Table Of ContentO(a 2) QCD corrections to the resonant
s
sneutrino / slepton production at LHC
2
1
0
2
n
a
J SwapanK Majhi∗
5
IndianAssociationfortheCultivationofScience,
] 2A&2BRajaSCMullickRoad,Kolkata700032,India
h
E-mail: [email protected]
p
-
p PrakashMathews
e SahaInstituteofNuclearPhysics,
h
1/AFBidhannagar,Kolkata700064,India
[
E-mail: [email protected]
1
v V Ravindran
6
RegionalCentreforAcceleratorbasedParticlePhysics,
0
1 Harish-ChandraResearchInstitute,ChhatnagRoad,Jhusi,
1 Allahabad211019,India.
.
1 E-mail: [email protected]
0
2
1 WepresentacompletenexttonexttoleadingorderQCDcorrectionstotheresonantproductionof
:
v sneutrinoandchargedsleptonattheTevatronandtheLargeHadronColliderwithinthecontext
Xi of R-parity violating supersymmetric model. We have demonstrated the role of NNLO QCD
r correctionsinreducinguncertaintiesresultingfromrenormalisationandfactorisationscalesand
a
therebymakingourpredictionsreliable.
10thInternationalSymposiumonRadiativeCorrections(ApplicationsofQuantumFieldTheoryto
Phenomenology)-Radcor2011
September26-30,2011
Mamallapuram,India
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
NNLOQCDcorrections SwapanKMajhi
1. Introduction
In supersymmetry[1], R-parity violation is one of the possible scenario in beyond standard
model physics. With many interest, weconsider only on the l ′ couplings which arises in lepton
ijk
numberviolating terminR-parityviolation superpotential givenbelow
W =m LH +l LL Ec+l ′ LQ Dc+l ′′ UcDcDc , (1.1)
i i 2 ijk i j k ijk i j k ijk i j k
where L and Q are the SU(2)-doublet lepton and quark superfields, Ec,Uc,Dc the singlet super-
i i i i i
fields and H the Higgs superfields. The subscripts i,j,k are generational indices. Note that l
i ijk
isantisymmetric under theinterchange ofthefirsttwoindices and l ′′ isantisymmetric under the
ijk
interchange ofthelasttwo. Thefirstthreetermsineqn.(1.1)violateleptonnumber(L)andthelast
termviolatesbaryon number(B)conservation.
Recently ATLAS group have been studied resonant production of heavy neutral scalar like
sneutrinoandsubsequentdecaytoem finalstate. Intheiranalysis,theyputtheboundsonsneutrino
masses(seeRef.[2])onthebasisofleadingorder(LO)result. Intevatron,bothCDF[3]andD0[4]
collaboration analyse theirdata(Run-IaswellasRun-IIdata)usingourfirstresults[5]onthenext
to leading order (NLO) QCD corrections to sneutrino and charged slepton productions at hadron
colliders. In their analysis to set bound on these R-parity violating couplings, cross section for
SMbackgroundprocessesnamelyDrell-Yanproductionofpairofleptons(sayl+l−,l±n )(seefirst
twopapersof[16])wasconsidered atthenexttonexttoleadingorder(NNLO)levelwhileforthe
R-parity violating effects only NLO corrected cross section was used. It was found that the NLO
QCD effects were quite large ∼10%−40% at both Tevatron as well as LHC Therefore, it is de-
sirabletocomputethecrosssectionsfortheresonantsneutrinoand/orchargedsleptonproductions
at NNLO in QCD. These results will quantitatively improve the analysis based on high statistics
data available in the ongoing and future experiments. From the theoretical point of view, higher
order radiative corrections provide a test of the convergence of the perturbation theory and hence
thereliablecomparisonofdatawiththetheorypredictionsispossible. Thefixedorderperturbative
resultsmostoftensufferfromlargeuncertaintiesduetothepresenceofrenormalisationandfactori-
sation scales. Theyget reduced as weinclude more and more terms in the perturbative expansion
thankstorenormalisationgroupinvariance. Inthisarticlewehavesystematicallyincludeditsscale
dependence through the renormalisation group equations and we discussed the impact of it in the
nextsections.
2. Brief discussionofNNLOcalculations
Inthissection, wedescribe verybriefly, thecomputation ofsecond order (a 2)QCDradiative
s
corrections toresonant production, inhadroncolliders, ofasneutrino/charged slepton. Wepresent
ourresultsinsuchawaythattheycanbeusedforanyscalar-pseudoscalar production whichisthe
maingoalofthiswork. Theinclusivehadronic crosssection forthereaction
H1(P1)+H2(P2)→f (pf )+X, (2.1)
isgivenby
s tfot = pl 1′22(Sm R2)a,b=(cid:229) q,q¯,gZt1dxx11 Zt/1x1 dxx22 fa(x1,m F2)fb(x2,m F2)D ab(cid:18)x1tx2,mf2,m F2,m R2(cid:19)
2
NNLOQCDcorrections SwapanKMajhi
m2
f
with t = , S=(P1+P2)2 , pf2 =mf2, (2.2)
S
where H and H denote the incoming hadrons and X represents an inclusive hadronic state. The
1 2
parton densities denoted by f (x,m 2) (c = q,q¯,g) depend on the scaling variables x (i = 1,2)
c i F i
through p =xP andthemassfactorizationscalem . Here p (i=1,2)arethemomentaofincom-
i i i F i
ing partons namely quarks, antiquarks and gluons. The coupling constant l ′ gets renormalised at
therenormalisation scale m duetoultraviolet singularities presentinthetheory. Thefactorisation
R
scale is introduced on the right hand side of the above equation to separate long distant dynamics
from theperturbatively calculable short distant partonic coefficient functions D . D depends on
ab ab
both m and m in such a way that the entire scale dependence goes away to all orders in pertur-
R F
bation theory when convoluted with appropriate parton densities. This is due to the fact that the
observableonthelefthandsideoftheaboveequationisrenormalisationgroup(RG)invariantwith
respect toboththescales. Thisimplies
ds f
m 2 tot =0, m =m ,m , (2.3)
dm 2 F R
d
m R2dm 2 l ′2(m R2)D ab x,m2f ,m F2,m R2 =0. (2.4)
Rh (cid:0) (cid:1)i
The partonic coefficient functions that appear in eqn.(2.2) are computable in perturbative QCD
in terms of strong coupling constant g . The ultraviolet singularities present in the theory are
s
regularised in dimensional regularisation and are removed in MS scheme, introducing the renor-
malisation scale m at every order in perturbative expansion. In addition, the Yukawacoupling l ′
R
alsogetsrenormalised duetostronginteraction dynamics. Hence,forourcomputation, werequire
only two renormalisation constants to obtain UV finite partonic coefficient functions, D . These
ab
constants are denoted by Z(m R) and Zl ′(m R), where the former renormalises the strong coupling
constantg andthelaterYukawacouplingl ′andboththecouplingsa (=g /(4p ))(andl ′)evolve
s s s
withscaletoNNLOthroughrenormalisation groupequations:
¥
m R2ddm 2lnas(m R2) = −(cid:229) ais(m R2)b i−1,
R i=1
¥
m R2ddm 2lnl ′(m R2) = −(cid:229) ais(m R2)gi−1. (2.5)
R i=1
wherecoefficientsb fori=0,...,3canbefoundin[6]forSU(N)QCD.Theanomalousdimensions
i
g for i=0,...,3 can be obtained from the quark mass anomalous dimensions given in [7]. The
i
perturbatively calculable D canbeexpanded inpowersofstrongcoupling constanta (m 2)as
ab s R
¥
D ab x,mf2,m F2,m R2 = (cid:229) ais(m R2)D (aib) x,mf2,m F2,m R2 .
(cid:0) (cid:1) i=0 (cid:0) (cid:1)
D getscontributions fromvariouspartonicreactions.
ab
Thecalculation ofvarious contributions fromthepartonic reactions involves careful handling
of divergences that result from one[8] and two loop[9] integrations in the virtual processes and
3
NNLOQCDcorrections SwapanKMajhi
two and three body phase space integrations in the real emission processes. The loop integrals
often give ultraviolet, soft and collinear divergences. But the phase space integrals give only soft
and collinear singularities. Soft divergences arise when the momenta of the gluons become zero
while the collinear diverges arise due to the presence of massless partons. We have regulated all
the integrals indimensional regularisation withspace timedimension n=4+e . Thesingularities
manifestthemselves aspolesine .
We have reduced all the one loop tensorial integrals to scalar integrals using the method of
Passarino-Veltman [10] in 4+e dimensions and evaluated resultant scalar integrals exactly. The
2-loop form factor, Ff (m2f ,m 2), is calculated using the dispersion technique [11]. Twoand three
body phase space integrals are done by choosing appropriate Lorentz frames[12]. Since we inte-
grateoverthetotalphasespacetheintegralsareLorentzinvariantandthereforeframeindependent.
SeveralroutinesaremadeusingthealgebraicmanipulationprogramFORM[13]inordertoperform
tensorial reduction ofoneloopintegrals andtwoandthreebodyphasespaceintegrals.
The UV singularities go away after performing renormalisation through the constants Z and
Zl ′. The soft singularities cancel among virtual and real emission processes[14] at every order in
perturbation theory. The remaining collinear singularities are renormalised systematically using
mass factorisation[15]. For more details on the computation of NNLO QCD corrections to pro-
cess of the kind considered here can be found in [16, 17]. The full analytical results for NNLO
calculation forsneutrino and/orchargeslepton canbefoundoutinouroriginal paper[17].
3. Results andDiscussion
We considered only the contributions from the first generation of quarks. Since at hadron
colliders, the resonant production is through the interaction term l ′ LQ Dc in the Lagrangian
ijk i j k
(see eq.(1.1)), for j,k = 2,3, the production rate will be suppressed due to the low flux of the
sea quarks. To obtain the production cross section to a particular order, one has to convolute the
partoniccoefficientfunctionsD withthecorrespondingpartondensities f ,bothtothesameorder.
ab a
Furtherthecouplingconstantsa (m )andl ′(m )shouldalsobeevaluatedusingthecorresponding
s R R
RGEs(eqn.(2.5))computedtothesameorder(moredetailsseeRef[7,18]). Wehaveusedthelatest
MSTWpartondensities[19]inournumericalcodeandthecorrespondingvaluesofa (M )forLO,
s Z
NLO and NNLOprovided with the sets. Since we are considering one l ′ non-zero, the LO and
i11
NLOcrosssectionsgetcontributions onlyfromdd ,dganddginitiatedsubprocesses andnoother
quark(antiquark)flavorscontributetothisorder. AtNNLOlevel,theincomingquarksotherthand
typequarkscanalsocontribute. Thetotalsneutrinoproductioncrosssectionasfunctionofitsmass
is plotted in fig.1for LHC(left panel) and RunIIof Tevatron (right panel) energies. Wehave set
therenormalisationscaletobethemassofthesneutrino, m R=mn˜. Thepairoflinescorrespondsto
thetwoextremechoices offactorisation scale: m F =10mn˜ (upper) and m F =mn˜/10(lower). The
plots clearly demonstrate that the NNLO contributions reduce the factorisation scale dependence
improving thetheoretical predictions forsneutrino production crosssection.
Thecross section falls off with the sneutrino mass due to the availability of phase space with
respect tothe mass, thechoice of m R =mn˜ and the parton densities. Thelatter effect, understand-
ably, ismorepronounced attheTevatronthanattheLHC.
4
NNLOQCDcorrections SwapanKMajhi
10
LO LO
1
NLO NLO
NNLO NNLO
m R = m˜n m R = m˜n
)
b
P 1
( 0.1
s
LHC (14 TeV)
Tevatron Run-II
0.1 0.01
0.1 0.14 0.18 0.22 0.26 0.3 0.1 0.14 0.18 0.22 0.26 0.3
m˜n (TeV) m˜n (TeV)
LO LO
0.01
NLO NLO
NNLO NNLO
0.1
m R = m˜n m R = m˜n
0.001
)
b
P
(
s
0.01 0.0001
Tevatron Run-II
LHC (14 TeV)
1e-05
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
m˜n (TeV) m˜n (TeV)
Figure1:Totalcross-sectionforthen˜ productionasafunctionofmn˜. Forsmallervaluesofsneutrinomass
theupper(lower)setoflinescorrespondtothefactorisationscalem F =10mn˜(0.1mn˜). Forlargervaluesof
sneutrinomassthelinescrosseachother.
Inorder toestimate the magnitude oftheQCDcorrections atNLOandNNLO,wedefinethe
K-factors asfollows:
K(1)=s f /s f , K(2)=s f /s f .
tot,NLO tot,LO tot,NNLO tot,LO
In fig.2, we have plotted both K(i) (i = 1,2) as a function of sneutrino mass. We have chosen
m F =m R =mn˜ forthisstudy. AttheLHC,TheK(1) variesbetween1.23to1.46andK(2) between
1.27 to 1.52 in the mass range 100 GeV ≤mn ≤1 TeV. At the Tevatron, we find that K(1) varies
between 1.55 to 1.53 and K(2) between 1.65 to 1.85 for the same mass range. Note that numbers
for K(1) differ from those given in our earlier work [5] due to the running of l ′ in the present
analysis. Thepresent analysis using running l ′ isthe correct way toreduce renormalisation scale
dependence in the cross section. We also observe that K factor is much bigger at the Tevatron
5
NNLOQCDcorrections SwapanKMajhi
1.6 2
K(1) K(1)
1.55 K(2) K(2)
1.5 1.9
1.45
1.8
1.4
K
1.35
1.7
1.3
1.25 1.6
1.2 LHC (14 TeV) Tevatron Run-II
1.15 1.5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
m˜n (TeV) m˜n (TeV)
Figure2: NLOK-factorK(1)andNNLOK-factorK(2)areplottedforsneutrinoproductionattheLHC(left
panel)andtheTevatronRun-II(rightpanel)asafunctionofitsmass.
compared to that of at the LHC. The reason behind this is attributed to the different behavior of
parton densities at the Tevatron and the LHC.Note that parton densities rise steeply as x→0 and
fall off very fast as x→1, which means the dominant contribution to the production results from
the phase space region where x∼t (=mn2˜/S) becomes small. t at Tevatron (0.05 <∼t <∼0.5) is
largercomparedtothatatLHC(0.007<∼t <∼0.07)(seealsofig.2). Becauseofthis,atTevatronthe
valencequarkinitiatedprocessesdominatewhilegluonandseaquarkinitiatedprocessesdominate
at the LHC. As the mass of the sneutrino increases, that is x approaches to unity, the K-factor at
Tevatronnaturally fallsoff. AtLHC,inthehighermassregion(∼1TeV),valencequarkdensities
start to dominate and hence it stays almost flat compared to Tevatron. We now turn to study the
impact of the factorisation scale (m ) and the renormalisation scale (m ) on the production cross
F R
section. The factorisation scale dependence for both LHC (left panel) and Tevatron (right panel)
are shown in upper panels of fig. 3, for mn˜ =300 GeV (LHC), mn˜ = 120 GeV (Tevatron). We
havechosen m R=mn˜ forboththeLHCandtheTevatron. Thefactorisation scaleisvariedbetween
m F =0.1 mn˜ and m F =10mn˜. Wefindthat the factorisation scale dependence decreases ingoing
fromLOtoNLOtoNNLOasexpected.
The dependence of the renormalisation scale dependence on the total cross sections for the
resonant production of sneutrino at the LHCand the Tevatron is shown in the lower panels of fig.
3. Note that the LOis already m dependent due to the coupling l ′(m ). Wehave performed this
R R
analysis for sneutrino mass mn˜ =300 GeV (LHC), mn˜ =120 GeV (Tevatron). We have set the
factorisation scale m F =mn˜ and the renormalisation scale is varied in the range 0.1≤ m R/mn˜ ≤
10. We find significant reduction in the m scale dependence when higher order QCDcorrections
R
are included. It is clear from both the panels of fig. 3 that our present NNLO result makes the
predictions almostindependent ofbothfactorisation andrenormalisation scales.
Wecouldnotdiscussorshowtheresultsofchargedsleptonduetopagelimitation. Werequest
readertofollowtheRef.[17].
Acknowledgement: SpeakeracknowledgesthefullsupportofSahaInstituteofNuclearPhysics,
Indiawherethisworkhasbeendone. SpeakeralsothankedRADCOR’sorganiser forinviting him
togiveaseminarinRADCOR2011.
6
NNLOQCDcorrections SwapanKMajhi
0.3 0.7
LO LO
0.28 NLO 0.65 NLO
NNLO NNLO
0.6
0.26
0.55
s (Pb) 00..2224 mm R˜n == m30˜n0 GeV 0 .04.55 mm R˜n == m12˜n0 GeV
0.2
0.4
0.18 0.35
Tevatron Run II
LHC (14 TeV)
0.16 0.3
0.1 1 10 0.1 1 10
m F/m˜n m F/m˜n
LO 0.7 LO
0.3 NLO NLO
0.65
NNLO NNLO
0.6
0.25 0.55
b) 0.5
P
( 0.45
s 0.2
0.4
0.15 mm F˜n == m30˜n 0 GeV 0.35 mm F˜n == m12˜n 0 GeV
LHC (14 TeV) 0.3 Tevatron Run II
0.25
0.1 1 10 0.1 1 10
m R/m˜n m R/m˜n
Figure3: Intheupperpanel,sneutrinoproductioncrosssectionsareplottedagainstthefactorisationscale
m F withafixedrenormalisationscalem R=mn˜ forbothLHCandTevatronenergies.Inthelowerpanel,they
areplottedagainsttherenormalisationscalem Rwithafixedfactorisationscalem F =mn˜ forbothLHCand
Tevatronenergies. Themassofthesneutrinoistakentobe300GeV(120GeV)atLHC(Tevatron).
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