaled squared invariant mass z = M2/s), transverse momentum qT, and rapidity y of the
slepton pair by the wellknown relations
s = xaxbS = M2/z,
?
u = M2−
Integration over qT requires the cancellation of soft and collinear singularities with virtual contributions in order to
arrive at the finite singledifferential partonic cross section
(39)
t = M2−
S(M2+ q2
T)xbey,(40)
?
S(M2+ q2
T)xae−y.(41)
dˆ σ(F.O.)
ab
dM2
(z;αs,µR,µF) = ˆ σ(0)
abδ(1 − z) +αs(µR)
π
ˆ σ(1)
ab(z;µR,µF) + O(α2
s),(42)
where the first term ˆ σ(0)
corrections can be found in Ref. [18].
The expansion of the resummed result reads
abis defined in Eqs. (7) and (8) and the second term including the full NLO SUSYQCD
d2σ(exp)
dM2dq2
T
(N,b;αs,µR,µF) =
?
a,b
fa/ha(N + 1;µF)fb/hb(N + 1;µF) ˆ σ(exp)
ab
(N,b;αs,µR,µF),(43)
where ˆ σ(exp)
ab
is obtained by perturbatively expanding the resummed component
ˆ σ(exp)
ab
(N,b;αs,µR,µF) =
?
c
ˆ σ(0)
c¯ c
?
δcaδ¯ cb+
∞
?
???
n=1
?αs(µR)
π
?n?
˜Σ(n)
ab→c¯ c(N,lnχ;µR,µF)
+ H(n)
ab→c¯ c
?
N;µR,µF
.(44)
Page 7
7
The perturbative coefficients˜Σ(n)are polynomials of degree 2n in lnχ, and H(n)embodies the constant part of the
resummed cross section in the limits b → ∞ and N → ∞. In particular, the firstorder coefficient˜Σ(1)is given by
˜Σ(1)
ab→c¯ c(N,lnχ) =˜Σ(1;2)
ab→c¯ cln2χ +˜Σ(1;1)
ab→c¯ c(N)lnχ,(45)
with
˜Σ(1;2)
ab→c¯ c= −2A(1)
cδcaδ¯ cb
and
˜Σ(1;1)
ab→c¯ c(N) = −2?B(1)
c δcaδ¯ cb+ δcaγ(1)
¯ c/b(N) + δ¯ cbγ(1)
c/a(N)
?
.(46)
IV.NUMERICAL RESULTS
We now present numerical results for the production of a righthanded selectron pair at the LHC for a centre
ofmass energy of
the top quark, we use the values mZ = 91.1876 GeV, mW = 80.403 GeV, ΓZ = 2.4952 GeV, ΓW = 2.141 GeV,
and mt = 174.2 GeV [5]. The electromagnetic fine structure constant α =
the improved Born approximation using the world average value GF = 1.16637 · 10−5GeV−2for Fermi’s coupling
constant, and sin2θW = 1 − m2
We choose the mSUGRA benchmark point BFHK B [31], which gives after the renormalization group evolution of
the SUSYbreaking parameters [32] a light ˜ eRof mass m˜ eR= 186 GeV and rather heavy squarks with masses around
800850 GeV. The topsquark mass eigenstate˜t1is slightly lighter, but does not contribute to the virtual squark loops
due to the negligible topquark density in the proton. For the LO (NLO and NLL) predictions, we use the LO 2001
[33] (NLO 2004 [34]) MRST sets of parton distribution functions. For the NLO and NLL predictions, αsis evaluated
with the corresponding value of Λnf=5
MS
= 255 MeV at twoloop accuracy. We allow the unphysical scales µF and µR
to vary between M/2 and 2M to estimate the perturbative uncertainty.
In Fig. 1, we present the transversemomentum spectrum of the selectron pair, obtained after integrating the
equations of Sec. II and Sec. III over M2, from the ˜ eR˜ e∗
energy. We plot the fixed order result at order αs (dashed line), the expansion of the resummed formula at the
same perturbative order (dotted line), the total NLL+LO matched result (solid line), and the uncertainty bands
from the scale variation. The fixed order result diverges as expected as qT tends to zero. The asymptotic expansion
of the resummation formula is in good agreement with the O(αs) result in this kinematical region, since the cross
section is dominated by the large logarithms that we are resumming. For intermediate values of qT, we can see that
the agreement between the expansion and the perturbative result is slightly worse. This effect was not present in
qTresummation for such qTvalues [13] and is thus related to the thresholdenhanced contributions important in the
largeM region. This can also be seen in Fig. 2, where we directly compare the jointly and qTmatched results, the
latter having been obtained with the qTresummation formalism of Ref. [25]. The two approaches lead to a similar
behaviour in the smallqTregion, but the jointlyresummed cross section is about 5%10% lower than the qTresummed
cross section for transverse momenta in the range 50 GeV < qT < 100 GeV. However, the effect of the resummation
is clearly visible in both cases, the resummationimproved result being even 40% higher than the fixedorder result
at qT = 80 GeV. In Fig. 1, we also estimate the theoretical uncertainties through an independent variation of the
factorization and renormalization scales between M/2 and 2M and show that the use of resummation leads to a clear
improvement with respect to the fixedorder calculation. In the small and intermediate qTregions the scale variation
amounts to 10% for the fixedorder result, while it is always less than 5% for the matched result.
The qTdistribution is affected by nonperturbative effects in the small qTregion coming, for instance, from partons
with a nonzero intrinsic transversemomentum inside the hadron and from unresolved gluons with very small trans
verse momentum. Global fits of experimental DrellYan data allow for different parameterizations of these effects,
which can be consistently included in the resummation formula of Eq. (23) through a nonperturbative form factor
FNP
ab. We include in our analysis three different parameterizations of this factor [35, 36, 37],
√S = 14 TeV. For the masses and widths of the electroweak gauge bosons and the mass of
√2GFm2
Wsin2θW/π is calculated in
W/m2
Z.
Rproduction threshold up to the hadronic centreofmass
FNP(LY −G)
ab
(b,M,x1,x2) = exp
?
?
?
−b2
?
?
?
¯ g1+¯ g2lnbmaxM
2
?
−b¯ g1¯ g3ln(100x1x2)
?
, (47)
FNP(BLNY )
ab
(b,M,x1,x2) = exp−b2
˜ g1+ ˜ g2lnbmaxM
2
M
+ ˜ g1˜ g3ln(100x1x2)
??
??
, (48)
FNP(KN)
ab
(b,M,x1,x2) = exp−b2
a1+ a2ln
3.2GeV+ a3ln(100x1x2).(49)
The most recent values for the free parameters in these functions can be found in Refs. [36, 37]. We show in the
Page 8
8
[GeV] [GeV]
TT
qq
00 5050 100100150 150200200
[ fb / GeV ]
/ dq
σ
d
T
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Matched
Fixed Order
Expansion
* at the LHC
R
e~
R
e~
→
0
, Z
γ
→
p p
2 M
≤
R
µ
,
F
µ ≤
M/2
[GeV] [GeV]
TT
qq
00101020 2030 30 4040
[%]
∆
30
25
20
15
10
5
0
5
LYG
BLNY
KN
FIG. 1: Transversemomentum distribution for the process pp → ˜ eR˜ e∗
(dotted) and asymptotically expanded results are shown, together with three different parameterizations of nonperturbative
effects (insert).
Rat the LHC. NLL+LO matched (full), fixed order
[GeV] [GeV]
TT
qq
00 50 50100100150 150200 200
[ fb / GeV ]
/ dq
σ
d
T
0
0.05
0.1
0.15
0.2
0.25
0.3
Jointly matched
matched
T
q
* at the LHC
R
e~
R
e~
→
0
, Z
γ
→
p p
= M
R
µ
=
F
µ
FIG. 2: Transversemomentum distribution of selectron pairs at the LHC in the framework of joint (full) and qT (dotted)
Page 9
9
upperright part of Fig. 1 the quantity
∆ =dσ(res.+NP)(µR= µF = M) − dσ(res.)(µR= µF = M)
dσ(res.)(µR= µF= M)
, (50)
which gives thus an estimate of the contributions from the different NP parameterizations (LYG, BLNY and KN).
They are under good control, since they are always less than 5% for qT> 5 GeV and thus considerably smaller than
the resummation effects.
The invariantmass distribution M3dσ/dM for ˜ eRpair production at the LHC is obtained after integrating the
equations of Sec. II and Sec. III over q2
Tand is shown in Fig. 3. The differential cross section dσ/dM has been
multiplied by a factor M3in order to remove the leading mass dependence of propagator and phase space factors. We
can see the Pwave behaviour relative to the pair production of scalar particles, since the invariantmass distribution
rises above the threshold at√s = 2m˜ eRwith the third power of the slepton velocity and peaks at about 200 GeV
above threshold (both for M3dσ/dM and the not shown dσ/dM differential distribution), before falling off steeply
due to the schannel propagator and the decreasing parton luminosity. In the largeM region, the resummed cross
section is 30% higher than the leading order cross section, but this represents only a 3% increase with respect to
the NLO SUSYQCD result. In the smallM region, much further then from the hadronic threshold, resummation
effects are rather limited, inducing a modification of the NLO results smaller than 1%. The shaded, horizontally,
and vertically hashed bands in Fig. 3 represent the theoretical uncertainties for the LO, NLO SUSYQCD, and the
jointlymatched predictions. At LO the dependence comes only from the factorization scale and increases with the
momentumfraction x of the partons in the proton (i.e. with M), being thus larger in the right part of the figure. This
dependence is largely reduced at NLO due to the factorization of initialstate singularities in the PDFs. Including
the dependence due to the renormalization scale in the coupling αs(µR), the total variation is about 7%11%. After
resummation, the total scale uncertainty is finally reduced to only 7%8% for the matched result, the reduction being
of course more important in the largeM region, where the resummation effects are more important.
In Fig. 4, we show the cross section correction factors
Ki=
dσi/dM
dσLO/dM
(51)
as a function of the invariantmass M. i labels the corrections induced by NLO QCD, NLO SUSYQCD, joint and
thresholdresummation (as obtained in [18]), these two last calculations being matched with the NLO SUSYQCD
result. At small invariant mass M, the resummation is less important, since we are quite far from the hadronic
threshold, as shown in the left part of the plot. At larger M, the logarithms become important and lead to a larger
increase of the resummed Kfactors over the fixedorder one. We also show the difference between threshold and joint
resummations, which is only about one or two percents. This small difference is due to the choice of the Sudakov
form factor G and of the Hfunction, which correctly reproduce transversemomentum resummation in the limit of
b → ∞, N being fixed, but which present some differences in the pure threshold limit b → 0 and N → ∞, as it was
the case for joint resummation for Higgs and electroweak boson production [22, 23]. However, this effect is under
good control, since it is much smaller than the theoretical scale uncertainty of about 7%.
V. CONCLUSIONS
With this work we complete our programme of performing precision calculations for slepton pair production at
hadron colliders. Together with the previous papers on transversemomentum [13] and threshold [18] resummation,
softgluon resummation effects are now consistently included in predictions for various distributions exploiting the qT,
threshold, and joint resummation formalisms. We found that the effects obtained from resumming the enhanced soft
contributions are important at hadron colliders, even far from the critical kinematical regions where the resummation
procedure is fully justified. The numerical results show a considerable reduction of the scale uncertainty with respect
to fixed order results and also a negligible dependence on nonperturbative effects, introduced through different
Gaussianlike smearings of the Sudakov form factors. These features lead to an increased stability of the perturbative
results and thus to a possible improvement of the slepton pair (sleptonsneutrino) search strategies at the LHC.
Acknowledgments
This work was supported by a Ph.D. fellowship of the French ministry for education and research.
Page 10
10
[GeV] [GeV]
RR
e~ e~
RR
e~
e~
MM
500 500100010001500 1500 200020002500250030003000
]
2
/dM [pb GeV
σ
d
3
M
100
1000
10000
Jointly matched
NLO SUSYQCD
LO
* at the LHC
R
e~
R
e~
→
0
, Z
γ
→
p p
FIG. 3: Invariantmass distribution M3dσ/dM of ˜ eR pairs at the LHC. We show the total NLL+NLO jointly matched (full),
as well as the fixedorder NLO SUSYQCD (dashed) and LO QCD (dotted) results, with the corresponding scale uncertainties
(vertically hashed, horizontally hashed, and shaded bands).
[GeV] [GeV]
RR
e~ e~
RR
e~
e~
MM
5005001000 100015001500 2000 20002500 250030003000
K
1.24
1.26
1.28
1.3
1.32
1.34
1.36
Jointly matched
Threshold matched
NLO SUSYQCD
NLO QCD
* at the LHC
R
e~
R
e~
→
0
, Z
γ
→
p p
FIG. 4: Kfactors as defined in Eq. (51) for ˜ eR pair production at the LHC. We show the total NLL+NLO jointly (full), and
Page 11
11
APPENDIX A: SFERMION MIXING
The soft SUSYbreaking terms Af of the trilinear Higgssfermionsfermion interaction and the offdiagonal Higgs
mass parameter µ in the MSSM Lagrangian induce mixings of the left and righthanded sfermion eigenstates˜fL,Rof
the electroweak interaction into mass eigenstates˜f1,2. The sfermion mass matrix is given by [2]
M2=
?m2
LL+ m2
mfmLR
f
mfm∗
m2
LR
RR+ m2
f
?
(A1)
with
m2
m2
LL= m2
RR= m2
˜ F+ (T3
˜ F′+ ef sin2θWm2
?cotβ
f− ef sin2θW)m2
Zcos2β,(A2)
Zcos2β,
for up − type sfermions.
tanβ for down − type sfermions.
(A3)
mLR = Af− µ∗
(A4)
It is diagonalized by a unitary matrix S˜ f, S˜ fM2S˜ f†= diag(m2
1,m2
2), and has the squared mass eigenvalues
m2
1,2= m2
f+1
2
?
m2
LL+ m2
RR∓
?
(m2
LL− m2
RR)2+ 4m2
fmLR2?
.(A5)
For real values of mLR, the sfermion mixing angle θ˜f, 0 ≤ θ˜f≤ π/2, in
S
˜f=
?
cosθ˜fsinθ˜f
−sinθ˜fcosθ˜f
?
with
?˜f1
˜f2
?
= S
˜f
?˜fL
˜fR
?
(A6)
can be obtained from
tan2θ˜f=
2mfmLR
m2
LL− m2
RR
.(A7)
If mLRis complex, one may first choose a suitable phase rotation˜f′
diagonalize it for˜fL and˜f′
fields, which couple to the uptype and downtype (s)fermions. The soft SUSYbreaking mass terms for left and
righthanded sfermions are m˜ Fand m˜ F′ respectively.
R= eiφ˜fRto make the mass matrix real and then
R. tanβ = vu/vd is the (real) ratio of the vacuum expectation values of the two Higgs
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