Resummed differential cross sections for top-quark pairs at the LHC

0.38 MB·

by  Benjamin D. Pecjak

IPPP/16/04
Resummed differential cross sections for top-quark pairs at the LHC
Benjamin D. Pecjakb, Darren J. Scottb, Xing Wanga, Li Lin Yanga,c,d
aSchool of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China
bInstitute for Particle Physics Phenomenology, University of Durham, DH1 3LE Durham, UK
cCollaborative Innovation Center of Quantum Matter, Beijing, China
dCenter for High Energy Physics, Peking University, Beijing 100871, China
We present state of the art resummation predictions for differential cross sections in top-quark
pair production at the LHC. They are derived from a formalism which allows the simultaneous
resummation of both soft and small-mass logarithms, which endanger the convergence of fixed-
order perturbative series in the boosted regime, where the partonic center-of-mass energy is much
largerthanthemasstothetopquark. WecombinesuchadoubleresummationatNNLL(cid:48) accuracy
with standard soft-gluon resummation at NNLL accuracy and with NLO calculations, so that our
results are applicable throughout the whole phase space. We find that the resummation effects on
the differential distributions are significant, bringing theoretical predictions into better agreement
with experimental data compared to fixed-order calculations. Moreover, such effects are not well
described by the NNLO approximation of the resummation formula, especially in the high-energy
tailsofthedistributions,highlightingtheimportanceofall-ordersresummationindedicatedstudies
of boosted top production.
6
1
0 INTRODUCTION ItisthereforeimportanttoassesstheeffectsofQCDcor-
2
rections even beyond NNLO, in order to see whether the

b gap between theory and data at high p can be bridged.
e The 8TeV run of the LHC delivered about 20fb−1 of T
F integrated luminosity to both the ATLAS and CMS ex-
5 periments. Among the many important results coming Forboostedtop-quarkpairswithhighpT therearetwo
from these data, the properties of the top-quark have classes of potentially large contributions. The first is the

] been measured with unprecedented precision. At the Sudakov-type double logarithms arising from soft gluon
h
same time, theoretical calculations of top-quark related emissions. Thesecondcomesfromgluonsemittednearly
p
- observableshaveseensignificantadvancementsinthelast paralleltothetopquarks,resultinginlargelogarithmsof
p few years. In particular, very recently the next-to-next- the form lnn(mt/mT), where mt is the top quark mass,
he to-leadingorder(NNLO)QCDcorrectionstodifferential and mT ≡ (cid:112)m2t +p2T is the transverse mass of the top
[ crosssectionsintop-quarkpair(tt¯)productionhavebeen quark. In [5], some of the authors of the current work
calculated[1]. In[2],theCMScollaborationperformeda developed a formalism for the simultaneous resumma-
2
comprehensive comparison between their measurements tionofbothtypeoflogarithmstoallordersinthestrong
v
0 [3] of the differential cross sections and various theoreti- coupling constant αs. In this Letter, we report the first
2 cal predictions, including those from the NNLO calcula- phenomenological applications of that formalism, giving
0 tion and those from Monte Carlo event generators with predictionsforthetop-quarkpT andthett¯invariantmass
7 next-to-leading order (NLO) accuracy matched to par- distributionsatthe8TeVLHC,andcomparingwithex-
0
ton showers. The overall agreement between theory and perimental measurements as well as the NNLO calcula-
.
1 data is truly remarkable, which adds to the success of tions when possible. With an eye to the future, we also
0 the Standard Model (SM) as an effective description of present predictions for the 13TeV LHC, where NNLO
6
Nature. results are not yet available.
1
: However,apersistentissueinthe8TeVresultsisthat
v
i the transverse momentum (pT) distribution of the top Our main finding is that the higher-order effects con-
X
quark is softer in the data than in theoretical predic- tained in our resummation formalism significantly alter
r tions, i.e., the experimentally measured differential cross the high-energy tails of the p and tt¯invariant mass dis-
a T
section at high p is lower than predictions from event tributions, softening that of the p distribution but en-
T T
generators or from NLO fixed-order calculations [3, 4]. hancingthatofthett¯invariantmassdistribution. These
While the NNLO corrections bring the fixed-order pre- effects bring our results into better agreement with the
dictions into better agreement with the CMS data, as experimental data compared to pure NLO fixed-order
notedin [1]and [2], there isstill somediscrepancy inthe calculations. Interestingly, for the case of the p distri-
T
high-p binswherep >200GeV. Giventheimportance bution,thissofteningofthespectrumisslightlystronger
T T
ofthett¯productionprocessasastandardcandleforval- than the similar effect displayed in recent NNLO results,
idating the SM and as an essential background for new andleadstoabettermodelingofthep >200GeVpor-
T
physics searches, it would be disconcerting ifthis feature tion of the CMS data [3]. We comment further on this
were to persist at higher p values and with more data. fact in the conclusions.
T
2
FORMALISM Mellin space) as well as the NLO fixed-order result cal-
culated in [10] and implemented in MCFM [11]. The
Our predictions are based on the factorization and precise matching formula can be found in [6]. After such
resummation formula derived in [5]. The technical de- a matching procedure, we denote the final accuracy of
tails will be given in a forthcoming article, although the our predictions, which are valid throughout phase space,
main elements have already been sketched out in [6]. In as NLO+NNLL(cid:48).
the kinematic situation where the top quarks are highly It would be desirable to match with the recent NNLO
boostedandtheeventsaredominatedbysoftgluonemis- results in [1] to achieve NNLO+NNLL(cid:48) accuracy. How-
sions, theresummedpartonicdifferentialcrosssectionin ever, at the moment NNLO results are only available
Mellin space can be written as for fixed (i.e., kinematics-independent) factorization and
renormalization scales µ ∼ µ ∼ m , whereas for the
f r t
(cid:34)
study of differential distributions over large ranges of
(cid:101)cij(N,Mtt¯,mt,µf)=Tr U(cid:101)ij(µf,µh,µs)Hij(Mtt¯,µh) phase space we consider it important to follow common
practice and use dynamical (i.e., kinematics-dependent)
(cid:18) M2 (cid:19)(cid:35) scale choices. Therefore, such an improvement over our
×U(cid:101)i†j(µf,µh,µs)×s(cid:101)ij lnN¯2µtt¯2,µs (1) result is not currently possible, and we leave it for the
s
future.
(cid:32) (cid:33)
m
×U(cid:101)D2(µf,µdh,µds)CD2(mt,µdh)s(cid:101)2D lnN¯µt ,µds ,
ds
PHENOMENOLOGY
where for simplicity, we have suppressed some variables
in the functional arguments which are unnecessary for In the following we present NLO+NNLL(cid:48) predictions
the explanations below. In the above formula, Mtt¯ is for the Mtt¯ and pT distributions at the LHC. In all
the invariant mass of the tt¯pair (which can be related our numerics we choose m = 173.2GeV and use
t
to the pT of the top quark in the soft limit through a MSTW2008NNLO PDFs [12]. For pT distributions, the
change of variables), N is the Mellin moment variable, default values for the factorization scale and the four
and N¯ ≡ NeγE with γE the Euler constant. The soft renormalizationscalesarechosenasµf =mT,µh =Mtt¯,
limit corresponds to N → ∞ in Mellin space. The four µs = Mtt¯/N¯, µdh = mt and µds = mt/N¯. For Mtt¯ dis-
coefficient functions Hij, s(cid:101)ij, CD and s(cid:101)D encode con- tributions, the only difference is µf =Mtt¯. We estimate
tributions from four widely-separated energy scales Mtt¯, scaleuncertaintiesbyvaryingthefivescalesaroundtheir
Mtt¯/N¯, mt and mt/N¯, respectively. The presence of the defaultvaluesbyfactorsoftwoandcombiningtheresult-
fourscalesleadstothetwotypesoflargelogarithmsdis- ingvariationsofdifferentialcrosssectionsinquadrature;
cussedintheintroduction. Incorrespondencewiththese we do not consider uncertainties from PDFs and α in
s
four physical scales, there are four unphysical renormal- this Letter. The hadronic differential cross sections are
izationscalesµh,µs,µdh andµds,oneforeachcoefficient first evaluated in Mellin space at a given point in phase
function. Thephilosophyofresummationistochoosethe space, and we then perform the inverse Mellin transform
four unphysical scales to be around their corresponding numerically using the Minimal Prescription [13]. This
physical scales, so that the four coefficient functions are procedure relies on an efficient construction of Mellin-
free of large logarithms and are well-behaved in fixed- transformed parton luminosities, for which we use meth-
order perturbation theory. One can then use renormal- ods outlined in [14, 15].
ization group (RG) equations to evolve these functions
The differential cross sections considered below span
to the factorization scale µ in order to convolute with
f severalordersofmagnitudewhengoingfromlowtohigh
the parton distribution functions (PDFs) and obtain the
valuesofpT orMtt¯. Inordertobetterdisplaytherelative
hadronic cross sections. The effects of the RG running
sizes of various results, we show in the lower panel of
areencodedinthetwoevolutionfactorsU(cid:101)ij (forHij and eachplotthedifferentialcrosssectionsnormalizedtoour
s(cid:101)ij) and U(cid:101)D (for CD and s(cid:101)D), which resum all the large default prediction, i.e., the ratio defined by
logarithms to all orders in α in an exponential form.
s
At the moment, the four coefficient functions are dσ
Ratio≡ . (2)
known to NNLO [5, 7, 8], while the two evolution fac- dσNLO+NNLL(cid:48)(cid:0)µ =µdefault(cid:1)
i i
tors are known to next-to-next-to-leading logarithmic
(NNLL) accuracy [5]. Such a level of accuracy is usu- Fig. 1 compares our NLO+NNLL(cid:48) resummed predic-
ally referredto asNNLL(cid:48) inthe literature, andweadopt tion for the normalized top-quark p distribution to
T
that nomenclature here. While the formula (1) is only the CMS measurement [3] in the lepton+jet channel at
√
applicable in the boosted soft limit, we can extend its the LHC with a center-of-mass energy s = 8TeV.
domainofvaliditybycombiningitwithinformationfrom Also shown is the NNLO result from [1], which adopted
NNLL soft gluon resummation derived in [9] (recast into by default the renormalization and factorization scales
3
ss-3/GeV) (10/dp) d(1/T 10 NNCLNMOLSmLmO+ Hf(t N =l=C+(N m j( )1d18Le7/ Lf2T 3'=,e.1 2Vm, 2Gt ) =em V1T73.3 GeV) s (pb/GeV)/dMdtt 101 mLmHft ==C ( 1187/ 2T3,e.12NNAV, 2GTLL)OOL eMAV+tStN N(l+LjL)'
1 10-1
10-2
10-1 10-3
Ratio 11..21 Ratio 11..24
1.0
1.0
0.8
0.9 0.6
0.8 0.4
0 50 100 150 200 250 300 350 400 400 600 800 1000 1200 1400 1600
p (GeV) M (GeV)
T tt
FIG.1. Resummedprediction(blueband)forthenormalized FIG. 3. Resummed prediction (blue band) for the absolute
top-quark pT distribution at the 8 TeV LHC compared with Mtt¯ distribution at the 8TeV LHC compared with ATLAS
CMS data (red crosses) [3] and the NNLO result (magenta data (red crosses) [16] and the NLO result (magenta band).
band) [1]. The lower panel shows results normalized to the
default NLO+NNLL(cid:48) prediction.
taluncertaintyisratherlargeduetolimitedstatistics, it
s (pb/GeV)/dpdT1100--21 mLHt =C 187 T3e.2NNAV GTLLOOLeAV+SNNLL' itmIsnioainFsnsstigelho.reeg2rsaetwri,intesghisnmhtcoosewbcioetscmuioscpmhaeexraepmceiootcrmtewepdriteahltrehivstaoahtnne.tthtaThetehhoseoirgefNthtNieacrLaneldOnpesrrrmeegsdiaueilcsllt--.
mf = (1/2,1,2) mT for such high pT values is not yet available, so we com-
10-3 pareinsteadwiththeNLOresultcomputedusingMCFM
withMSTW2008NLOPDFsanddynamicalrenormaliza-
tion and factorization scales, whose default values are
10-4
µ = µ = m . Scale uncertainties of the NLO results
r f T
areestimatedthroughvariationsofµ =µ byafactorof
r f
10-5 two around the default value. From the plot one can see
Ratio 11..48 that the NLO result calculated in this way does a good
job in estimating the residual uncertainty from higher
1.0
order corrections, as the resummed band lies almost in-
0.6
300 400 500 600 700 800 900 1000 1100 1200 side the NLO one up to p = 1.2TeV. On the other
p (GeV) T
T
FIG. 2. Resummed prediction (blue band) for the absolute hand, the inclusion of the higher-order logarithms in the
p distributionatthe8TeVLHCintheboostedregioncom- NLO+NNLL(cid:48) result significantly reduces the theoretical
T
pared with the ATLAS data (red crosses) [4] and the NLO uncertainty, which is crucial for future high precision ex-
result (magenta band). periments at the LHC.
Our formalism is flexible and can be applied to other
differential distributions as well. To demonstrate this
µr = µf = mt, and also used a slightly different top- fact, in Fig. 3 we show the NLO+NNLL(cid:48) resummed pre-
quark mass, mt = 173.3GeV. At low pT, it is clear dictionforthetop-quarkpairinvariantmassdistribution
that both the NLO+NNLL(cid:48) and the NNLO results de- alongwithameasurementfromtheATLAScollaboration
scribe the data fairly well. With the increase of pT, it [16] at the 8TeV LHC. Since the NNLO result in [1] for
appearsthattheNNLOpredictionsystematicallyoveres- this distribution has an incompatible binning, it is cur-
timatesthedata,althoughthereisstillagreementwithin rently not possible to include it in the plot, so we show
errors. Ontheotherhand,withthesimultaneousresum- instead the NLO result computed with the same input
mation of the soft gluon logarithms and the mass log- as in Fig. 2, but this time with the default scale choice
arithms and also with the dynamical scale choices, our µr =µf =Mtt¯. OnecanseefromtheplotthattheNLO
NLO+NNLL(cid:48) resummed formula produces a softer spec- result with this scale choice is consistently lower than
trum which agrees well with the data. the experimental data. The resummation effects signif-
In[4],theATLAScollaborationcarriedoutameasure- icantly enhance the differential cross sections, especially
mentofthetop-quarkpT spectruminthehighly-boosted at high Mtt¯. As a result, the NLO+NNLL(cid:48) prediction
regionusingfat-jettechniques. Althoughtheexperimen- agrees with data quite well. We have found that choos-
4
s (pb/GeV)/dpdT11001--0211 mLmHft ==C 1(1173/32 T.,21NNe ,VGLL2OO)e mV+NTNLL' Relative corrections 0001....4680 mLmHft ==C 1M173tt3 /T2.2e VGeV aBNeyNoLnOd cNoNrrLeOction
0.2
10-3
0.0
10-4 500 1000 1500 2000 2500 3000 3500 4000
M (GeV)
tt
Ratio11100011---..76524 Relative corrections-000...011 mLmHft ==C 1m173T3 T.2e VGeV aBNeyNoLnOd cNoNrrLeOction
1.0
0.80 200 400 600 800 1000 1200 1400 1600 1800 2000 -0.2 400 600 800 1000 1200 1400 1600 1800 2000
p (GeV) p (GeV)
T T
FIG. 5. Relative sizes of the corrections at approximate
(pb/GeV)Mtt 101 NNLLOO+NNLL' NEqN.L(O3)(abnludet)haenedxpbleaynoantdion(bslathcker)e, wfoirthprreecsipseecdtetfionNitiLoOns..See
s/dd 10-1 mLHt =C 11733 T.2e VGeV
10-2 mf = (1/4,1/2,1) Mtt
10-3
up to pT =2TeV and Mtt¯=4.34TeV, contrasted with
10-4 the NLO results. Note that for the Mtt¯distribution, we
10-5 have changed the default µf to a lower value Mtt¯/2 for
10-6 the reasons explained above. The plots exhibit similar
patterns as observed at 8TeV, namely that the higher-
10-7
Ratio 0111....8024 oprTdedrisrterisbuumtimonatbiounteeffnehcatnscseetrhveattoofstohfteeMn ttt¯heditsatriliboufttiohne
0.6 compared to a pure NLO calculation.
0.4
0.2
500 1000 1500 2000 2500 3000 3500 4M00 0(GeV) Asmentionedbefore,wewouldliketomatchourcalcu-
tt
FIG. 4. Resummed predictions (blue bands) for the p and lations with the NNLO results when they become avail-
T
Mtt¯distributionsatthe13TeVLHCcomparedwiththeNLO able in the future. We end this section by discussing
results (magenta bands). the expected effects of such a matching, by estimating
the size of resummation corrections beyond NNLO. We
do this in Fig. 5, where the relative sizes of the beyond-
ing the default renormalization and factorization scales NNLO corrections generated through the resummation
to be half the invariant mass increases the fixed-order formula are displayed as a function of Mtt¯ or pT with
cross section and therefore mimics to some extent the the default scale choices. The exact NNLO results for
resummationeffects. Infact,thisprocedurehasbeenex- these scale choices are not yet available, so we show in
tensively employed in the literature for processes such as comparison the relative sizes of the approximate NNLO
Higgsproduction[17],wherehigher-ordercorrectionsare (aNNLO) corrections obtained by expanding and trun-
also large. Consequently, it may be advisable to employ cating our resummation formula to that order. More
a renormalization and factorization scale of the order of precisely, the blue and black curves in Fig. 5 correspond
Mtt¯/2infixed-ordercalculations(andMonteCarloevent to
generators), and we shall use this choice when studying
the Mtt¯distribution at the 13 TeV LHC below.
The LHC has started the 13TeV run in 2015. So far dσaNNLO−dσNLO
aNNLO correction≡ , (3)
there are only two CMS measurements [18, 19] of dif- dσNLO
ferential cross sections for tt¯production, based on just dσNLO+NNLL(cid:48) −dσaNNLO
42pb−1 of data. The resulting experimental uncertain- Beyond NNLO≡ dσNLO ,
ties are therefore quite large and it is not yet possible to
probe higher pT or Mtt¯values. Nevertheless, in the near
future there will be a large amount of high-energy data, where dσaNNLO refers to the approximate NNLO result.
whichwillenablehigh-precisionmeasurementsoftt¯kine- The figure clearly shows that corrections beyond NNLO
maticdistributions,alsointheboostedregime. InFig.4 are significant in the tails of the distributions, especially
we show our predictions for the pT and Mtt¯ spectrum in the case of the Mtt¯distribution.
5
CONCLUSIONS AND OUTLOOK for collaboration on many related works. X. Wang and
L.L.YangaresupportedinpartbytheNationalNatural
ScienceFoundationofChinaunderGrantNo. 11575004.
In this Letter we have presented new resummation
D. J. Scott is supported by an STFC Postgraduate Stu-
predictions for differential cross sections in tt¯ produc-
dentship.
tion at the LHC. The predictions include the simul-
taneous resummation to NNLL(cid:48) accuracy of both soft
and small-mass logarithms, which endanger the conver-
genceofthefixed-orderperturbativeseriesintheboosted
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