The Zγ transverse-momentum spectrum at NNLO+N3LL

The

Z

γ

transverse-momentum

spectrum

at

NNLO+N

3

LL

Marius Wiesemann

a

,

∗

,

Luca Rottoli

b

,

c

,

Paolo Torrielli

d

a_{Max-Planck-InstitutfürPhysik,FöhringerRing6,80805München,Germany}

b_{DipartimentodiFisicaG.Occhialini,UniversitàdegliStudidiMilano-BicoccaandINFN,PiazzadellaScienza3,20126Milano,Italy} c_{ErnestOrlandoLawrenceBerkeleyNationalLaboratory,UniversityofCalifornia,Berkeley,CA94720,USA}

d_{DipartimentodiFisicaandArnold-ReggeCenter,UniversitàdiTorinoandINFN,SezionediTorino,ViaP.Giuria1,I-10125,Turin,Italy}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received 22 June 2020

Received in revised form 1 August 2020 Accepted 20 August 2020

Available online 25 August 2020 Editor: G.F. Giudice

We considerthe transverse-momentum (pT)distributionof Zγ pairsproduced inhadroniccollisions.

Logarithmically enhanced contributionsat small pT are resummed toall ordersin QCDperturbation

theoryandcombinedwiththefixed-orderprediction.Weachievethemostadvancedpredictionforthe

Zγ pT spectrum bymatching next-to-next-to-next-to-leadinglogarithmic (N3LL) resummationtothe

integrated crosssectionatnext-to-next-to-leadingorder(NNLO). Byconsidering +−

γ

productionat the fullydifferentiallevel,includingspincorrelations,interferencesandoff-shell effects,arbitrarycuts canbeappliedtotheleptonsandthephoton.WepresentresultsattheLHCinpresenceoffiducialcuts andfindagreementwiththe13 TeVATLASdataatthefew-percentlevel.

©2020TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

Vector-boson pair production processes are an integral part of the rich physics programme at the Large Hadron Collider (LHC). They play a crucial role in both precision measurements of Standard-Model (SM) rates and the search for new-physics phenomena. In particular, production processes of neutral vector bosons,like Z

γ

production,provide veryclean experimental sig-naturesinthe Z

→ 

+



−decaychannels,sincethefinal statecan befullyreconstructed.Theirpureexperimentalsignaturesand rel-ativelylargecross sectionsrenderthem well suitedto searchfor anomalous couplings. For instance, the measurement of a non-zero Z Z

γ

coupling, which is absent in the SM, would be direct evidence of physics beyond the SM (BSM). Z

γ

production con-tributesalsoasirreduciblebackgroundtodirectsearchesforBSM resonancesandtoHiggsbosonmeasurements,seee.g.Ref. [1]. Al-though the decay into a Z

γ

pair of the Higgs boson is a rare loop-inducedprocess intheSM,new-physicsextensionsmay sig-nificantlyenhancethisdecaychannel.

Thepreciseknowledgeofratesanddistributionsin Z

γ

produc-tionprovides a strong test ofthe gauge structure ofelectroweak (EW)interactions andthemechanism ofEWsymmetry breaking. Measurementsof Z

γ

productionhavebeencarriedoutattheLHC at7 TeV [2–7], 8 TeV [8–11], and 13 TeV [12,13]. The latest mea-surementofRef. [13] is the first dibosonanalysis touse the full

*

Corresponding author.

E-mailaddresses:marius.wiesemann@cern.ch(M. Wiesemann), luca.rottoli@unimib.it(L. Rottoli), torriell@to.infn.it(P. Torrielli).

Run IIdatasetandachievesremarkablysmallexperimental uncer-tainties.

To matchthe precision achieved by the experimentsa signif-icant effort hasbeenmadeto advancetheoretical predictions for Z

γ

productioninthepastyears.The next-to-leadingorder(NLO) QCD cross section has been known for some time both for on-shell Z bosons[14] and includingtheir leptonic decays [15].The loop induced gluon-fusioncontribution was thefirst contribution to thenext-to-next-to-leadingorder(NNLO) QCD crosssection to be computed [16–18]. InRef. [19] the NLO crosssection, includ-ing photon radiationoff theleptons, andthe loop-inducedgluon fusioncontributionwerecombined.ThecompleteNNLOQCD cor-rections to



+



−

γ

productionat the fullydifferential level were firstcalculatedinRefs. [20,21] andlaterconfirmedbyan indepen-dentcalculation[22].Electroweak(EW)correctionswerepresented inRefs. [23,24].

Twodifferentmechanismsarerelevanttoproduceisolated pho-tonsinthefinalstate:aperturbativeonethroughdirect production in the underlying hard subprocess, and a non-perturbative one throughfragmentation ofaquarkoragluon.Thelatterproduction mechanism requires the knowledge of the respective fragmenta-tion functions to absorb singularities related to collinear photon emissions,andthosefunctionsaredeterminedfromdatawith rel-ativelylarge uncertainties.In experimental analysesthe fragmen-tation component is typically suppressed by the criteriaused to isolate photons. On the theoretical side, the separation between thetwoproductionmechanismsisdelicate,assharplyisolatingthe https://doi.org/10.1016/j.physletb.2020.135718

0370-2693/©2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by

Fig. 1. Feynman

diagrams for the production of two charged leptons and a photon: (a-b) sample tree-level diagrams in the quark-annihilation channel contributing at LO; (c)

sample loop-induced diagram in the gluon-fusion channel contributing at NNLO.

photonfromthepartonswouldspoilinfrared(IR)safety. Remark-ably,byexploitingFrixionesmooth-conephotonisolation [25] the fragmentationcomponentcanbecompletelyremovedinanIR-safe manner,whichhasthefurtheradvantageofsubstantially simplify-ingtheoreticalcalculationsofphotonprocessesbeyondtheleading order(LO).Experimentally,thefinitegranularityofthecalorimeter preventsacompleteimplementationofthesmooth-coneisolation. Asa consequence,experimental analysesrelyinstead onisolation criteriawith a fixed cone. To facilitatedata–theory comparisons, the smooth-cone parameters are typically tuned in comparisons with calculations including fragmentation functions in order to mimic the fixed-cone isolation criteria of the experiments, see e.g. Ref. [26]. Dueto thelargescale separationbetweenthe pho-ton energy and the hadronic energy within the isolation cone, thepresenceofisolationcutsinducespotentiallylarge non-global (NG) logarithms, whose resummation is known up to leading-logarithmic(LL)accuracy[27,28] (seealsoRef. [29]).

Inthispaperweconsiderthetransverse-momentum (pT) dis-tributionof Z

γ

pairs.Thisdistributionisamongthemost impor-tantdifferentialobservablesin Z

γ

production,andithasrecently beenmeasured ata precision of a few percent by usingthe full Run II data set [13]. For the first time, we perform transverse-momentum resummationof Z

γ

pairs at next-to-next-to-next-to-leading logarithmic (N3LL) accuracy and match it to the NNLO integrated cross section. To this end, we calculate the process pp

→ 

+



−

γ

with off-shelleffects andspin correlations by con-sistently including all resonant and non-resonant topologies. Our computationisfullydifferentialinthemomentaofthefinal-state leptonsandthephoton,whichallowsustoapplyarbitraryfiducial cuts.

Weemploythe Matrix+RadISH interface[30],whichcombines NNLO calculations within Matrix [31,32] with the RadISH re-summationformalismofRefs. [33–35].Alltree-levelandone-loop amplitudesare evaluated with OpenLoops2 [36–38].At two-loop level we use the qq

¯

→

V

γ

amplitudes of Ref. [39]. NNLO accu-racy is achieved by a fully general implementation of the qT -subtractionformalism [40] within Matrix. TheNLO parts therein (for Z

γ

and Z

γ

+1-jet) are calculated by Munich1 _{[43], which} uses the Catani–Seymour dipole subtractionmethod [44,45]. The MatrixframeworkfeaturesNNLOQCDcorrectionstoalarge num-ber ofcolour-singletprocesses athadron colliders. Ithas already beenusedtoobtainseveralstate-of-the-artNNLOQCDpredictions [20,21,46–52],2 _{andformassivedibosonprocessesithasbeen} ex-tendedtocombineNNLOQCDwithNLOEW corrections[58] and withNLOQCDcorrectionstotheloop-inducedgluonfusion contri-bution[59,60].Throughtherecentlyimplemented Matrix+RadISH interface[30] itisnowalsopossibletodealwiththeresummation oftransverseobservablessuchasthetransversemomentumofthe colour-singletfinalstate.

1 _{The Monte Carlo program Munich features a general implementation of an}

ef-ficient, multi-channel based phase-space integration and computes both NLO QCD and NLO EW [41,42] corrections to arbitrary SM processes.

2 _{It was also used in the NNLO+NNLL computation of Ref. [53], and in the NNLOPS}

computations of Refs. [54–57].

Weconsidertheprocess

pp

→ 

+



−

γ

+

X

for massless leptons



∈ {

e

,

μ

}

.Although ourcalculation also ap-plies to theprocess pp

→

ν

ν γ

¯

+

X , we donot consider it here, as the transverse momentum of the Z

γ

pair in that case can-not be experimentally reconstructed. Representative LOdiagrams are shown in Fig. 1(a-b). They are driven by quark annihilation inthe initialstate andinvolvesingle-resonantt-channel Z

γ

pro-duction (panel (a)) andsingle-resonant s-channel Drell–Yan(DY) topologies (panel(b)).Fig.1(c)showsaloop-induceddiagramthat is drivenby gluon fusionin theinitial state andentersthe cross sectionatNNLO.Theloop-inducedgluon-fusioncontributionis ef-fectively onlyLO accurate and hasBorn kinematics. Therefore, it contributestriviallytotheZ

γ

transverse-momentum(pT,γ ) dis-tribution. Furthermore,itscontribution israthersmall, beingless than 10% of theNNLO corrections andwell below1% of the full Z

γ

crosssectionatNNLO[31].Wethusrefrainfromincludingthe loop-inducedgluon-fusioncontributioninourcalculation.

The perturbative description ofthe Z

γ

transverse-momentum spectrum atfixed orderbreaksdown inkinematicregimes domi-natedbysoftandcollinearQCDradiation,i.e.atsmallpT,γ ,due tothepresenceoflargelogarithmsL

=

ln

(

mγ

/

pT,γ

)

,withmγ beingtheinvariantmassoftheZ

γ

pair.Overthelastfourdecades, avarietyofformalismshasbeendevelopedtoperformthe resum-mation of large logarithmic contributions in the transverse mo-mentum pT ofcolour-singletprocesses [33,34,61–70].Weemploy the RadISH formalismofRefs. [33–35] toresumtherelevant loga-rithmictermstoallorders.Thelogarithmicaccuracyiscustomarily definedintermsofthelogarithm ofthecumulativecrosssection ln

σ

(

pT

)

. The dominant terms

α

_SnLn+1 are referred to as leading logarithmic, terms of

α

n

SLn as next-to-leading logarithmic (NLL),

terms of

α

n

SLn−1 as next-to-next-to-leading logarithmic (NNLL),

andsoon.Weperform theresummationofthe Z

γ

pT spectrum up to N3LL basedon theformulæ presentedin Ref. [34].The re-summation formalismhasbeenimplemented inthe RadISH code forHiggsandDrell-Yanproduction.Theapplicationto more com-plex colour-singletprocesses, suchas Z

γ

production,isachieved throughthe Matrix+RadISH interface[30].WenotethattheLL re-summation of the loop-induced gluon-fusion contribution to the Z

γ

crosssectionisformallyofthesameorderasN3_{LLcorrections}

totheqq channel,

¯

andthatbothcontributionscanbetreated com-pletely independently. Theproper treatmentof theformerwould require to go beyond an effective LO+LL accuracy, by combining NLO QCD corrections to the loop-induced gluon-fusion contribu-tion with NNLL resummation. Given its small numerical impact, weleavesuchstudyforfuturework.

In order for the theoretical prediction to be reliable over the entirespectrum,theresummationoflargelogarithmsatsmallpT mustbecombinedwiththefixed-ordercrosssection,validathigh pT.WeconsistentlymatchN3LLresummationforthepT,γ spec-trumwithNNLO correctionsatthe levelofcumulativecross sec-tion,definedas(

κ

=NNLO,N3_LL)

pT,1≥30 GeV, pT,2≥25 GeV, |η| ≤2.47, m≥40 GeV,

pT,γ≥30 GeV, |ηγ| ≤2.37, m+mγ≥182 GeV, Rγ>0.4,

Frixione isolation with n=2,δ0=0.1, and=0.1, pcone0T .2/pT,γ<0.07.

σ

κ

(

pvetoT,γ

)

≡

pveto T,γ



0 dpT,γ d

σ

κ

(

pT,γ

)

dpT,γ

.

(1)

Thecrosssectionsshouldbeunderstoodasbeingfullydifferential intheBornphasespace,whichallowsustoapplyarbitraryIR-safe cutsonthekinematicsoftheleptonsandthephoton.

Thereisacertainleveloffreedomwhendefiningmatching pro-cedures that differ from one another only by terms beyond the formalaccuracyofthecalculation.Westudytwodifferent match-ingschemes.Thefirstschemeweconsiderisacustomaryadditive scheme,whichatNNLO+N3LLisdefinedas

σ

add.match.

NNLO+N3_LL

(

pvetoT,γ

)

=

σ

NNLO

(

pvetoT,γ

)

−



σ

_N3_LL

(

pveto_T,γ

)



NNLO

+

σ

_N3_LL

(

pveto_T,γ

) .

(2)

The notation

[. . .]

_Nk_LO is used to indicate that theexpression

in-side thebracketis expandedin

αS

andtruncatedatNk_{LO. Thus,} thesecond termcorresponds to theexpansion of the resummed cumulative cross section

σ

N3_LL

(

pveto_T,γ

)

up to NNLO, i.e.

O(

α

2_S

)

, whichsubtractsalllogarithmicallyenhancedcontributionsatsmall pveto_{T,γ from}thefixed-ordercomponent. Thistermisnecessaryto renderEq. (2) finite in the p_Tveto_,γ

→

0 limit andto remove the doublecountingbetweenthefirstandthethirdterm.

Thesecondschemeweconsiderisamultiplicativescheme [71,

72],definedas

σ

_NNLOmult.₊match_N3_LL.

(

pvetoT,γ

)

=

σ

N3LL

(

p veto T,γ

)

σ

asym. N3_LL

⎡

⎢

⎣

σ

_Nasym3_LL.

σ

NNLO

(

pvetoT,γ

)



σ

_N3_LL

(

pveto_T,γ

)



NNLO

⎤

⎥

⎦

NNLO

,

(3)

where

σ

_Nasym3_LL. is the asymptotic (pvetoT,γ

→ ∞

) limit of the re-summedcrosssection. Inthe limit pveto_T,γ

→

0,Eq. (3) yieldsthe resummed prediction, while for pveto_T,γ

→ ∞

it reproduces the fixed-order result. The detailed matching formulæ forthe multi-plicative scheme are reported in appendix A of ref. [71]. Since inboth matching schemes the cumulative crosssection tends to

σ

NNLOwhen pvetoT,γ

→ ∞

,byconstructionthedifferential distribu-tionfulfilstheunitarityconstraint,i.e.itsintegralyieldstheNNLO crosssection.

WepresentpredictionsfortheLHCat13 TeV.The EW param-eters are evaluated through the Gμ scheme by setting the EW coupling to

α

=

√

2 GFm2W



1

−

m2_W

/

m2_Z



/

π

andthe mixing an-gle to cos

θ

_W2

= (

m2_W

−

i



WmW

)/(

m2Z

−

i



ZmZ

)

, employing the complex-massscheme [73] throughout. We choose the PDG [74] values for the input parameters: GF

=

1

.

16639

×

10−5GeV−2, mW

=

80

.

385 GeV,



W

=

2

.

0854 GeV, mZ

=

91

.

1876 GeV,



Z

=

2

.

4952 GeV.Foreachperturbativeorderweusethecorresponding setof Nf

=

5 NNPDF3.0 [75] partondistributions with

αS

(

mZ

)

=

0

.

118.The renormalizationscale (

μ

R) andthefactorization scale (

μ

F)arechosendynamicallyas

μ

R

=

μ

F

=

μ

0

≡

m2_

+

p2_T,γ

,

(4)

whiletheresummationscale( Q )issetto

Q

=

Q0

≡

1

2mγ

.

(5)

Uncertainties from missing higher-order contributions are esti-matedfromcustomary7-pointrenormalization- and factorization-scale variations by a factor of two around

μ0

for Q

=

Q0 with

the constraint 0

.

5

≤

μ

R

/

μ

F

≤

2, and by varying Q by a factor oftwo around Q0 for

μ

F

=

μ

R

=

μ

0.The totalscale uncertainty

is evaluatedas theenvelopeof theresulting nine variations.The resummation is turned off at high pT,γ by means of modified logarithms asdefinedinRef. [34], withexponent p

=

4. Wehave checkedthatourpredictionshaveanegligibledependenceonthe valueofp.Non-perturbativecorrectionshavenotbeenincludedin ourresults.

We study predictions for the pT,γ distribution in two se-tups that involve different phase-spaceselection cuts, defined in Table 1: The first is a loose selection that solely aims at pre-ventingQEDsingularities,includingtransverse-momentumand ra-pidity requirements for the photon, a lower invariant mass cut on the lepton–photon system, a Z -mass window for the lepton pair, and Frixione smooth-cone isolation [25]. This setup will be referred to as“inclusive”in thefollowing. Thesecond setup cor-responds to the fiducial selection of the 13 TeV ATLAS analysis of Ref. [13], which uses a tighter requirement forthe transverse momentumofthephoton,alowerinvariant-masscut onthe lep-ton pair,transverse-momentumandrapidity requirementsonthe leading andsubleadinglepton,a lower boundforthe sumofthe invariantmassesoftheleptonpairandthe



γ

system,alepton– photon separation in



R

=



φ

2

₊

_η

2_{,anda two-foldphoton}

isolation: in addition to a Frixione isolation with a rather small cone,thetransverseenergyofhadronscollimatedwiththephoton isrequirednottoexceedasmallfractionofitstransverse momen-tum.Inourparton-levelcalculationwedefine pcone0.2

T asthesum

ofthetransversemomentaofallpartonswithinaconeofR

=

0

.

2 aroundthephoton.Thesecondsetupisreferredtoas“fiducial”in thefollowing.Oneshouldbearinmindthatsuchisolationcriteria induceNGlogarithmiccorrections,whichwedonotresuminour formalism. We will estimate their effect on the pT,γ spectrum below.

Westartthediscussionofourresultsbycomparingthe expan-sion ofthe resummation withthe fixed-order spectrum at small pT,γ in Fig.2,whichprovidesa strongcheckofourcalculation. The plots demonstrateata remarkable precision that the expan-sionoftheresummed crosssection matchesthefixed-ordercross section atsmalltransverse momentaboth fortheinclusivesetup inpanel(a)and(b)andthefiducialsetupinpanel(c)and(d).As canbeseenfromthelowerframeinpanel(a)and(c),therelative difference



rel betweenthe NNLOdistributionandthe NNLO

Fig. 2. Panel

(a) and (c): transverse-momentum spectrum of the

Zγpair at NLO (purple, dot-dashed) and NNLO (orange, dashed), and the expansion of the NLL (blue, dotted)

and N3_{LL (red, solid) cross section. The lower frame shows the relative difference between the fixed-order cross section and the expansion, normalized to the latter. Panel (b)}

and (d): the upper frame shows the difference at the cumulative level between NLO and NLL expansion (blue, dotted), and between NNLO and N3_{LL expansion (red, solid).}

The lower frame shows the same results for the derivative of the cumulative cross section with respect to ln(pT,γ/GeV).

curve)vanishes downto pT,γ

=

0

.

01 GeVwithin the numerical errors atthe permille level.In theupper frame ofpanel (b) and (d)we show thedifference ofthe NNLO crosssection andNNLO expansion ofthe N3LLcross section at the cumulative level (red solidcurve).Sincetheir differencetendstozeroatlowtransverse momenta,alsoconstanttermsinpT,γ matchbetweenNNLOand N3LL.Thefactthat atNLOthedifference withtheNLLexpansion (bluedotted curve)tendsto aconstant differentfromzerois ex-pected, since our NLL result does not include the constant NLO termsinpT,γ .Finally,thelowerframeinpanel(b)and(d)shows thattheabsolutedifferencebetweenthefixed-orderresultandthe expansionoftheresummationaftertakingthederivativeofthe cu-mulativecrosssectionswithrespecttoln

(

pT,γ

/

GeV

)

yieldszero withinnumericaluncertainties atsmalltransversemomenta.This indicates that all logarithmic terms in pT,γ are correctly pre-dicted.Notonlydothesecomparisonsprovidestringentchecksof

the validity of our calculation, butthey also show the excellent precisionthatournumericalframeworkcanachieve.

Wefurthernoticefromtheseplotsthatthelogarithmically en-hancedcontributions becomedominantovertheregular termsat smaller values of transverse momentum compared to other pro-cesses (cf. Ref. [30] for instance).Especially inthe fiducial setup, regular contributions become non-negligible already at pT,γ

∼

1GeV.3 Indeed,it has been shownbefore [31,76] that processes withidentifiedphotonsinthefinal statereceiveratherlarge cor-rectionsfrompower-suppressedtermsatsmalltransverse

momen-3 _{We notice that the fiducial cuts used in Ref. [13] enhance u- andt-channel}

production modes, see Fig.1(a), while suppressing the s-channel

contribution, see

Fig.1(b). Therefore, the fiducial setup enhances power corrections stemming from

Fig. 3. Panel

(a) and (b):

pT,γ spectrum at NLO (black, dotted), NLL (brown, dash-double-dotted), and NLO+NLL (magenta, dash-dotted) in the small-pT,γ (panel (a))

and large-pT,γ (panel (b)) region. The lower frames show the ratio to the central NLO+NLL prediction. Panel (c) and (d): pT,γ spectrum at NNLO (red, dashed), N3LL

(green, double-dash-dotted), and NNLO+N3_{LL (blue, solid) in the small-p}

T,γ (panel (a)) and large-pT,γ (panel (b)) region. The lower frames show the ratio to the central

NNLO+N3_{LL prediction.}

tum.InthemultiplicativeschemeEq. (3) thosearesuppressedby

σ

_N3_LL

(

pveto_T,γ

)

at small pT,γ . Although such effects are beyond the nominal accuracy, this suppression may induce numerically relevantcorrections,in particularinthefiducialsetup considered here. Thisbehaviour is undesirable, since onthe one handthese powercorrectionsare agenuine non-singularcontribution tothe crosssection, andontheotherhandthesuppressionofthelatter inducedby Eq. (3) isnot theonedictatedby anall-order resum-mationofsuchnon-singularterms.Forthisreasonwebelievethat themultiplicative schemeinEq. (3), although formallycorrect, is notidealwhenthefixed-ordercrosssectionfeatureslarge power-suppressedcorrections, andconsequently we choose the additive schemeinEq. (2) asthedefaultthroughoutthispaper.4

4 _{We point out that, although the multiplicative scheme in Eq. (3) (which is the}

default in Matrix+RadISH) is suboptimal with respect to the additive scheme in Eq. (2) in cases where large power corrections are present, it should not be thought

of as being suboptimal in general. In particular, for processes or setups where power corrections are moderate, their (undesirable) suppression, induced by the multi-plicative matching, has very little impact on the physics results. Conversely a

mul-We nowturnto discussingtheresummed transverse-momen-tum spectrum of the Z

γ

pair. Fig. 3 shows results inthe inclu-sive setup and compares the matched NLO+NLL spectrum to the NLL and the NLO results in panel (a) and(b), and the matched NNLO+N3LLspectrum tothe N3LLandtheNNLO resultsinpanel (c)and(d).At large transversemomenta (panel(b) and(d)),the matched results nicely converge towards the fixed-order predic-tions. At smalltransverse momenta (panel(a) and(c)), the NLO andNNLOpredictions becomeunreliable, whilethe resummation yields physical results. The matched predictions are very close to the purely resummed ones atsmall transverse momenta and then progressively move farther apart at larger pT,γ . Looking at the scale uncertainties, we observe a substantial reduction in the size of the respectivebands when moving from NLO+NLL to NNLO+N3LL: Atlarge pT,γ they decreaseby roughly a factorof

tiplicative matching is numerically more stable at small transverse momenta, and, more importantly, it includes the constant contributions in pT through the match-ing when those are not available in the resummation component, cf. Refs. [30,71].

Fig. 4. pT,γ spectrum at NLO+NLL (magenta, dash-dotted) and NNLO+N3LL (blue, solid) in the inclusive (panel (a)) and fiducial (panel (b)) setup. The lower frames show

the ratio to the central NNLO+N3_{LL prediction.}

two,from20%to10%.AtsmallpT,γ thereductionisevenmore significant.ForpT,γ



20 GeVtheNLO+NLLuncertaintyincreases between about 10% to more than 30%, while it is at the few-percent level at NNLO+N3LL, reaching at most

∼

8% in the first bin.

Fig.4comparesdirectlytheresultsatNLO+NLLandNNLO+N3LL atsmalltransversemomenta,bothintheinclusivesetupinpanel (a) and in the fiducial setup in panel (b). Higher-order correc-tions movethe peak by 1–2 GeVtowards largervalues of pT,γ . The substantial reduction ofscale uncertainties has alreadybeen pointed out for the inclusive case, and we find a quite similar picture in the fiducial case. For pT,γ



10 GeV NLO+NLL and NNLO+N3LLresultsagreewithinuncertaintieswitheachother, al-thoughthecorrectionsatpT,γ

=

10 GeVarealreadyabout15%in both setups.With increasing valuesof pT,γ thecorrections be-comeprogressivelylarger,reachingabout30%at pT,γ

=

50 GeV. For pT,γ



10 GeVthehigher-ordercorrections arenot covered by the scale-uncertainty band of the NLO+NLL prediction,which appears to be significantly underestimated. Thisbehaviour is not unexpected,asitisdirectlyinheritedfromthefixed-order calcula-tion,wheretherelativelysmallNLOuncertaintiesdonotcoverthe substantialNNLOcorrectionsinthetail.Westress thatinthetail ofthe pT,γ distributionthe(N)NLOpredictioniseffectivelyonly (N)LO accurate,whichexplains thisobservation, asLO uncertain-ties generally tend to underestimate higher-order effects. While we finda fairly similar patternin the two setups, theadditional fiducialcutstendto slightlyincrease therelative sizeof the cor-rections.

InFig.5wecompareourdefaultNNLO+N3LLpredictionsinthe additivematching schemeofEq. (2) withNNLO+N3_{LLpredictions}

inthemultiplicative matchingschemeofEq. (3) for theinclusive case in panel (a) and (b), and for the fiducial case in panel (c) and(d).Forlargetransversemomenta(panel(b)and(d)),thetwo predictionsareingoodagreementwithinuncertainties,sinceboth eventuallyapproachtheNNLOresultinthetailofthedistribution. At smalltransverse momenta(panel (a) and(c)),the situationis differentfor thetwo setups. Byand large,in the inclusivesetup wefindgoodagreementbetweenthetwomatchingschemeswith overlapping uncertaintybands andat most2%differences inthe centralvalue. Thedifferencecan beunderstoodasanuncertainty relatedtotheinclusion oftermsbeyondnominalaccuracyinthe matchedprediction.Inthefiducialsetupthedifferencesare some-what larger. Multiplicative and additive schemes differ by up to

∼

8% for pT,γ between 4 GeV and 20 GeV, and there is a gap between their uncertainty bands. This is in line with the large powercorrectionsobservedinthefiducialsetupinFig.2(c),which are suppressedinthemultiplicativeschemeandpreservedinthe additive one. As already stressed above, the suppression ofsuch genuine non-singular contributionsis undesirable, which justifies our preference forthe additive scheme, especially inthe fiducial setup, andwe refrainfromusingthe matchingsystematicsasan additionaluncertainty.

We continue by studying the impactof NG logarithmic terms stemming fromphotonisolation inFig. 6.Such termsare not in-cluded inourresummation approachandenter onlythrough the matching to fixed order.Fig. 6(a) compares theNNLO+N3LL pre-dictionsinthefiducialsetupwithandwithoutthepcone0.2

T

/

pT,γ

<

0

.

07 isolation cut (which is additional tothe smooth-cone isola-tion). Their ratioindicates that at pT,γ valuesaround the peak andsmaller,theadditionalisolationhasaminimal impact,as ex-pected since it is power suppressed, while it induces effects of

O(

10%

)

in the tail of the distribution. We show the same ratio atNNLOinthelowerframe,whichisessentiallyindistinguishable fromthe oneat NNLO+N3LL.Inother words,isolation effects are adopted purelyfromthe fixed-orderprediction.Infact, thesmall effectatpT,γ



10 GeVindicatesthattheresummationofthose corrections should have a minor impact in that region. Further-more,weestimatetheall-ordereffectsofincludingNGlogarithmic contributions in the fiducialsetup using the Pythia8 [77] parton shower (PS)matchedtoNLO calculationsintheMC@NLOscheme [78] within MadGraph5_aMC@NLO [79]. To this end, Fig. 6(b) shows NLO+PSresults withandwithout pcone0.2

T

/

pT,γ

<

0

.

07 re-quirement inthe mainframe andtheir ratio inthe lower frame. For comparison we show the same ratio at LO+PS and at NLO. The effects of the additional isolation are vanishingly small at LO+PS, which can be considered a lower bound for the impact that NG logarithmic terms stemming fromphoton isolation have on the all-order prediction of pT,γ . The ratios at NLO+PS and at NLO are very similar to each other, with the matching to PS slightly reducing the effects duethe additional isolation require-ment.Theirdifference canbe regardedasan estimate ofthe size of theNG logarithmic correctionsbeyondfixed order induced by the pcone0.2

T

/

pT,γ

<

0

.

07 requirement.Sincethedifference isvery smallatlow pT,γ andatmost

∼

2%inthematchingregion,we neglectsucheffectfromnowon.Wenotethatitisless straightfor-wardtoestimatetheNGlogarithmiccontributionsfortheFrixione

Fig. 5. pT,γ spectrum at NNLO+N3LL in the additive matching scheme (blue, solid) and in the multiplicative matching scheme (green, long-dashed) in the inclusive (panel

(a) and (b)) and fiducial (panel (c) and (d)) setup, showing the small-pT,γ (panel (a) and (c)) and large-pT,γ (panel (b) and (d)) region. The lower frames show the ratio

to the central prediction in the additive scheme.

smooth-cone isolation, which for IR safety cannot be removed. However, we have verified that by varying the smooth-cone ra-diusdownto

δ

0

=

0

.

01 theanalogousdifferenceisonlymoderately

affectedandremainsnegligibleatandbelowthepeakofthe spec-trum.

WeconcludeouranalysisbycomparingourNNLO+N3_LL

predic-tionsto13 TeVATLAS data[13] inFig.7.The analysisofRef. [13] is the first diboson measurement that includes the full Run II data set. The agreement is truly remarkable, especially with the precision of both theoretical prediction and data being at the few-percent level.The shape ofthe distribution is very well de-scribed by the predictedspectrum, andnone of the data points are morethan one standard deviationaway from the theoretical uncertaintyband.Weobservethatresummationandmatchingare crucialnotonlyatsmallpT,γ ,butalsointheintermediateregion 40



pT,γ



200 GeV, where the comparison to data is signifi-cantlyimprovedwithrespecttotheNNLOcomparisoncarriedout inRef. [13].Furthermore,ourresultsareaclearimprovementover thecomparisonagainstNLO+PSpredictionsinRef. [13].In conclu-sion,our resummed results not only constitute the most precise

predictionofthespectrumtodate,buttheyalsoprovidethemost accuratedescriptionofthe13 TeVATLASdata.

To summarize, we have presented the first calculation of the transverse-momentum spectrum of Z

γ

pairs at NNLO+N3LL. At hightransversemomentaweexploitthemostaccuratefixed-order predictionknowntodate,whileatsmalltransversemomentawe performtransverse-momentumresummationatN3_{LLaccuracyfor}

the first time. Furthermore, our matching approach respects the unitarity of the spectrum, so that its integral yields exactly the total cross section at NNLO. Our results show that higher-order correctionsinboth thefixed-orderandthelogarithmic seriesare mandatorytoobtainareliabledescriptionofthedistribution. Com-paring NLO+NLL to NNLO+N3_{LL predictions we find corrections}

of more than 30% in the tail ofthe distribution both in our in-clusive and our fiducial setup. Those are inherited directly from the large NNLO corrections. Around the peak of the spectrum, we findcorrectionsbetween10% and20%witha clearchange in shapeofthedistribution.Moreover,atNNLO+N3LLthepeakmoves by 1–2 GeV towards larger transverse momenta with respect to NLO+NLL. The inclusion of higher-order corrections substantially

Fig. 6. Panel

(a):

pT,γ spectrum at NNLO+N3LL in the fiducial setup (blue, solid) and without the additional pcone0T .2/pT,γ isolation (purple, dash-dotted). The lower frame

shows the ratio to the central prediction with the pcone0.2

T /pT,γ<0.07 requirement as well as the result when taking the same ratio for the central predictions at NNLO

(red, dashed). Panel (b): Same results for central predictions at NLO+PS with pcone0.2

T /pT,γ<0.07 requirement (light blue, solid) and without (grey-blue, long-dashed). The

lower frame shows the ratio to the former as well as the same ratios at NLO (black, dotted) and at LO+PS (brown, dash-dotted).

Fig. 7. NNLO+N3_{LL prediction of the p}

T,γ spectrum (blue, solid) compared to

AT-LAS data [13] (green data points). The lower frame shows the ratio to the central

NNLO+N3_{LL prediction.}

reducesscaleuncertainties,especiallyintheregionofsmall trans-versemomenta. Furthermore,bymeans ofan NLO+PSsimulation wehaveestimatedtheimpactofincludingNGlogarithmic contri-butionsbeyondfixedorderandfoundittobeminorwithrespect tothe NNLO+N3_{LLscale uncertainties.Finally,we havecompared}

ourbestpredictionatNNLO+N3_{LLtoATLASdataat13 TeVforthe}

transverse-momentum spectrumof the Z

γ

pair, andfound a re-markableagreementwithinuncertaintiesatthefew-percentlevel. We reckon that our results will play a crucial role in the rich physicsprogramme thatis basedonprecisionstudies of Z

γ

pro-ductionattheLHC.

Declarationofcompetinginterest

None.

Acknowledgements

We are indebted to Pier Monni,Massimiliano Grazzini, Stefan Kallweit and Emanuele Re for stimulating discussions and com-ments on the manuscript. MW would like to thank Giulia Zan-derighi andDanieleLombardi forusefuldiscussions. Thework of LR is supported bythe European ResearchCouncil Starting Grant 714788REINVENT.

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