1、MITCTP 4530DESY 14-008NIKHEF 2014-002Dissecting Soft Radiation with FactorizationIain W. Stewart,1 Frank J. Tackmann,2 and Wouter J. Waalewijn3, 41Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA2Theory Group, Deutsches Elektronen-Synchrotron (DESY), D-
2、22607 Hamburg, Germany3Nikhef, Theory Group, Science Park 105, 1098 XG, Amsterdam, The Netherlands4ITFA, University of Amsterdam, Science Park 904, 1018 XE, Amsterdam, The NetherlandsAn essential part of high-energy hadronic collisions is the soft hadronic activity that underlies theprimary hard int
3、eraction. It can receive contributions from soft radiation from the primary hardpartons, secondary multiple parton interactions (MPI), and factorization violating e ects. Theinvariant mass spectrum of the leading jet in Z+jet and H+jet events is directly sensitive to thesee ects. We use a QCD factor
4、ization theorem to predict the dependence on the jet radius R, jetpT, jet rapidity, and partonic process for both the perturbative and nonperturbative components ofprimary soft radiation. The nonperturbative contributions involve only odd powers of R, and thelinear R term is universal for quark and
5、gluon jets. The hadronization model in Pythia8 agreeswell with these properties. The perturbative soft initial state radiation (ISR) has a contributionthat depends on the jet area in the same way as the underlying event. This degeneracy is broken bythe jet pT dependence. The size of this soft ISR co
6、ntribution is proportional to the color state of theintial partons, yielding the same positive contribution for gg!Hg and gq!Zq, but a negativeinterference contribution for q q!Zg. Hence, measuring these dependencies allows one to separatehadronization, soft ISR, and MPI contributions in the data.So
7、ft hadronic activity plays a role in practically allbut the most inclusive measurements at the LHC. It isoften an important yet hard-to-quantify source of uncer-tainty, so improving its theoretical understanding is vital.One can consider four conceptually di erent sources forthe e ects that are expe
8、rimentally associated with softhadronic activity and the underlying event (UE):1. Perturbative soft radiation from the primary hardpartons within factorization2. Nonperturbative soft e ects within factorizationassociated with hadronization3. Multiple parton interactions (MPI) at lower scalesin the s
9、ame proton-proton collision4. Factorization breaking contributionsFor any given observable, the question is how much ofeach of these sources is required to describe the data.For example, it is known that including higher-order per-turbative corrections (source 1) in parton-shower MonteCarlos can giv
10、e a nontrivial contribution to traditionalUE measurements 1, 2.Traditionally, the UE activity is measured in regions ofphase space away from hard jets 212. These results areused to tune the MPI models that are used to describethe UE in Monte Carlo programs 1318. These modelsare then extrapolated int
11、o the jet region, where they areused to describe various jet observables, including thejet mass spectrum in dijet and Drell-Yan events 19, 20,which is an important benchmark jet observable at theLHC.In this paper, we directly consider the jet region andgive a eld-theoretic description of primary sof
12、t e ects(sources 1 and 2). Using the jet mass spectrum and itsrst moment, we show how to cleanly distinguish sources1, 2, and 3 via their characteristic dependence on the jetradius R, jet momentum pJT, and participating partons.We will not consider factorization breaking e ects here(see e.g. Ref. 21
13、).We consider the jet mass spectrum in exclusive pp!Z+1-jet and pp ! H+1-jet events. The factorizationformula for mJ pJT that includes sources 1 and 2 isgiven by 2224d dm2Jd 2 =X;a;bH ( 2)ZdkSdkB (I aaI bb fafb)(kB)J J(m2J 2pJTkS)S (kS;pcut kB;yJ;R): (1)Here, 2 = fpJT;yJ;Yg, where Y is the rapidity
14、ofthe Z=H+jet system, and denotes the partonic chan-nel. The H ( 2) contains the perturbative matrix el-ements for the hard process, the I aa describe pertur-bative collinear initial-state radiation from the incomingprimary partons, and the fa are the parton distributionfunctions. For the normalized
15、 jet mass spectrum, thedependence on pcut, which vetoes additional jets, largelydrops out. As a result, the shape of the jet mass spectrumis determined by the jet function J J, describing ener-getic nal-state radiation from the outgoing primary par-ton, and by the soft function S . See also Refs. 25
16、, 26.The soft function S describes the primary initial andnal state soft radiation. It depends on the jet throughyJ and R but not pJT, and can be factorized as 2729S (kS;kB;yJ;R) =ZdkSpert (kS k;kB;yJ;R) (2)F (k;yJ;R) 1 +O QCD=kB ;2PYTHIA8AU2qgfiZqH7TeVLyJ 50 GeV. The jets are de ned usinganti-kT 37
17、, 38. In Fig. 1, we show the jet mass spectrumfor quark and gluon jets with R = 1 after parton shower-ing (black dotted) and including both hadronization andMPI (blue dashed). Using Eq. (3) with Eq. (1) predictsthat for m2J QCDpJT the nonperturbative correctionsshift the tail of the jet mass spectru
18、m bym2J = (m2J)pert + 2pJT (R): (4)We can regard the partonic result from Pythia8 as thebaseline purely perturbative result. Shifting it in thisway with = 2:4 GeV for qg!Zq and = 2:7 GeV forq q!Zg yields the green dot-dashed curves in Fig. 1. Wesee that the e ect of both hadronization and MPI in the
19、tail is well captured by a shift, so ! had + MPI . Forhadronization, combining Eq. (2) with Eq. (1) predicts aconvolution with a nonperturbative function,d dm2J =Zdk d partonic dm2J (m2J 2pJTk) F (k): (5)The result of this convolution, shown by the red solidcurves in Fig. 1, yields excellent agreeme
20、nt with thehadronization+MPI result over the full range of the jetmass spectrum!1 Both hadronization and MPI populatethe jet region with a smooth background of soft parti-cles, which can explain why the MPI e ect is reproducedalongside the hadronization by a convolution of the formEq. (5). This appa
21、rant degeneracy motivates us to de-termine the calculable behavior of the jet mass spectrum1 Here, F (k) = (4k= 2 )e 2k= ; the simplest ansatz that sat-is es the required properties of being normalized, vanishing atk = 0, falling o exponentially for k ! 1, and having a rstmoment with the above value
22、s. Fixing the value of fromthe tail, we nd similar levels of agreement across all values ofpJT, yJ, R, for all partonic channels, as well as for di erent jetveto cuts (including no jet veto).3qgRArrowZqqqRArrowZgggRArrowHgPYTHIA8AU2LParen1partRArrowhadRParen1EcmEqual7TeV,VertBar1yJVertBar1Less2300Le
23、sspTJLess400GeVBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidU
24、pTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidU
25、pTriangleSolidUpTriangleMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSol
26、idDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamond0.0 0.5 1.0 1.50.00.51.01.52.02.53.03.54.0RCapOmegahadLParen1RRParen1Slash1LParen1RSlash12RParen1LBracket1GeVRBracket1 qgRArrowZqqqRArrowZggg
27、RArrowHgHERWIGPlusPlusLParen1partRArrowhadRParen1EcmEqual7TeV,VertBar1yJVertBar1Less2300LesspTJLess400GeVBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletSolidUpTriangleSolidU
28、pTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidU
29、pTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleSolidUpTriangleMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSol
30、idDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamondMedSolidDiamond0.0 0.5 1.0 1.50.00.51.01.52.02.53.03.54.0RCapOmega
31、hadLParen1RRParen1Slash1LParen1RSlash12RParen1LBracket1GeVRBracket1FIG. 2. The R dependence of had (R) extracted from the rst jet mass moment in Pythia8 (left panel) and Herwig+ (rightpanel). It follows the form in Eq. (10) predicted from factorization. The small-R behavior only depends on whether t
32、he jet isinitiated by a quark (blue dashed) or gluon (orange solid and green dotted).due to primary perturbative and nonperturbative soft ra-diation within factorization, study its dependence on pJT,yJ, and R, and compare these results to Monte Carlocontributions for soft ISR, hadronization, and MPI
33、.We consider the rst moment in m2J,M1 = 1 Zdm2Jm2J d dm2J; (6)which tracks the shift observed in Fig. 1. Taking the rstmoment of Eq. (1) combined with Eqs. (2) and (3), wecan compute the dependence of primary soft radiation onpJT, yJ, R, and partonic channel,M1 = Mpert1 (pJT;yJ;R) + 2pJT (R): (7)Her
34、e, Mpert1 (pJT;yJ;R) contains all perturbative contri-butions, while (R) encodes the shift due to nonper-turbative e ects.To describe the results for M1 from Pythia8 and Her-wig+, including their partonic, hadronization, andMPI contributions, we writeM1 = Mpartonic1 (pJT;yJ;R) + 2pJT had (R)+ 2pJThM
35、PI(yJ;R) + MPI (yJ;R)i: (8)Here, Mpartonic1 is the partonic contribution, had is givenby partonic ! hadronic, MPI is de ned by partonic !partonic+MPI, and MPI is the small remainder whichensures these contributions add up to the full partonic!hadronic+MPI. Equation (8) encodes the dependenceon pJT,
36、yJ, observed in Pythia8 and Herwig+ andthe fact that their hadronization and MPI are each indi-vidually described by shifts to M1.For pp ! H=Z+jet, the nonperturbative parameter(R) in Eq. (7) is given by the vacuum matrix elementof lightlike soft Wilson lines Ya, Yb, and YJ YJ(yJ; J)along the beam a
37、nd jet directions,(R) =Z 10drZ 11dyZ 2 0d f(r;y yJ; J;R)0 TYyJYybYya ET(r;y; )TYaYbYJ 0 : (9)Here, the rapidity y, azimuthal angle , and transversevelocity r = pT=mT are measured with respect to thebeam axis. The color representation of the Wilson linesdepends on the partonic channel, which induces
38、the dependence of . The jet mass measurement func-tion is f(r;y; ;R) = (coshy rcos ) b(y; ;r) 0. The soft function contains a contribution due tointerference between ISR from the two beams 41,Spert (kS) sC R2 1 kS+; (12)which contributes to Mpert1 as (pJT)2R4 with color factorCqg!q = Cgg!g = CA2 = 3
39、2 ;Cq q!g = CF CA2 = 16 : (13)In Fig. 3, we compare our NLL and NNLL predic-tions 24 for Mpert1 to the corresponding Mpartonic1 fromPythia8 and Herwig+ as a function of R, dividingby the leading R2 dependence. The e ect of is seenin the slight negative slope at NLL, and is also presentin the Monte C
40、arlos. The R4 contribution only entersat NNLL and is seen in the rise at large R for qg!Zq(left panel). This soft ISR e ect is partially modelledby soft emissions in the parton shower, which explainsthe similar R4 contribution for qg ! Zq in Pythia8and Herwig+. For q q !Zg (right panel) Eqs. (12)and
41、 (13) predict the R4 contribution from soft ISR tobe negative, which we observe at NNLL. This negativeinterference e ect is not captured by the Monte Carlos.In Fig. 4, we show the pJT dependence of the R4 com-ponent, c 4, of the partonic moment, obtained by ttingMpart1 2pJTR2 = c2 R+c 4 R2: (14)5The
42、 MPI contribution to the moment, MPI=R2 R2,is shown as well. The apparent ambiguity between thesedi erent R4 contributions is demonstrated by the notabledi erences between the various tunes for c 4 and MPI,while their sum c 4 + MPI is much closer. However, it isclearly resolved by the pJT dependence
43、: c 4 pJT (as pre-dicted), whereas MPI is independent of pJT. Althoughthe destructive soft interference for q q!Zg is not mod-elled by the Monte Carlos, the value of c 4 is signi cantlysmaller than in qg!Zq, in particular for Herwig+.The channel dependence provides an additional handleto separate so
44、ft ISR from MPI: c 4 depends on the colorchannel as in Eq. (13), whereas MPI is channel indepen-dent. As shown in Ref. 41, the yJ dependence of softISR is quite di erent between Herwig+ and Pythia8.To conclude, we have used QCD factorization to pre-dict the properties of the perturbative and nonpert
45、urba-tive components of primary soft radiation for jet mass inpp!H=Z+jet. We have shown that the nonperturba-tive soft e ects involve odd powers of R and are universalfor quark and gluon jets for R 1. Hadronization mod-els in Monte Carlos agree with these predictions. Theperturbative soft radiation
46、has a contribution that scaleslike R4, just like the contribution from MPI. These com-ponents depend di erently on pJT and on the partonicprocess. Hence, separately measuring quark and gluonchannels in Drell-Yan and in di erent bins of pJT pro-vides the possibility to clearly distinguish between MPI
47、and primary soft radiation.We thank Jesse Thaler and Simon Pl atzer for helpfulconversations. This work was supported in part by theO ce of Nuclear Physics of the U.S. Department of En-ergy under Grant No. DE-FG02-94ER40818, the DFGEmmy-Noether Grant No. TA 867/1-1, and the MarieCurie Fellowship PII
48、F-GA-2012-328913. We thank theErwin Schr odinger Institute for hospitality while portionsof this work were completed.1 M. Cacciari, G. P. Salam, and S. Sapeta, JHEP, 1004,065 (2010), arXiv:0912.4926 hep-ph.2 S. Chatrchyan et al. (CMS Collaboration),Eur. Phys. J. C, 72, 2080 (2012), arXiv:1204.1411he
49、p-ex.3 T. A older et al. (CDF Collaboration), Phys. Rev. D,65, 092002 (2002).4 D. Acosta et al. (CDF Collaboration), Phys. Rev. D, 70,072002 (2004), hep-ex/0404004.5 R. Field and R. C. Group (CDF Collaboration), (2005),hep-ph/0510198.6 T. Aaltonen et al. (CDF Collaboration), Phys. Rev. D,82, 034001 (2010), arXiv:1003.3146 hep-ex.7 R. Field, (2010), arXiv:1010.3558 hep-ph.8 G. Aad et al. (ATLAS Collaboration), (2010), ATL-PHYS-PUB-2010-014.9 G. Aad et al. (ATLAS Collaboration), Phys. Rev. D, 83,112001 (2011), arXiv
