Published for SISSA by Springer
Received: March 18, 2016 Revised: May 30, 2016 Accepted: July 16, 2016 Published: July 22, 2016
Impact of jet veto resummation on slepton searches
Frank J. Tackmann,a Wouter J. Waalewijnb,c and Lisa Zeunec
aTheory Group, Deutsches Elektronen-Synchrotron (DESY),
D-22607 Hamburg, Germany
bITFA, University of Amsterdam,
Science Park 904, 1018 XE, Amsterdam, The Netherlands
cTheory Group, Nikhef,
Science Park 105, 1098 XG, Amsterdam, The Netherlands
E-mail: mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected] , [email protected]
Abstract: Several searches for new physics at the LHC require a xed number of signal jets, vetoing events with additional jets from QCD radiation. As the probed scale of new physics gets much larger than the jet-veto scale, such jet vetoes strongly impact the QCD perturbative series, causing nontrivial theoretical uncertainties. We consider slepton pair production with 0 signal jets, for which we perform the resummation of jet-veto logarithms and study its impact. Currently, the experimental exclusion limits take the jet-veto cut into account by extrapolating to the inclusive cross section using parton shower Monte Carlos. Our results indicate that the associated theoretical uncertainties can be large, and when taken into account have a sizeable impact already on present exclusion limits. This is improved by performing the resummation to higher order, which allows us to obtain accurate predictions even for high slepton masses. For the interpretation of the experimental results to bene t from improved theory predictions, it would be useful for the experimental analyses to also provide limits on the unfolded visible 0-jet cross section.
Keywords: Supersymmetry Phenomenology, Jets
ArXiv ePrint: 1603.03052
Open Access, c
[circlecopyrt] The Authors.
Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP07(2016)119
Web End =10.1007/JHEP07(2016)119
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Contents
1 Overview 1
2 Jet veto resummation 42.1 Factorization formula 52.2 Hard scattering process 72.3 Estimating the theory uncertainty 9
3 Results 103.1 Slepton production at 8 TeV 103.2 Slepton production at 13 TeV 13
4 Conclusions 15
A Fixed-order ingredients 16A.1 Hard function 16A.2 Beam function 17A.3 Soft function 18A.4 Nonsingular contributions 18
B RGE ingredients 19B.1 Anomalous dimensions 20B.2 Pro les scales 21
1 Overview
A crucial challenge at the LHC is to discriminate a faint Beyond-the-Standard Model (BSM) signal from large Standard Model (SM) backgrounds, since for most BSM searches no \smoking gun" signature exists. To eliminate SM backgrounds containing jets, many analyses require a xed number of hard jets corresponding to the expected number of signal jets in the hard-interaction process. This amounts to placing a veto on additional jets above a certain transverse momentum pcutT arising from QCD initial-state or nal-state radiation.
Typical examples are supersymmetry (SUSY) searches for third generation squarks requiring two signal jets and vetoing a third jet [1{3], or electroweakino/slepton searches usually requiring 0 signal jets [4{8]. Jet vetoes are also applied in other BSM searches, including anomalous triple-gauge couplings [9], unparticles [10], large extra dimensions and dark matter candidates in mono-photon, mono-Z and mono-jet events [11{13]. In this paper, we concentrate on slepton (selectron and smuon) searches, focusing in particular on the analysis in ref. [5], which is representative of analyses with no nal state jets. Searches
{ 1 {
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with jets in the nal state are more complicated, as the jet transverse momenta introduce additional kinematic scales in the cross section, and are left for future work.
Exclusion limits require reliable predictions for the expected BSM cross section. So far, the focus of theory calculations has mostly been on the total production cross section, while the e ect of exclusive phase-space cuts like jet vetoes has not been much investigated. However, since jet vetoes impose a strong restriction on additional QCD emissions, they can signi cantly alter the cross section and pose an important source of theory uncertainty, as was observed some time ago in the context of Higgs production [14, 15].
The jet veto introduces large logarithms in the 0-jet cross section, schematically,
0(pcutT) = a00 + s
a12 ln2 pcutT
Q + a11 lnpcutT
Q + a10 [parenrightbigg]
a24 ln4 pcutT
Q + a23 ln3pcutT
Q + a22 ln2pcutT
Q + a21 lnpcutT
Q + a20 [parenrightbigg]
+ [notdef] [notdef] [notdef] + (terms suppressed by pcutT=Q) ; (1.1) where amn are coe cients and Q denotes the hard-interaction scale, which is set by the (typical) partonic invariant mass, e.g. twice the slepton mass. For pcutT Q, the logarithmic terms produce large corrections leading to a poor perturbative convergence. This can become a large e ect for SUSY particle production for which Q can easily be 1 TeV or more, and it will only get more important as the measurements continue to probe higher BSM scales.
The actual experimental limit is on the visible cross section in the ducial phase space including all experimental reconstruction e ciencies and acceptance cuts, and in particular including the jet veto. Its interpretation in terms of the exclusion limits quoted by the experiments involves the extrapolation from the measured 0-jet cross section to the inclusive cross section using parton shower Monte Carlos. An important outcome of our approach is that we are able to obtain a reliable estimate of the theory uncertainty associated with the jet veto, which parton showers typically do not provide. For this reason, the jet-veto uncertainties, which we nd to have a sizeable impact, are also not taken into account in the current results that involve a jet veto.
To obtain accurate theoretical predictions and assess the theoretical uncertainties, the logarithmic terms in eq. (1.1) can be systematically summed up to all orders in s. This resummation for jet vetoes in hadronic collisions has been well-developed in the context of Drell-Yan and Higgs production [14, 16{29], and the same methods have also been used to study diboson processes [30{35].
The amn coe cients in eq. (1.1) are not all independent, and their structure allows the logarithmic series to be rewritten as
0(pcutT) = b0 + b1 s + [notdef] [notdef] [notdef]
exp
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+ 2s
[bracketleftbigg] [summationdisplay]
c0m + c1m s + [notdef] [notdef] [notdef]
ms lnm+1 pcutT Q
[bracketrightbigg]
m 1
+ (terms suppressed by pcutT=Q) : (1.2)
Each of the series inside round brackets is now free of logarithms, and so can be computed order by order in s. Doing so then amounts to systematically performing the resummation
{ 2 {
to higher logarithmic order. The resummation orders relevant for our discussion include all terms in eq. (1.2) as follows:
LL: b0; c0m ; NLL: b0; c0m; c1m ; NLL[prime]: b0; b1; c0m; c1m : (1.3)
The b1 term, rst included at NLL[prime], is important as it incorporates the full one-loop virtual corrections into the resummation, including both QCD and SUSY-QCD corrections. The remaining terms suppressed by pcutT=Q in eqs. (1.1) and (1.2) start at O( s) and vanish
as pcutT=Q ! 0. At NLL[prime]+NLO we include them at O( s), which then reproduces the
inclusive NLO cross section in the limit pcutT ! 1.
We now give a preview of our main results, leaving details of the calculation to section 2 and the appendices. A more extensive discussion with additional plots and results for ~
R~
R
production are given in section 3. Figure 1 shows our resummed predictions for the slepton production cross section with a jet veto at NLL (green band, dotted line) and at NLL[prime]+NLO order (red band, solid line) as a function of the slepton mass m~[lscript] for 8 TeV (left plot) and13 TeV (right plot). In the left plot we use pcutT = 20 GeV, as in the ATLAS analysis [5], and in the right plot we choose pcutT = 25 GeV and 100 GeV as representative values.
The bands show the perturbative uncertainties (but no parametric PDF uncertainties), which are systematically estimated by varying resummation and renormalization scales, as discussed in detail in section 2.3. The overlap between the bands and the reduction in uncertainties demonstrate the excellent stability of the resummed calculation, allowing us to obtain precise predictions even up to high slepton masses, see right panel, where the impact of the jet veto increases.
To investigate the implications for the exclusion limit, we extract the 95% CL upper limit on the visible 0-jet cross section from the experimental results by using ATOM [36] and CheckMATE [37] to determine the signal region e ciencies excluding the jet veto. These are shown in the left panel as the dotted and dashed black curves. We translate this into a 95% CL exclusion limit shown as error bars in the bottom panel, using our NLL prediction (green) or NLL[prime]+NLO prediction (red). This can be compared to the exclusion limit provided by ATLAS (blue) [5], for which the total NLO cross section [38{40] was multiplied with the signal region e ciencies (including the jet veto) obtained using HERWIG++ [41]. (A more consistent combination of the inclusive NLO cross section with parton showers was obtained in ref. [42].) The ATLAS exclusion accounts only for the theory uncertainty associated with the total production cross section, following ref. [43], but does not take into account the uncertainty associated with the jet veto.
The perturbative precision of the parton shower is formally at most that of our NLL results, and hence the perturbative uncertainties due to the jet veto in the experimental limits could easily be as large as that. This has a sizeable impact: using our NLL result the exclusion would go down to m~[lscript]L [similarequal] 270 GeV. Note that even with our NLL[prime]+NLO
predictions the uncertainty on the exclusion is still larger than the one obtained by ATLAS. In the future, it would be advantageous to separate out theory-sensitive acceptance cuts in the experimental results for example by quoting the observed limit on the visible 0-jet cross section with unfolded detector e ciencies. This avoids folding a dominant theory
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L production as a function of m~[lscript] at 8 TeV (left plot) and13 TeV (right plot). The results at NLL are shown by the green (light) band and dotted lines and at NLL[prime]+NLO by the orange (dark) band and solid lines. In the left plot, we use pcutT = 20 GeV and the dotted and dashed black lines show the experimental 95% CL upper limit on the visible 0-jet cross section, which are extracted from the ATLAS results in ref. [5] using ATOM [36] and CheckMATE [37]. The error bars in the bottom panel give the resulting 95% CL exclusion limits on m~[lscript] using our NLL prediction (green) and NLL[prime]+NLO prediction (red). This is compared to the 95% CL exclusion limit provided by ATLAS (blue), which does not take into account the jet-veto uncertainty. In the right plot, we show predictions for the 0-jet cross section for two representative values of pcutT (25 GeV and 100 GeV), where the cross section is rescaled by a normalization factor for better visibility.
dependence directly into the quoted exclusion limits and allows the experimental results and their interpretation to easily bene t from future improvements in theoretical predictions.
Finally, we note that soft gluon (threshold) resummation for the total slepton production cross section has been studied extensively in refs. [39, 40, 44, 45]. We emphasize that this type of resummation is separate and can be considered in addition to the jet veto re-summation we discuss here. For current values of slepton masses under investigation at the LHC, the e ect on the total cross section and uncertainty is rather small, and we therefore do not include it here. The perturbative description with a jet at large pT present in the nal state can also be improved by considering the slepton-pair plus jet process [46].
2 Jet veto resummation
In this section, we discuss the calculation in some detail. We utilize the jet-pT resummation of ref. [25] using soft-collinear e ective theory (SCET) [47{52].
In section 2.1, we present the factorization formula for the process, pp !
Figure 1. The 0-jet cross section for ~
L~
~ ! ~01~01,
and discuss how it is used to resum the jet-veto logarithms. Section 2.2 discusses the hard function that describes the underlying short-distance interaction for slepton pair production. In particular, we show that correlations between the jet veto and other kinematic selection cuts are negligible, which will allow us to ignore the slepton decay. In
{ 4 {
section 2.3, we explain how the theoretical uncertainties are estimated through resummation and renormalization scale variations. All xed-order perturbative ingredients are collected in appendix A, while the anomalous dimensions and scale choices are summarized in appendix B.
2.1 Factorization formula
The SCET factorization formula for the 0-jet cross section is given by [19, 20]
0(pcutT; mSUSY; cuts) = [integraldisplay]
dQ2 dY Hq q(Q2; Y; mSUSY; cuts; )
[notdef] Bq(pcutT; xa; ; ) B q(pcutT; xb; ; ) Sq q(pcutT; ; )+ nons0(pcutT; mSUSY; cuts) : (2.1)
Here Q and Y are the total invariant mass and rapidity of the sleptons, and
xa = QEcm eY ; xb =
~ ! ~01~01.
It contains all the analysis cuts applied on the slepton nal state but not the jet veto. The relevant SUSY masses are summarized by mSUSY, which in addition to the slepton
and neutralino masses also includes the squark and gluino masses at one-loop order (see gure 2(c)). The hard function will be discussed in section 2.2 and appendix A.1.
Due to the jet veto, the real QCD radiation is restricted to be collinear to the beam axis or soft. The beam function Bq (B q) describes the e ect of the jet veto on collinear initial-state radiation from the colliding (anti)quark with momentum fraction xa (xb),
and combines the nonperturbative parton distribution functions (PDFs) with perturbative initial-state radiation [16]. The restriction of the jet veto on soft radiation is encoded in the soft function Sq q. The required NLO results for the beam and soft functions are given in appendix A.2 and appendix A.3. The dependence on the jet algorithm and jet radius e ects rst appear at NNLL in ln(pcutT=Q) and O( 2s) [19{21] and are beyond the order we consider here.
The nonsingular cross section nons in eq. (2.1) only consists of the O(pcutT=Q) sup
pressed terms already mentioned in eq. (1.2) and vanishes for pcutT ! 0. In appendix A.4 we describe how the nonsingular terms are obtained.
Eq. (2.1) factorizes the large jet veto logarithms. For example, the leading double logarithm in the NLO cross section splits up as
ln2 pcutTQ = ln2
Q
+ 2 ln
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Q
Ecm eY : (2.2)
The hard function Hq q describes the short-distance scattering process, q
q !
pcutT
pcutT
Q + ln
pcutT
ln
ln
2 ; (2.3) where the three terms on the right-hand side are the contributions from the NLO hard, beam, and soft functions, respectively. The key to obtaining a resummed prediction for the cross section is that each individual term can be made small by an appropriate choice of the renormalization scale and rapidity renormalization scale , namely
H Q 2m~[lscript]; B S pcutT ; B Q 2m~[lscript]; S pcutT : (2.4)
{ 5 {
(a) Leading order
q
q
q
g
g
g
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q
q
q
(b) One-loop QCD corrections
q
q
q
q
q
q
(c) One-loop SUSY-QCD corrections
Figure 2. Leading order and one-loop virtual corrections to slepton pair production.
Hqq
H 2m~`
RGE
S pcutT
Sqq
Bq
B pcutT
RGE
S pcutT
B 2m~`
Figure 3. The hard, beam, and soft functions are evolved in virtuality from their natural scales H 2m~[lscript] and B S pcutT. The beam and soft functions are also evolved in rapidity from
their natural scales B 2m~[lscript] and S pcutT.
By evaluating each of the hard, beam, and soft functions at their natural scale, they contain no large logarithms. The logarithms in the cross section are then e ciently resummed by evolving each of the functions using their renormalization group evolution (RGE) for and the rapidity RGE for [53, 54] to the common (and arbitrary) scales and at which the cross section in eq. (2.1) is evaluated. The RGE is illustrated in gure 3 and the formulae needed for carrying it out are collected in appendix B.
{ 6 {
2.2 Hard scattering process
We now discuss the hard function, which contains the hard scattering process q
q !
~01~01 including the tree-level and virtual loop corrections shown in gure 2. We consider a simpli ed (R-parity conserving) model where all SUSY particles except for the slepton ~
and the lightest neutralino ~01 are heavy and B(
~
! ~01) = 1. We will argue that we
can simply calculate inclusive slepton production with a jet veto, without considering the subsequent decay of the sleptons, since the jet veto is uncorrelated with the other cuts on the slepton decay products. The resulting hard function is given in appendix A.1.
The jet veto is factorized from the other cuts in eq. (2.1), since only the soft and beam functions depend on the jet veto, whereas the hard function depends on the other cuts.1 Hence, the only possibility to introduce correlations between the jet veto and other cuts is through the common variables Q2 and Y .2 If the cuts were to induce sizeable changes in the Q2 and Y dependence of the hard function, then the pcutT-dependent beam and soft functions would get weighted in a cut-dependent way when integrated over Q2 and Y .
We have investigated this using MadGraph (version 2.3.2) [56] for the signal regions SR-mT2 of ref. [5], which consist (besides the jet veto) of the following cuts:
Two (same- avor) leptons with pT > 35 GeV and pT > 20 GeV. The pseudorapidity
of each lepton is required to be [notdef] [notdef] < 2:47 for electrons and [notdef] [notdef] < 2:4 for muons.
The dilepton invariant mass m[lscript][lscript] > 20 GeV and [notdef]m[lscript][lscript] mZ[notdef] > 10 GeV.
Three possible cuts on the stransverse mass [57, 58] mT2 > 90, 120, or 150 GeV. The resulting tree-level cross section, corresponding to the tree-level hard function, is shown in gure 4 for a selectron mass of 250 GeV and a neutralino mass of 20 GeV. The gray line shows the number of events per bin without cuts and the colored lines show the number of events after the signal region cuts. The bands indicate the statistical uncertainty due to the number of simulated events. The top and bottom rows show the Q and Y dependence, respectively. In the right column, each bin is normalized to the total number of events in that bin, i.e., showing the acceptance of the cuts in each Q and Y bin. We can see that the cut acceptance is essentially at in Q and Y , so these cuts do not a ect the shape in Q and Y but only the normalization. The Y dependence is no longer at for [notdef]Y [notdef] > 1:5,
but this corresponds to only 8% of the total cross section. This implies that to very good approximation we can treat the other cuts as a Q and Y independent multiplicative correction which we can factor out from eq. (2.1). This treatment is completely su cient for our purposes, since in order to compare to the experimental measurements we will also have to include experimental reconstruction e ciencies, which we are anyway only able to
1The kinematic cuts considered here a ect only the hard kinematics. If one would have additional IR-sensitive cuts that resolve the hadronic nal state, the factorization in eq. (2.1) would get more complicated, see e.g. ref. [55] for a prototypical situation.
2In principle, the hard function is independent of the boost Y , however the cuts are not. In addition, the nonsingular corrections nons0 depend on both the jet veto and the other cuts, but these corrections are negligible in the relevant region of pcutT Q.
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Figure 4. The e ect of the signal region cuts (besides the jet veto) on the Q (upper row) and Y (lower row) dependence of the cross section. The left column shows the number of events per bin, before (gray) and after (blue, green and orange) cuts. The right column shows the acceptance per bin.
do approximately. Hence, we focus our attention on the jet-veto cut, which receives large QCD corrections, without considering the other cuts.
Once we restrict ourselves to only calculating the jet veto, the assumption that B(
~
!
~01) = 1 allows us to focus on slepton production without the subsequent decay. We do not consider mixing in the slepton sector and we separately discuss ~
L~
L and ~
R~
R production.3
This is a good approximation for sleptons of the rst two generations, which we focus on here. For staus, mixing e ects are relevant and can be easily included.
At tree level, slepton pairs are produced via a q
q-initiated s-channel exchange of a photon or a Z boson, as shown in gure 2(a). The leading-order hard function is simply equal to the corresponding partonic cross section, which has been calculated in refs. [59{62]. Since the intermediate =Z decays into a noncolored nal state, the one-loop QCD corrections a ect only the q
qV production vertex and are identical to those of the Drell-Yan process [63], see gure 2(b). The one-loop SUSY-QCD corrections are shown in gure 2(c). They have been calculated in ref. [38] neglecting squark mixing and in ref. [64] including squark mixing. In the simpli ed model considered here, the squarks are heavy and SUSY-
3 ~
[lscript]L (~
[lscript]R) denotes the superpartner of a left-handed (right-handed) lepton [lscript] and will be referred to as a left-handed (right-handed) slepton.
{ 8 {
QCD corrections are small compared to the QCD corrections. Mixing e ects in the squark sector are therefore neglected. The resulting NLO hard function is given in appendix A.1. If squark mixing e ects become relevant, they can be straightforwardly included in the hard function. Note also that at one-loop order gluon-initiated slepton production is in principle also possible via a Higgs or quartic scalar coupling [65, 66]. However, the corresponding cross section is very small (except in the resonance region) and is therefore not considered here (or in Prospino).
2.3 Estimating the theory uncertainty
In this section, we discuss the resummation scales that are used to obtain the central value for the cross section and to assess the perturbative uncertainty, with additional details relegated to appendix B.2. We have also evaluated the parametric PDF uncertainty for the resummed 0-jet cross section, which is explained in the discussion of gure 10 in section 3 below.
In SCET, resummation is performed by evaluating the hard, beam, and soft functions at their natural virtuality and rapidity resummation scales and then evolving them to common and scales using their virtuality and rapidity RG equations, as illustrated in gure 3. The resummation is crucial for pcutT Q 2m~[lscript], but must be switched o for large pcutT to correctly reproduce the xed-order cross section in that region. The smooth transition between the resummation and xed-order regions is achieved by using pcutT-depended resummation scales, called pro le scales. Pro le scales were rst introduced to study the B ! Xs spectrum [67] and the thrust event shape in e+e collisions [68]. They
have since been applied in many resummed calculations and a variety of di erent contexts (see e.g. refs. [14, 23, 25, 67{78]) and are established as a reliable method to assess the perturbative uncertainty in resummed predictions. Our pro le scales are constructed by considering the relative size of the singular and nonsingular cross section contributions, as discussed in appendix B.2. They are shown in gure 5, where solid curves correspond to the central scale choice and dotted curves correspond to variations that are used to estimate the perturbative uncertainty, as discussed below.
Our procedure for estimating the perturbative uncertainty using pro le scale variations follows ref. [25]. The perturbative uncertainty 0 on the 0-jet cross section is given by
20 = ( [notdef]0)2 + 2resum ; (2.5)
where [notdef]0 reproduces the standard xed-order uncertainties in the limit of large pcutT, whereas the resummation uncertainty resum associated with the jet veto vanishes in the large pcutT region. Both [notdef]0 and resum are estimated via pro le scale variations, shown in gure 5.
The set of pro le variations V[notdef] contributing to [notdef]0 are displayed in the left panel of gure 5. They vary the overall scale by a factor 1=2 and 2 as well as the parameters that control the transition points between resummation and xed-order regions. For each pro le vi in V[notdef] we calculate the 0-jet cross section vi0, from which we obtain [notdef]0 by taking the
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200
ml = 250 GeV
S, S, B
0 0 100 200 300 400 500
pTcut [LBracket1]GeV[RBracket1]
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Figure 5. Pro le functions and their variations used to determine the theory uncertainty, as explained in the text. Left: pro le functions for H; B in blue and for B; S; S in red. Solid lines show the central scale choice, while dotted lines show the variations contributing to [notdef]0 where the yellow shading is between the pro les belonging to the same value of FO. Right: variations of B (green lines and shading) and B; S; S (red lines and yellow shading) contributing to resum.
(symmetrized) envelope,
[notdef]0(pcutT) = max
vi2V[notdef][vextendsingle][vextendsingle]vi
0 (pcutT) central0(pcutT)
[vextendsingle][vextendsingle]
: (2.6)
The pro le scale variations Vresum contributing to resum are shown in the right panel of gure 5. They separately vary each of the beam and soft and scales up and down but keep the hard scale H = FO xed. They thus directly probe the size of the logarithms and the associated resummation uncertainty, while smoothly turning o as the resummation itself is turned o . This yields the following estimate for resum,
resum(pcutT) = max
vi2Vresum[vextendsingle][vextendsingle]vi
0 (pcutT) central0(pcutT)
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[vextendsingle][vextendsingle]
: (2.7)
For additional details on the pro le variations we refer to appendix B.2 and ref. [25].
3 Results
In this section, we discuss our results for the 0-jet cross section, 0, for slepton production at 8 and 13 TeV and discuss the implications on current slepton exclusion limits, using the ATLAS analysis in ref. [5] as a representative example.
3.1 Slepton production at 8 TeV
We start by presenting our 8 TeV results. In gure 6, we show the pcutT dependence of the 0-jet cross section. This allows us to discuss the transition between the resummation and xed-order regions, as well as the perturbative convergence and uncertainties. We consider the implications for the ATLAS exclusion limit in gure 7. The CTEQ6L1 PDFs [79] are used for the plots in this section, to remain consistent with the ATLAS analysis [5]. We show separate results for the direct production of left-handed and right-handed sleptons, focusing on the edge of the 8 TeV exclusion limits [5, 6].
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R (right) production at 8 TeV as a function of the jet veto, pcutT. We compare the results at NLL (green band, dotted line), NLL[prime]+NLO (orange band, solid line), and xed NLO (gray, dashed line), where the bands show the respective perturbative uncertainties.
0.5
Figure 6. The 0-jet cross section for ~
L~
L (left) and ~
R~
2.0
JHEP07(2016)119
3.0
3.0
2.5
2.5
0[LParen1]p Tcut[RParen1][LBracket1]fb[RBracket1]
0[LParen1]p Tcut[RParen1][LBracket1]fb[RBracket1]
2.0
1.5
1.5
1.0
1.0
260 280 300 320 ml [LBracket1]GeV[RBracket1]
0.5
200 220 240 260 280 ml [LBracket1]GeV[RBracket1]
Figure 7. The 0-jet cross section for ~
L~
L (left) and ~
R~
R (right) production as a function of m~[lscript] at 8 TeV. Shown are our NLL (green band, dotted line) and NLL[prime]+NLO (red band, solid line) predictions, as well as the observed 95% CL upper limit on the visible 0-jet cross section, using ATOM (black dotted line) and CheckMATE (black dashed line) to determine the signal region e ciencies. The error bars in the lower panels show the 95% CL exclusion limits obtained from our NLL prediction (green) and NLL[prime]+NLO prediction (red), and for comparison the limit provided by
ATLAS (blue).
Our predictions for the 0-jet cross section at 8 TeV are shown in gure 6 as a function of pcutT for ~
L~
L (left panel) and ~
R~
R (right panel) production. We take m~[lscript] = 250 GeV as a representative value,4 and treat other SUSY particles as decoupled. The predictions are shown at NLO (gray band, dashed line), NLL (green band, dotted line) and NLL[prime]+NLO (red band, solid line). The scale choice for the central value (line) and method for estimating
4For right-handed selectrons or smuons the exclusion limits are 200 GeV, whereas for left-handed
sleptons they are 275 GeV.
{ 11 {
the perturbative uncertainty (band) were discussed in section 2.3 and appendix B.2 for the resummed predictions. For the NLO prediction, we use the xed-order scale FO =
2m~[lscript] = 500 GeV for the central value and estimate the perturbative uncertainty with the ST method [15]. The latter avoids that the naive xed-order scale variations typically underestimate the perturbative uncertainty in the xed-order predictions for small pcutT due to cancellations between perturbative corrections to the total cross section and those related to the jet veto.
In the region pcutT Q, the large logarithms spoil the applicability of the xed-order
perturbative expansion and eventually drive the NLO cross section negative. A jet veto of 20 GeV, as used in the ATLAS analysis [5], sits deep inside this resummation region. We observe that our best prediction at NLL[prime]+NLO is signi cantly lower than the xed
NLO result. On the other hand, xed-order perturbation theory does provide a reliable prediction at large values of pcutT, where the resummation must be turned o . Accordingly, the NLL[prime]+NLO prediction smoothly merges into the NLO result, for which the nonsingular contribution to the cross section is important, as discussed in appendix A.4. We have veri ed that in the limit of large pcutT our NLL[prime]+NLO prediction exactly reproduces the
NLO total cross section of Prospino.5 Comparing the NLL and NLL[prime]+NLO uncertainty bands, we nd that the increased resummation and matching order leads to a substantial reduction of the uncertainties with the NLL[prime]+NLO band fully inside the NLL uncertainty band (except in the xed-order region where the uncertainties match those of the xed-order total cross section).
Next, we investigate the implications of our resummed 0-jet slepton production cross section for the ATLAS exclusion limit [5]. In their results, the visible cross section in signal region a is calculated as
vis = (pp !
~ ) [notdef] (a) ; (3.1) where (a) contains both the reconstruction e ciencies and the acceptance for the cuts of signal region a. They use the total cross section (pp !
~ ) at NLO from Prospino2.1 [38], checked against Resummino [39, 40], and determine (a) using events generated by HERWIG++ v2.5.2 [41] using the CTEQ6L1 PDF set. The resulting vis is then compared to the measured 95% CL upper limit on the visible BSM cross section 95vis in the signal region a.
To compare the 95vis reported by ATLAS to our predictions, we determine the upper limit on the visible 0-jet cross section as
950,vis = 95vis
(anoJV) ; (a) = (anoJV) JV : (3.2)
Here, (anoJV) is the signal region e ciency including reconstruction e ciencies and acceptance cuts but excluding the jet veto cut. In other words, we separate the total signal region e ciency (a) into the product of (anoJV) and the jet veto e ciency JV. Excluding the latter e ectively avoids having to rely on the Monte Carlo to correctly describe the e ect of the jet veto. The resulting 950,vis is now de ned without reconstruction e ciencies and without acceptance cuts other than the jet veto. To model the ATLAS analysis and
5The default value for the xed-order scale in Prospino is m~[lscript], which we changed to 2m~[lscript] for this comparison.
{ 12 {
JHEP07(2016)119
determine the signal region e ciencies, we employ ATOM [36] and CheckMATE [37].6 Using the cut- ow tables provided by ATLAS for m~[lscript] = 250 GeV and m~[notdef]01 = 10 GeV, we validated both the ATOM and CheckMATE results for (anoJV) for the signal regions a = m120T2 and m150T2 of ref. [5] and found agreement at the 5{10% level.
Figure 7 shows the results for 950,vis as a function of the slepton mass (for a neutralino with m~[notdef]01 = 20 GeV), obtained with ATOM (dotted black line) and CheckMATE (dashed
black line). This can be directly compared to our resummed predictions for 0-jet slepton production at NLL (green band, dotted line) and at NLL[prime]+NLO (red band, solid line). Note that we show here the combined cross section for mass degenerate selectrons and smuons, whereas all other plots (except the left panel of gure 1) are for one generation of sleptons.
The ATLAS exclusion limits were determined using the signal region with the highest expected sensitivity, which is m150T2 (m120T2) near the exclusion for left-handed (right-handed)
sleptons around m~[lscript]L 300 GeV (m~[lscript]R 250 GeV). We chose these signal regions in g
ure 7, neglecting the possibility that the signal region with the highest expected sensitivity might change within the plotted range. The intersections of the 950,vis curves with our resummed predictions set our NLL and NLL[prime]+NLO exclusion limits,7 shown by the green and red error bars in the lower panels of the plots. The blue error bars in the lower panels show for comparison the current exclusion limits as quoted by ATLAS, which account for the theory uncertainty on the total cross section (including PDF uncertainties) following ref. [43]. However, this does not include the uncertainty induced by the jet veto, which could easily be as large as our NLL uncertainty, since the perturbative precision of parton showers to model the jet veto is at best NLL.8 At NLL the exclusion limits are noticeably weaker and would go down to 270 GeV for left-handed sleptons and 210 GeV for right-
handed sleptons. Even our NLL[prime]+NLO results (without including PDF+ s uncertainties) yield somewhat larger uncertainties. Encouragingly, the overall central values of our best exclusion limits are similar to those obtained by ATLAS. They agree well in the left panel (~
L~
L) and are slightly lower in the right plot (~
R~
R). However, the overall central values should be treated with some caution as they rely on the signal e ciencies from ATOM and CheckMATE, which have 5{10% uncertainties. To draw any rm conclusions on the nal limits, the experimental analyses would need to provide results for 950,vis or to directly implement our improved theoretical predictions and uncertainties in their interpretations.
3.2 Slepton production at 13 TeV
We continue our discussion with the 0-jet cross section for slepton production at 13 TeV. Following the PDF4LHC recommendations [86], we use the PDF4LHC15 nlo mc PDF set in this section. In gure 8, we show our resummed results for the 0-jet cross section as a
6Both programs use FastJet [80] and utilize the mT2 variable [57, 58, 81, 82]. CheckMATE applies Delphes 3 [83] for detector simulation, whereas ATOM builds on RIVET [84]. A detailed description and validation of ATOM can be found in ref. [85].
7We simply exclude the regions where the calculated 0-jet cross section is larger than the upper limit, 950,vis, without calculating a CLs value.
8Note that the Monte Carlo predictions are reweighted to the total NLO cross section. This is equivalent to rescaling the NLL green band in gure 6 to match the NLO result at large pcutT [greaterorsimilar] 2m~[lscript] and does not
improve the resummation precision.
{ 13 {
JHEP07(2016)119
0.5
pp l
[LParen1]13 TeV[RParen1]
10 100 1000
0.20
10 100 1000
l
0.4
0.15
0[LParen1]p Tcut [RParen1][LBracket1]fb[RBracket1]
0[LParen1]p Tcut [RParen1][LBracket1]fb[RBracket1]
0.3
0.10
0.2
0.05
0.1
0.0
0.00
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pTcut [LBracket1]GeV[RBracket1]
JHEP07(2016)119
R (right) production as a function of pcutT for m~[lscript] = 500 GeV at 13 TeV. The bands show the perturbative uncertainties.
pp l
Figure 8. The 0-jet cross section for ~
L~
L (left) and ~
R~
0.5
0.20
500GeV[RParen1]4 0 [LParen1]p Tcut [RParen1][LBracket1]fb[RBracket1]
0.4
0.15
0.3
500GeV[RParen1]4 0 [LParen1]p Tcut [RParen1][LBracket1]fb[RBracket1]
0.10
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0.05
[LParen1]m l
pTcut = 25 GeV
0.0 300 400 500 600 700 800 900 1000
0.1
Ll
L [LParen1]13 TeV[RParen1]
NLL+NLO
NLL
pTcut = 100 GeV
[LParen1]m l
pTcut = 25 GeV
0.00 300 400 500 600 700 800 900 1000
pp l
Rl
R [LParen1]13 TeV[RParen1]
NLL+NLO
NLL
pTcut = 100 GeV
ml
[LBracket1]GeV[RBracket1]
ml
[LBracket1]GeV[RBracket1]
R (right) production as a function of m~[lscript] for pcutT = 25 GeV and 100 GeV at 13 TeV. Shown are the NLL (green band, dotted line) and
NLL[prime]+NLO (red band, solid line) predictions with their perturbative uncertainty. We multiply the cross section by (m~[lscript]=500 GeV)4 for better visibility.
function of pcutT for m~[lscript] = 500 GeV. Comparing this to the 8 TeV results with m~[lscript] = 250 GeV in gure 6, we observe an increase in the perturbative uncertainties. This is expected due to the higher slepton mass, which leads to larger logarithms in the cross section.
In gure 9, we show the resummed 0-jet cross section as a function of m~[lscript] for pcutT =25 GeV and 100 GeV. The nonsingular contribution is small enough that we can neglect it in this plot.9 The overlap between the NLL and NLL[prime]+NLO bands illustrates again the excellent stability of our resummed calculation.
In gure 10 we focus on the uncertainties, normalizing all results to the central NLL[prime]+NLO result. The 0-jet cross section for left-handed slepton production is shown for pcutT = 25 GeV (left panel) and pcutT = 100 GeV (right panel) at NLL (green band, dotted lines) and NLL[prime]+NLO (red band, solid lines). Furthermore, the yellow band shows the PDF uncertainty of the NLL[prime]+NLO result, obtained using the standard deviation approach
9Even for pcutT = 100 GeV and m~[lscript] = 250 GeV where it is least suppressed the nonsingular correction is only 1%. For larger m~[lscript] and smaller pcutT the nonsingular contribution is signi cantly smaller.
{ 14 {
Figure 9. The 0-jet cross section for ~
L~
L (left) and ~
R~
1.4
1.4
pp l
Ll
pp l
Ll
NLL'+NLO [LParen1]p Tcut [RParen1]
NLL'+NLO [LParen1]p Tcut [RParen1]
1.2
1.2
1.0
1.0
0[LParen1]p Tcut [RParen1] 0
0[LParen1]p Tcut [RParen1] 0
0.8
0.8
0.6
NLLNLL+NLO Dpdf [LParen1]PDF4LHC15[RParen1]
0.6
NLLNLL+NLO Dpdf [LParen1]PDF4LHC15[RParen1]
300 400 500 600 700 800 900 1000
300 400 500 600 700 800 900 1000
ml
[LBracket1]GeV[RBracket1]
ml
[LBracket1]GeV[RBracket1]
Figure 10. The 0-jet cross section for ~
L~
L as a function of m~[lscript] for pcutT = 25 GeV (left) and 100 GeV (right) at 13 TeV. The predictions are normalized to the NLL[prime]+NLO central value. The NLL and
NLL[prime]+NLO perturbative uncertainty are shown by the green and orange band, respectively. The yellow band shows in addition the PDF uncertainty for the NLL[prime]+NLO results, determined following ref. [86].
in ref. [86].10 The perturbative uncertainty is still larger than the PDF uncertainty, so we are not yet limited by the latter, though they become comparable for pcutT = 100 GeV.
In this gure, the increase of the perturbative uncertainty when going to higher slepton masses is clearly visible. For pcutT = 25 GeV the relative NLL uncertainty increases from 24% at m~[lscript] = 300 GeV to 38% at m~[lscript] = 1000 GeV. Going from NLL to NLL[prime]+NLO, we observe a signi cant improvement. The NLL[prime]+NLO uncertainty is roughly a factor of three to four smaller, and increases from 5:8% at m~[lscript] = 300 GeV to 11:2% at m~[lscript] = 1000 GeV.
The corresponding results for ~
R~
R production are very similar. Finally, we note that it is certainly feasible if necessary to further reduce the perturbative uncertainties by going one order higher to NNLL[prime].
4 Conclusions
To maximize their sensitivity, several LHC searches for new physics require a speci c number of signal jets and veto additional jets with transverse momentum above a certain value pcutT, typically around 20-50 GeV. This jet veto introduces large logarithms of pcutT over the scale of new physics in the cross section, which requires resummation to obtain the best possible predictions.
We have presented the rst predictions of a SUSY cross section including the higher-order resummation of jet-veto logarithms. Focusing on slepton (selectron and smuon) production, where a 0-jet sample is selected, we carry out resummation at NLL[prime] order and
10An alternative method to calculate the PDF uncertainties is given in eq. (24) of ref. [86]. Here the uncertainty is determined by reordering the cross sections obtained from the member PDFs and taking the spread between 68% most central ones, which is particularly suitable when the departure from the Gaussian regime is sizeable. We have checked that this method leads to slightly smaller uncertainties in our case. E.g. for pcutT = 25 GeV and m~[lscript] = 600 GeV, the PDF uncertainty obtained from the standard deviation is 4.3%, whereas the PDF uncertainty calculated with the reordering method is 4.0%.
{ 15 {
JHEP07(2016)119
match our resummed results to the NLO cross section. Here we utilize the SCET framework for jet veto resummation developed in Higgs production. Our analysis can also be extended to other new physics processes, including those with nal-state jets (e.g. stop/sbottom production), which however also pose additional challenges due to the additional scales involved.
A central aspect of our study is a systematic and thorough assessment of the theory uncertainty associated with the jet veto, which we estimate using resummation pro le scales. At the low resummation order provided by parton showers, this uncertainty is substantial and not accounted for in current exclusion limits quoted by ATLAS and CMS. The higher-order resummed predictions provide much improved precision and will thus bene t the interpretation of the experimental observations. One possibility to easily utilize these (and future) theoretical improvements, is for the experimental analyses to also provide results for 950,vis.
At the 13 TeV LHC run II the slepton mass reach is expected to increase up to
500 GeV and beyond with 100 fb1 (see e.g. refs. [87, 88]). Our results show that the impact of the jet veto increases further at higher slepton masses, as expected. We provide precise resummed predictions for the 0-jet slepton cross sections at 13 TeV up to slepton masses of 1 TeV. Our predictions are available upon request. We hope that these results will allow the experimental analyses to continue relying on and bene ting from jet vetoes in optimizing the experimental sensitivity to new physics. And once discovered, accurate theory predictions will be important to reveal the nature of any new particle.
Acknowledgments
We thank Kazuki Sakurai for helpful discussions and all the ATOM authors for providing us with a version of their code. We also thank Stefan Liebler and Piotr Pietrulewicz for comments on the manuscript. This work was supported by the German Science Foundation (DFG) through the Emmy-Noether Grant No. TA 867/1-1, by the Netherlands Organization for Scienti c Research (NWO) through a VENI grant, and the D-ITP consortium, a program of the NWO that is funded by the Dutch Ministry of Education, Culture and Science (OCW).
A Fixed-order ingredients
A.1 Hard function
The hard function consists of the Born cross section B and virtual corrections,
Hq q(Q2; mSUSY; ) = B(1 + V ) : (A.1)
The Born cross section for slepton production is (see gure 2(a))
B = 2em 9Q2
{ 16 {
JHEP07(2016)119
1
E2cm
1 4m2~[lscript]sQ2
3/2h~[lscript]s~[lscript]s (A.2)
where the index s = L; R labels the slepton state. The couplings enter in
h~[lscript]s~
[lscript]s = Q2qQ2[lscript] + QqQ[lscript]
(gq+g+q)(g[lscript] sL+g+[lscript] sR) 1 m2Z=Q2
+ (gq2+g+q2)(g[lscript]2 sL+g+[lscript]2 sR)
2(1 m2Z=Q2)2
; (A.3)
where Qq and Q[lscript] are the electric charges, and g[notdef]q; g[notdef][lscript] are the couplings to the Z boson
gf =
I3f sin2 W Qf
sin W cos W ; g+f =
sin W Qf
cos W ; (A.4)
For the one-loop virtual corrections from QCD and SUSY-QCD, which are shown in gures 2(b) and 2(c), we get
V = s( )CF4 (VQCD + VSUSY) + h:c:
VQCD = ln2 [parenleftbigg]
JHEP07(2016)119
Q2
2
[parenrightbigg]
+ 3 ln
Q2 2[parenrightbigg] 8 + 72 6
VSUSY = 1 +
2 m2~g 2m2~q Q2
B0(Q2; m2~q; m2~q) B0(0; m2~g; m2~q)
[bracketrightbig]
+ B0(Q2; m2~q; m2~q)
+ 2m4~g + (Q2 2m2~q) m2~g + m4~qQ2 C0(0; 0; Q2; m2~q; m2~g; m2~q)
B0(0; m2~g; m2~q) + (m2~q m2~g)B[prime]0(0; m2~g; m2~q) ; (A.5) where we have neglected squark mixing. This is in agreement with the expressions in refs. [45, 89{91]. B0 and C0 are the scalar one-loop integrals, for which we use the
LoopTools conventions [92]. Note that VSUSY has no IR divergences in the full theory and hence does not have an explicit dependence and therefore cannot change the anomalous dimensions of the SCET hard function for Drell-Yan.
Since we consider a simpli ed model with heavy squarks and gluinos, the SUSY-QCD corrections are much smaller than the QCD corrections. In our numerical results we choose m~g = m~q = 4 TeV, though the precise value in this region is irrelevant.
A.2 Beam function
The (anti)quark beam function can be computed as a convolution of perturbative matching coe cients, Iqj, and the standard PDFs, fj,
Bq(pcutT; x; ; ) =
Xj
[integraldisplay]
1
dzz Iqj(pcutT; z; ; ) fj[parenleftbigg]
xz ;
: (A.6)
x
The matching coe cients expanded to NLO are
Iqj(pcutT; z; ; ) = qj (1 z) +
s( )
4 I(1)qj(pcutT; z; ; ) + O( 2s) : (A.7)
The rapidity-renormalized O( s) matching coe cients were extracted from the calculations
in ref. [25],
I(1)qq(pcutT; z; ; ) = 2CF
ln
[bracketleftbigg][parenleftbigg]
4 ln
pcutT
Q + 3
(1 z) 2Pqq(z)
[bracketrightbigg]+ Iqq(z)
;
I(1)qg(pcutT; z; ; ) = 2TF [bracketleftbigg]
2 ln
Pqg(z) + Iqg(z)
[bracketrightbigg]
; (A.8)
pcutT
{ 17 {
with
Pqq(z) =
(1 z) 1 z
+(1 + z2) + 32 (1 z)
Pqg(z) = (1 z)
(1 z)2 + z2 Iqq(z) = 1 zIqg(z) = 2z(1 z) : (A.9)
These agree with the results in refs. [22, 93, 94].
A.3 Soft function
The NLO soft function is obtained from ref. [25] using Casimir scaling
Sq q(pcutT; ; ) = 1 + s( )
4 CF
JHEP07(2016)119
8 ln pcutT
ln pcutT 2 ln pcutT
[parenrightbigg] 23[bracketrightbigg]: (A.10)
A.4 Nonsingular contributions
The xed-order cross section can be split into a singular part and a nonsingular part,
FO0(pcutT) = sing0(pcutT) + nons0(pcutT) ; (A.11)
where we suppress the dependence on the SUSY masses for simplicity. The logarithmically enhanced terms in the singular cross section, sing0(pcutT), are contained in the resummed part in eq. (2.1). The nonsingular cross section, nons0(pcutT), contains terms which scale as
O(pcutT=Q) and vanishes for pcutT ! 0. In this section we discuss how to extract nons0(pcutT),
which is essential to reproduce the correct xed-order cross section for large pcutT.
As suggested by eq. (A.11), the NLO nonsingular cross section can be extracted from the full NLO cross section and the NLO singular cross section. We achieve this using
nons0(pcutT) = [integraldisplay]
pcutT
[epsilon1]!0
dpjetT
dFO0dpjetT dsing0 dpjetT
: (A.12)
The left panel of gure 11 shows the NLO results for dFO0=dpjetT (red solid), ds0=dpjetT (blue dashed) and their di erence dnons0=dpjetT (green dotted). We determine the NLO singular cross section, by setting all scales in the NLL[prime] result equal to FO, thus switching o the resummation. The full NLO cross section, di erential in pjetT, is obtained by generating about 3 million events for pp !
~ + j using Madgraph 2.3.2 [56] with a lower cuto on pjetT of 0:2 GeV. For small pjetT a precise cancellation between large values of dFO0=dpjetT and ds0=dpjetT is needed to obtain a reliable result for the nonsingular cross section, see gure 11. This is achieved using a large number of Monte Carlo events and tting the nonsingular to the functional form
dnons0
dpjetT
= a ln pjetT
+ b + c pjetT
ln pjetT 2m~[lscript]
+ d pjetT
; (A.13)
2m~[lscript]
2m~[lscript]
2m~[lscript]
{ 18 {
pp l
Ll
0.001
0.30
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Ll
= 250 GeV
= 250 GeV
perbin[LBracket1]pbGeV[RBracket1]
0.25
10-4
nons [LBracket1]fb[RBracket1]
0.20
singular
nonsingular
0 100 200 300 400 500
10-5
full NLO
0.15
0
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0.05
0.00 0 100 200 300 400 500
pTjet [LBracket1]GeV[RBracket1]
pTcut [LBracket1]GeV[RBracket1]
JHEP07(2016)119
Figure 11. Left: singular (blue dashed) and nonsingular (green dotted) contributions to the full NLO (red solid) di erential cross section for ~
L~
L production. Right: the (integrated) nonsingular
cross section for ~
L~
L at NLO.
which has the correct leading behavior for the di erential spectrum for pjetT ! 0. In this t all points with pjetT < x are included, where the default is x = 2m~[lscript]. As an important cross check, we ensure that the tted result is stable under varying x. The left panel of gure 11 shows that for pjetT [greaterorsimilar] m~[lscript], the nonsingular contributions are of the same size as the singular contributions, requiring their inclusion to correctly reproduce the full xed-order cross section. Our nal results for the NLO nons0(pcutT) can be seen in the right panel of gure 11. The band indicates the perturbative uncertainty, and is obtained by calculating the nonsingular terms three times, evaluating the ingredients at FO = m~[lscript]; 2m~[lscript] and 4m~[lscript].
The nonsingular for right-handed slepton production is obtained in the same manner.
B RGE ingredients
As explained in section 2.1, the resummation of large logarithms is achieved in SCET by rst evaluating the functions in the factorized cross section eq. (2.1) at their natural virtuality ( H; B; S) and rapidity ( B; S) scales, and then RG evolving them to (arbitrary)
common scales and . Writing this evolution out explicitly, eq. (2.1) for inclusive slepton production becomes
0(pcutT; mSUSY) =
[integraldisplay]
dQ2 dY Hq q(Q2; mSUSY; H)
[notdef] Bq(pcutT; xa; B; B) B q(pcutT; xb; B; B) Sq q(pcutT; S; S)
[notdef] U0(pcutT; Q2; H; B; S; B; S) + nons0(pcutT; mSUSY) : (B.1)
At NLL[prime] (NLL) order, we have to include the NLO (LO) results for the hard, beam and soft functions, given in appendix A. The evolution factor U0 is given by the product of the individual evolution factors that evolve each of the functions from their natural scale to
{ 19 {
the common scales and ,
U0(pcutT; Q2; H; B; S; B; S) =
[vextendsingle][vextendsingle][vextendsingle][vextendsingle]exp
[bracketleftbigg][integraldisplay]
[notdef] [notdef] [prime] qH(Q2; [prime])[bracketrightbigg][vextendsingle][vextendsingle][vextendsingle][vextendsingle]
d [prime]
2
[notdef] exp
[bracketleftbigg][integraldisplay]
d [prime]
[prime] 2 qB(Q; [prime]; )[bracketrightbigg]
d [prime]
exp
[bracketleftbigg][integraldisplay]
[notdef] [prime] qS( [prime]; )[bracketrightbigg]
ln B q (pcutT; B) + ln S q (pcutT; S)[bracketrightbigg]: (B.2)
The anomalous dimensions entering here are collected in the next subsection. Note that due to RGE consistency the dependence on the arbitrary scales and exactly cancels between the di erent factors in eq. (B.2).
B.1 Anomalous dimensions
The anomalous dimension of the hard, beam, and soft functions that enter in the evolution kernel in eq. (B.2) have the following general structure [20, 89]
qH(Q2; ) = qcusp[ s( )] ln
[notdef] exp
JHEP07(2016)119
Q2
2 + qH[ s( )] ;
qB(Q; ; ) = 2 qcusp[ s( )] ln
Q + qB[ s( )] ;
+ qS[ s( )] ;
q (pcutT; ) = 4 q (pcutT; ) + q [ s(pcutT)] ; (B.3) where the exact path-independence of the evolution in ( ; ) space [54] is ensured by
q ( 0; ) = [integraldisplay]
qS( ; ) = 4 qcusp[ s( )] ln
d [prime]
[notdef] [prime] qcusp[ s( [prime])] : (B.4)
The exact independence of the cross section is equivalent to the RG consistency relation
2 qH(Q2; ) + 2 qB(Q; ; ) + qS( ; ) = 0 : (B.5)
We give the cusp and noncusp anomalous dimensions in terms of an expansion in s,
qcusp( s) =
1
Xn=0 qn
s 4
n+1; qX( s) =
1
Xn=0 qX n
s 4
n+1: (B.6)
At NLL (and NLL[prime]) we require the one-loop noncusp anomalous dimensions qX 0 and the two-loop cusp anomalous dimension, q0, q1, as well as the two-loop running for s. At
NNLL we would need each at one order higher, which we also give below.The coe cients for the cusp anomalous dimension are [95, 96]
q0 = 4CF ;
q1 = 4CF
[bracketleftbigg][parenleftbigg]
67
9
2 3
CA 209 TF nf
[bracketrightbigg];
[bracketleftbigg][parenleftbigg]
245
1342
114
q2 = 4CF
6
27 +
45 +
22 3
3
C2A +
41827 +40227 56 3 3
CA TF nf
+
553 + 16 3
CF TF nf 1627 T 2F n2f[bracketrightbigg]: (B.7)
{ 20 {
The hard noncusp anomalous dimension is those of the quark form factor [97, 98]. The noncusp anomalous dimension coe cients for the soft function and rapidity evolution follow from ref. [25] using Casimir scaling, and those for the beam function then follow from the consistency relation in eq. (B.5). This leads to
qH 0 = 6CF ;
qH 1 = CF [bracketleftbigg][parenleftbigg]
82
9 52 3
CA + (3 42 + 48 3)CF + [parenleftbigg]659 + 2[parenrightbigg] 0[bracketrightbigg];
qB 0 = 6CF ;
qB 1 = CF
(3 42 + 48 3)CF + (14 + 16(1 + 2) ln 2 96 3)CA
+
JHEP07(2016)119
193 432 +803 ln 2
0[bracketrightbigg];
qS 0 = 0 ;
qS 1 = 8CF
[bracketleftbigg][parenleftbigg]
52
9 4(1 + 2) ln 2 + 11 3[parenrightbigg]
CA +
29 +7212 203 ln 2
0[bracketrightbigg];
q 0 = 0 ;
q 1 = 16CF [bracketleftbigg][parenleftbigg]
0[bracketrightbigg]+ C2(R) ; (B.8)
where C2(R) = 16CF CA(2:49 ln R20:49)+O(R2) denotes the clustering correction from
the jet algorithm [25]. For completeness, in our convention we have
0 = 11
3 CA
43 TF nf ; CA = Nc ; CF =
17
9 (1 + 2) ln 2 + 3[parenrightbigg]
CA +
49 +212 53 ln 2
1
2 ; (B.9)
where Nc = 3 is the number of colors and nf = 5 is the number of active quark avors.
B.2 Pro les scales
In this appendix we give the expressions for the scales H; B; S and B; S employed for our central value and uncertainty estimate. A discussion of our pT -dependent pro le scales is given in section 2.3, and includes plots and our procedure for estimating the perturbative uncertainty.
At small values of pcutT the full NLO cross section is governed by the singular cross section containing the logarithmic terms which need to be resummed; see the left panel of gure 11 and its discussion. From the anomalous dimensions in eq. (B.3) we can read o the canonical scales already given in eq. (2.4) for which the logarithms in the functions are minimized,
H = 2m~[lscript] Q ; B = pcutT ; B = 2m~[lscript] Q ;
S = pcutT ; S = pcutT : (B.10)
These are the appropriate scale choices in the resummation region.
{ 21 {
N2c 1
2Nc =
43 ; TF =
At large values pcutT Q, singular and nonsingular contributions are of similar size
and there are large cancellations between them. This can be observed in the left panel of gure 11, where for pcutT [greaterorsimilar] 300 GeV the singular and nonsingular contributions have larger magnitudes (and opposite signs) than the full result. To reproduce this cancellation and thus the xed-order result, resummation must be turned o at this point. This is achieved by evaluating all functions in the factorized cross section at a common xed-order scale
H = B = S = B = S = FO = 2m~[lscript]; (B.11)
which is also the scale used for the nonsingular corrections. The value FO = 2m~[lscript] Q is chosen to agree with the value of H used at small pcutT. In the intermediate region, both resummation and xed order terms are relevant. In this region, the scales are chosen to smoothly interpolate between the resummation region at small pcutT values and the xed-order region at large pcutT values.
We follow ref. [25] and choose our (central) pro le scales according to
H = B = FO ;
B = S = S = FO [notdef] frun pcutT=(2m~[lscript]) [parenrightbig]
1 + (x=x0)2=4[bracketrightbig]x 2x0 nonperturbative region x 2x0 x x1 resummation region
x + (2x2x3)(xx1)
2
2(x2x1)(x3x1) x1 x x2 transition from resummation
1 (2x1x2)(xx3)
2
2(x3x1)(x3x2) x2 x x3 transition to xed order 1 x3 x xed-order region
The values for x1, x2, x3 determine where the transition from resummation to xed-order region happens. They are chosen as
{x1; x2; x3[notdef] = [notdef]0:15; 0:4; 0:65[notdef] (B.14)
by considering the relative size of the singular and nonsingular terms in gure 11. Below x1 we have exact canonical running, eq. (B.10), while above x3 the resummation is fully turned o . For 2m~[lscript] = 500 GeV this corresponds to [notdef]75, 200, 325[notdef] GeV. In addition we
choose x0 = 2:5 GeV= FO. The resulting central scales are shown as solid blue ( H; B) and red ( B; S; S) lines in gure 5.
To estimate the perturbative uncertainties in the resummed prediction, variations of the pro le scales are considered, as discussed in section 2.3. Here we very brie y summarize the variations; more details on their derivation can be found in ref. [25]. The set of variations V[notdef] determining [notdef]0 has 14 pro le scale variations, which are all possible combinations of
1. an overall up and down variation of the xed-order scale FO by factors of 2 and 1/2,
{ 22 {
; (B.12)
with
frun(x) =
JHEP07(2016)119
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
x0
(B.13)
2. four variations for the transition points x1; x2; x3
{x1; x2; x3[notdef] : [notdef]0:1; 0:3; 0:5[notdef] ; [notdef]0:2; 0:5; 0:8[notdef] ; [notdef]0:04; 0:4; 0:8[notdef] ; [notdef]0:2; 0:35; 0:5[notdef] : (B.15)
The set of variations Vresum of B; S; B, S determining resum are combinations of
upi(pcutT) = centrali(pcutT) [notdef] fvary pcutT=(2m~[lscript]) [parenrightbig]
downi(pcutT) = centrali(pcutT) = fvary pcutT=(2m~[lscript])
upi(pcutT) = centrali(pcutT) [notdef] fvary pcutT=(2m~[lscript])
downi(pcutT) = centrali(pcutT) = fvary pcutT=(2m~[lscript])
The multiplicative variation factor is de ned as
fvary(x) =
;
[parenrightbig]
;
[parenrightbig]
[parenrightbig]
;
: (B.16)
JHEP07(2016)119
8
>
>
<
>
>
:
2(1 x2=x23) 0 x x3=2 ;
1 + 2(1 x=x3)2 x3=2 x x3 ; 1 x3 x ;
(B.17)
which approaches a factor of 2 for pcutT ! 0 and turns o for x ! x3. Out of the 80 possible
combinations of variations, all combinations leading to arguments of logarithms which are more then a factor of 2 di erent from their central values are not considered. This leaves a total of 35 pro le scale variations in Vresum.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
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SISSA, Trieste, Italy 2016
Abstract
Abstract
Several searches for new physics at the LHC require a fixed number of signal jets, vetoing events with additional jets from QCD radiation. As the probed scale of new physics gets much larger than the jet-veto scale, such jet vetoes strongly impact the QCD perturbative series, causing nontrivial theoretical uncertainties. We consider slepton pair production with 0 signal jets, for which we perform the resummation of jet-veto logarithms and study its impact. Currently, the experimental exclusion limits take the jet-veto cut into account by extrapolating to the inclusive cross section using parton shower Monte Carlos. Our results indicate that the associated theoretical uncertainties can be large, and when taken into account have a sizeable impact already on present exclusion limits. This is improved by performing the resummation to higher order, which allows us to obtain accurate predictions even for high slepton masses. For the interpretation of the experimental results to benefit from improved theory predictions, it would be useful for the experimental analyses to also provide limits on the unfolded visible 0-jet cross section.
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