# Joint resummation for slepton pair production at hadron colliders

**ABSTRACT** We present a precision calculation of the transverse-momentum and invariant-mass 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 next-to-leading logarithmic accuracy with a process-independent Sudakov form factor, thus ensuring a universal description of soft-gluon 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 non-perturbative 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 non-perturbative contributions remain small.

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**ABSTRACT:**We present a first precision analysis of the transverse-momentum spectrum of gaugino pairs produced at the Tevatron and the LHC with center-of-mass energies of 1.96 and 10 or 14 TeV, respectively. Our calculation is based on a universal resummation formalism at next-to-leading logarithmic accuracy, which is consistently matched to the perturbative prediction at O(αs). Numerical results are given for the “gold-plated” associated production of neutralinos and charginos decaying into three charged leptons with missing transverse energy as well as for the pair production of neutralinos and charginos at two typical benchmark points in the constrained MSSM. We show that the matched resummation results differ considerably from the Monte Carlo predictions employed traditionally in experimental analyses and discuss the impact on the determination of SUSY mass parameters from derived transverse-mass spectra. We also investigate in detail the theoretical uncertainties coming from scale and parton-density function variations and non-perturbative effects.Physics Letters B 07/2013; · 4.57 Impact Factor - SourceAvailable from: link.springer.com[show abstract] [hide abstract]

**ABSTRACT:**Motivated by hints for a light Standard Model-like Higgs boson and a shift in experimental attention towards electroweak supersymmetry particle production at the CERN LHC, we update in this paper our precision predictions at next-to-leading order of perturbative QCD matched to resummation at the next-to-leading logarithmic accuracy for direct gaugino pair production in proton-proton collisions with a center-of-mass energy of 8 TeV. Tables of total cross sections are presented together with the corresponding scale and parton density uncertainties for benchmark points adopted recently by the experimental collaborations, and figures are presented for up-to-date model lines attached to them. Since the experimental analyses are currently obtained with parton showers matched to multi-parton matrix elements, we also analyze the precision of this procedure by comparing invariant-mass and transverse-momentum distributions obtained in this way to those obtained with threshold and transverse-momentum resummation.Journal of High Energy Physics 07/2012; 2012(10). · 5.62 Impact Factor - SourceAvailable from: export.arxiv.org[show abstract] [hide abstract]

**ABSTRACT:**We present an implementation for slepton pair production at hadron colliders in the POWHEG BOX, a framework for combining next-to-leading order QCD calculations with parton-shower Monte-Carlo programs. Our code provides a SUSY Les Houches Accord interface for setting the supersymmetric input parameters. Decays of the sleptons and parton-shower effects are simulated with PYTHIA. Focussing on a representative point in the supersymmetric parameter space we show results for kinematic distributions that can be observed experimentally. While next-to-leading order QCD corrections are sizable for all distributions, the parton shower affects the color-neutral particles only marginally. Pronounced parton-shower effects are found for jet distributions.Journal of High Energy Physics 08/2012; 2012(10). · 5.62 Impact Factor

Page 1

arXiv:0709.3057v1 [hep-ph] 19 Sep 2007

Joint resummation for slepton pair production at hadron colliders

Giuseppe Bozzi

Insitut f¨ ur Theoretische Physik, Universit¨ at Karlsruhe, Postfach 6980, D-76128 Karlsruhe, Germany

Benjamin Fuks and Michael Klasen∗

Laboratoire de Physique Subatomique et de Cosmologie,

Universit´ e Joseph Fourier/CNRS-IN2P3, 53 Avenue des Martyrs, F-38026 Grenoble, France

(Dated: February 5, 2008)

We present a precision calculation of the transverse-momentum and invariant-mass 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 imple-

ment the joint resummation formalism at the next-to-leading logarithmic accuracy with a process-

independent Sudakov form factor, thus ensuring a universal description of soft-gluon 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 non-perturbative 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 non-perturbative contributions remain small.

Rproduction and compare the

PACS numbers: 12.60.Jv,13.85.Ni,14.80.Ly

I.INTRODUCTION

KA-TP-15-2007

LPSC 07-72

SFB-CPP-07-26

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 SUSY-breaking 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 non-mixing slepton pairs has been calculated in [6, 7, 8, 9],

while the mixing between the interaction eigenstates was included in [10]. The next-to-leading order (NLO) QCD

corrections have been calculated in [11], and the full SUSY-QCD corrections with non-mixing squarks in the loops

have been added in [12]. Recently, an accurate calculation of the transverse-momentum (qT) spectrum including

soft-gluon resummation at the next-to-leading 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 threshold-enhanced 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 transverse-momentum and threshold

resummations, i.e. the soft-gluon emission by the initial state, it would be desirable to have a formalism capable to

handle at the same time the soft-gluon contributions in both the delicate kinematical regions, qT ≪ M and M2∼ s,

M being the slepton pair invariant-mass and s the partonic centre-of-mass 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

impact-parameter (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 heavy-quark 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 soft-gluon 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 transverse-momentum resummation in [25]. The inverse

transforms from the Mellin and impact-parameter 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 (transverse-momentum, threshold, and joint), showing

their impact on the qT-spectrum and on the invariant-mass distribution. Our results are summarized in Sec. V.

II.JOINT RESUMMATION AT THE NEXT-TO-LEADING 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 Drell-Yan (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 centre-of-mass 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 N-space, 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 N-space will be specified in Sec. III and the N-moments 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 N-space 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 lowest-order cross section ˆ σ(0)

the final state F is assumed to be colourless.

For slepton pair and slepton-sneutrino 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 right-handed (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 CKM-matrix 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 left-handed and T3

f= 0 for right-handed (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 fixed-order 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 N-moments of the Altarelli-Parisi 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 next-to-leading 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 fixed-order 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 A-term, while the B-term accounts for the difference between the eikonal approx-

imation and the full partonic cross section in the threshold region, i.e. the flavour-conserving collinear contributions.

In the large-N limit, these coefficients are directly connected to the leading terms in the one-loop 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 β-function 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 transverse-momentum 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 χ-dependence appearing in the C-coefficient 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 first-order 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 Drell-Yan resummation scheme and take Hq¯ q(αs,µR) ≡ 1. The C-coefficients are

then given by

a/b(N) and H(1)

c¯ c(µR) are in principle resummation-scheme 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 b-space, 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 b-space

[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 b-contour,

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 N-moments 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 fixed-order calculation and to avoid possible double-counting 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 fixed-order 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 double-differential 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 well-known 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 single-differential 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 SUSY-QCD

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 first-order 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 right-handed selectron pair at the LHC for a centre-

of-mass 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 SUSY-breaking parameters [32] a light ˜ eRof mass m˜ eR= 186 GeV and rather heavy squarks with masses around

800-850 GeV. The top-squark mass eigenstate˜t1is slightly lighter, but does not contribute to the virtual squark loops

due to the negligible top-quark 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 two-loop 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 transverse-momentum 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

qT-resummation for such qT-values [13] and is thus related to the threshold-enhanced contributions important in the

large-M region. This can also be seen in Fig. 2, where we directly compare the jointly- and qT-matched results, the

latter having been obtained with the qT-resummation formalism of Ref. [25]. The two approaches lead to a similar

behaviour in the small-qTregion, but the jointly-resummed cross section is about 5%-10% lower than the qT-resummed

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 resummation-improved result being even 40% higher than the fixed-order 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 fixed-order calculation. In the small and intermediate qT-regions the scale variation

amounts to 10% for the fixed-order result, while it is always less than 5% for the matched result.

The qT-distribution is affected by non-perturbative effects in the small qT-region coming, for instance, from partons

with a non-zero intrinsic transverse-momentum inside the hadron and from unresolved gluons with very small trans-

verse momentum. Global fits of experimental Drell-Yan data allow for different parameterizations of these effects,

which can be consistently included in the resummation formula of Eq. (23) through a non-perturbative 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 centre-of-mass

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

LY-G

BLNY

KN

FIG. 1: Transverse-momentum distribution for the process pp → ˜ eR˜ e∗

(dotted) and asymptotically expanded results are shown, together with three different parameterizations of non-perturbative

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: Transverse-momentum distribution of selectron pairs at the LHC in the framework of joint (full) and qT (dotted)

Page 9

9

upper-right 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 (LY-G, 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 invariant-mass distribution M3dσ/dM for ˜ eR-pair 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 P-wave behaviour relative to the pair production of scalar particles, since the invariant-mass 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 s-channel propagator and the decreasing parton luminosity. In the large-M 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 SUSY-QCD result. In the small-M 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 SUSY-QCD, and the

jointly-matched predictions. At LO the dependence comes only from the factorization scale and increases with the

momentum-fraction 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 initial-state 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 large-M 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 invariant-mass M. i labels the corrections induced by NLO QCD, NLO SUSY-QCD, joint- and

threshold-resummation (as obtained in [18]), these two last calculations being matched with the NLO SUSY-QCD

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 K-factors over the fixed-order 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 H-function, which correctly reproduce transverse-momentum 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 transverse-momentum [13] and threshold [18] resummation,

soft-gluon 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 non-perturbative effects, introduced through different

Gaussian-like 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 (slepton-sneutrino) 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 SUSY-QCD

LO

* at the LHC

R

e~

R

e~

→

0

, Z

γ

→

p p

FIG. 3: Invariant-mass distribution M3dσ/dM of ˜ eR pairs at the LHC. We show the total NLL+NLO jointly matched (full),

as well as the fixed-order NLO SUSY-QCD (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 SUSY-QCD

NLO QCD

* at the LHC

R

e~

R

e~

→

0

, Z

γ

→

p p

FIG. 4: K-factors 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 SUSY-breaking terms Af of the trilinear Higgs-sfermion-sfermion interaction and the off-diagonal Higgs

mass parameter µ in the MSSM Lagrangian induce mixings of the left- and right-handed 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

f|mLR|2?

.(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 up-type and down-type (s)fermions. The soft SUSY-breaking mass terms for left- and

right-handed 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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