Probing the hardest branching of jets in heavy ion collisions
aa r X i v : . [ h e p - ph ] A ug Probing the hardest branching of jets in heavy ion collisions
Yang-Ting Chien a,b and Ivan Vitev a a Theoretical Division, T-2, Los Alamos National Laboratory, Los Alamos, NM 87545, USA b Erwin Schr¨odinger International Institute for Mathematical Physics, Universit¨at Wien, Wien, Austria
We present the first calculation of the momentum sharing and angular separation distributionsbetween the leading subjets inside a reconstructed jet in heavy ion collisions. These observablesare directly sensitive to the hardest branching in the process of jet formation and are, therefore,ideal for studying the early stage of the in-medium parton shower evolution. The modification ofthe momentum sharing and angular separation distributions in lead-lead relative to proton-protoncollisions is evaluated using the leading-order medium-induced splitting functions obtained in theframework of soft-collinear effective theory with Glauber gluon interactions. Qualitative and inmost cases quantitative agreement between theory and preliminary CMS measurements suggeststhat the parton shower in heavy ion collisions can be dramatically modified early in the branchinghistory. We propose a new measurement which will illuminate the angular distribution of the hardestbranching within jets in heavy ion collisions.
The dramatic suppression of hadron and jet cross sec-tions observed at the Relativistic Heavy Ion Collider(RHIC) [1–6] and the Large Hadron Collider (LHC) [7–14] signals the strong modification of parton showerswithin strongly-interacting matter. This jet quenchingphenomenon has been an essential tool to study the prop-erties of the quark-gluon plasma (QGP) produced inultrarelativistic nucleus-nucleus (A+A) collisions. Theemergence of the in-medium parton branching, qualita-tively different from the one which governs the jet forma-tion in e + + e − , e + + p , p + p collisions, is at the heartof all jet modification studies. Although the traditionalenergy loss picture has been very successful in describingthe suppression of cross section, to disentangle the de-tailed jet formation mechanisms in the medium requirescomprehensive studies of jet substructure observables.In the past few years there has been a proliferation ofjet substructure measurements in A+A collisions [15–18],which gave differential and correlated information abouthow quark and gluon radiation is redistributed due tomedium interactions. It is now definitively establishedthat the jet shape [19] and the jet fragmentation func-tion [20], which describe the transverse and longitudi-nal momentum distributions inside jets, are modified inheavy ion collisions. Both of these observables dependstrongly on the partonic origin of jets, and their nontriv-ial modification patterns are partly due to the increase ofthe quark jet fraction in heavy ion collisions [21–24]. Tobetter understand the jet-by-jet modifications for theseobservables, one can devise strategies to isolate purerquark or gluon jet samples.Another collinear type of jet substructure observable,called the groomed momentum sharing, has been stud-ied in the context of the soft drop jet grooming pro-cedure [25] and Sudakov safety [26]. This observableprobes the hard branching in the jet formation and isdominated by the leading-order Altarelli-Parisi splittingfunctions [27]. Given a jet reconstructed using the anti- k T algorithm [28] with radius R , one reclusters the jet / xk ⊥ θ = Rθ = ∆ x = z cut x = 1 − z cut x − xθ x, k ⊥ k ⊥ = ω tan θ x (1 − x ) ω FIG. 1: Illustration of the phase space regions for the z g distribution calculation constrained by R , ∆ and z cut . Atleading order a collinear parton splits into partons with mo-menta k = ( xω, k ⊥ /xω, k ⊥ ) and p − k . Depending on thekinematics of the splitting, the soft branch can be either ofthe partons. using the Cambridge-Aachen algorithm [29, 30] and goesthrough the clustering history, grooming away the softbranch at each step until the following condition is satis-fied, z cut < min( p T , p T ) p T + p T ≡ z g , (1)i.e., the soft branch is not carrying less than a z cut frac-tion of the sum of the transverse momenta therefore isnot dropped. Note that by definition z cut < z g < . Dueto detector granularity one also demands that the angu-lar separation between the two branches be greater thanthe angular resolution ∆,∆ < ∆ R ≡ r g . (2)More generally, by selecting the angular separation∆ R , defined as the groomed jet radius r g , one couldalso examine the momentum sharing distribution p ( z g )at different splitting angles and the p ( r g ) distribution.For jets with small radii [31–34], the z g distribution canbe described by the collinear parton splitting functions.At leading order, for a parton i with collinear momentum p = ( ω, ,
0) [54] splitting into partons j, l with momenta k = ( xω, k ⊥ /xω, k ⊥ ) and p − k , the splitting functionsin vacuum P vaci → jl ( x, k ⊥ ) are well-known and their non-singular parts are reproduced below, P vacq → qg = α s ( µ ) π C F − x ) x k ⊥ , (3) P vacg → gg = α s ( µ ) π C A h − xx + x − x + x (1 − x ) i k ⊥ , (4) P vacg → q ¯ q = α s ( µ ) π T F n f h x + (1 − x ) i k ⊥ , (5) P vacq → gq = P vacq → qg ( x → − x ) . (6)The z g distribution is calculated by integrating the split-ting functions over the partonic phase space constrainedby R , ∆ and z cut and shown in FIG. 1, p i ( z g ) = R k R k ∆ dk ⊥ P i ( z g , k ⊥ ) R / z cut dx R k R k ∆ dk ⊥ P i ( x, k ⊥ ) . (7)Here, k ∆ = ωx (1 − x ) tan ∆2 , k R = ωx (1 − x ) tan R and P i ( x, k ⊥ ) = X j,l h P i → j,l ( x, k ⊥ ) + P i → j,l (1 − x, k ⊥ ) i . Note that for anti- k T jets the angle θ between the twofinal state partons satisfies ∆ < θ < R . The effect ofrunning coupling can be taken into account by setting µ = k ⊥ in the splitting function. The final z g distributionis then weighted by the jet production cross sections, p ( z g ) = 1 σ total X i = q,g Z P S dηdp T dσ i dηdp T p i ( z g ) , (8)with the phase space cuts ( P S ) on the jet p T and η im-posed in experiments.The z g distributions for quark-initiated and gluon-initiated jets are very similar throughout the whole z g region. The color factors C F = 4 / C A = 3 forquarks and gluons cancel and the distributions follow ap-proximately 1 /z g , the leading behavior of the splittingfunctions in Eqs. (3) and (4) for x < /
2. The insen-sitivity of z g to the partonic origin of jets implies thatits modification in heavy ion collision is not significantlyaffected by the change of the quark/gluon jet fraction asone observes in the jet shape and the jet fragmentationfunction.In the presence of the medium, P i → jl ( x, k ⊥ ) = P vaci → jl ( x, k ⊥ ) + P medi → jl ( x, k ⊥ ) , (9)which is the sum of the vacuum and medium-induced splitting functions. The later were calculated using soft-collinear effective theory [35–39] with Glauber gluon in-teractions (SCET G ) [40–43] in a QGP model consisting ofthermal quasi-particles undergoing longitudinal Bjorken-expansions [44]. SCET G is an effective field theory ofQCD suitable for describing jets in the medium. It goesbeyond the traditional parton energy loss picture in thesoft gluon limit, and it provides a systematic frameworkfor resumming jet substructure observables and consis-tently including medium modifications. The medium-induced splitting functions used in this paper have beenpreviously applied to describe and predict several hadronand jet observables in heavy ion collisions [22, 24, 45, 46].It can be seen analytically and confirmed numericallythat in the region of interest x < /
2, the leading behav-ior of the in-medium splitting functions follows approxi-mately 1 /x [42]. A testable hypothesis is that the mo-mentum sharing distribution will show enhancement atthe smallest values of z g and suppression near z g = 1 / z g is insensitive to the fla-vor of jet-initiating partons, the effect from the changeof quark/gluon jet fractions due to the different amountsof cross section suppression is minor.For the cross section calculations, we use the CTEQ5Mparton distribution functions [48] and the leading-order O ( α s ) QCD partonic cross section results. We estimatethe theoretical uncertainty by varying the coupling be-tween the jet and the QCD medium g = 2 . ± . α s ( m Z ) = 0 . τ br [fm] = 0 .
197 GeV fm z g (1 − z g ) ω [GeV] tan ( r g /
2) (10)suggests that for typical jets with ω = 2 p T = 400 GeV, r g = 0 . z g = 0 .
1, the branching time τ br < √ s NN = 5 .
02 TeV [18]. In both proton-proton and lead-lead (Pb+Pb) collisions, the jets are reconstructed usingthe anti- k T algorithm with R = 0 . β = 0 and z cut = 0 .
1. Another cut on ∆ R > . R is thedistance between the two branches in the pseudorapidity-azimuthal angle plane. The requirement also effectivelyselects jets with the branching angle greater than 0 . z g and its normalizeddistribution p ( z g ) = 1 N jet dNdz g , (11)are measured. The jets are selected with the followingcuts on the jet transverse momentum ( p T ) and pseudora-pidity ( η ): p T >
140 GeV and | η | < .
3. The in-mediummomentum sharing modification is quantified by takingthe ratio of the z g distributions in proton-proton andlead-lead collisions, R p ( z g ) AA = p ( z g ) P bP b (cid:14) p ( z g ) pp . (12)The modification patterns are examined across a widerange of p T bins with different collisional centralities.FIG. 2 shows the result for the ratio of the momentumsharing distributions of inclusive jets in 0-10% centralPb+Pb and p+p collisions at √ s NN = 5 .
02 TeV. Weconsider two p T bins 140 GeV < p T <
160 GeV (up-per panel) and 250 GeV < p T <
300 GeV (lower panel)to study the modification pattern as a function of thejet transverse momentum. The preliminary CMS datashows a strong modification of the momentum sharingdistribution for jets with lower p T in central collisions,and the modification decreases quite quickly when thejet p T becomes higher. The red bands correspond to thetheoretical calculations with the variation of g = 2 . ± . p T increases. However, the p T dependence in our theorycalculation is not as strong as suggested in the prelimi-nary CMS measurements, with the amount of modifica-tion around z g = 0 . p T jets. For jets with higher p T , our calculationis consistent with the preliminary CMS data within theexperimental uncertainties.FIG. 3 shows the modification of the momentum shar-ing distribution for inclusive jets in mid-peripheral lead-lead collisions with centrality 30-50% at √ s NN = 5 . < p T <
160 GeV bin since the modification is larger for lower p T jets. Both the CMS preliminary data and our calculationshow moderate modifications of the z g distributions, andwe are consistent with each other. The medium modi-fication of the z g distribution decreases with collisionalcentrality.Predictions for the momentum sharing distribution ra-tios for inclusive jets in proton-proton and central lead-lead collisions at √ s NN = 5 .
02 TeV are shown in FIG. 4. s NN = = È Η È < < p T <
160 GeVCentrality: 0 - %Β= z cut = D R > = H ± L z gp ( z g ) PbPb p ( z g ) pp s NN = = È Η È < < p T <
300 GeVCentrality: 0 - %Β= z cut = D R > = H ± L z gp ( z g ) PbPb p ( z g ) pp FIG. 2: Comparison of theoretical calculations and prelimi-nary CMS data for the ratio of momentum sharing distribu-tions of inclusive anti- k T R = 0 . √ s NN = 5 .
02 TeV. Jets are soft-droppedwith β = 0, z cut = 0 . R > .
1. Bands correspondto the theoretical uncertainty estimated by varying the cou-pling between the jet and the medium ( g = 2 . ± . < p T <
160 GeVand | η | < .
3. Lower panel: modification for jets with250 GeV < p T <
300 GeV and | η | < . s NN = = È Η È < < p T <
160 GeVCentrality: 30 - %Β= z cut = D R > = H ± L z gp ( z g ) PbPb p ( z g ) pp FIG. 3: Comparison of theoretical calculations and prelimi-nary CMS data for the momentum sharing modification ofinclusive jets in proton-proton and lead-lead collisions at √ s NN = 5 .
02 TeV. Shown are the same studies as in FIG.2 for anti- k T R = 0 . < p T <
160 GeVand | η | < . s NN = = È Η È < - %Β= z cut = D R > < p T <
80 GeV250 GeV < p T <
300 GeVg = H ± L z gp ( z g ) PbPb p ( z g ) pp FIG. 4: Theoretical calculations for the momentum sharingdistribution ratio of inclusive jets in proton-proton and centrallead-lead collisions at √ s NN = 5 .
02 TeV. Jets are soft-droppedwith β = 0, z cut = 0 . R > .
2. We study its jet p T dependence and provide results for 60 GeV < p T <
80 GeV(red band) and 250 GeV < p T <
300 GeV (blue band).
We consider the p T bins 60 GeV < p T <
80 GeV (redband) and 250 GeV < p T <
300 GeV (blue band). How-ever, whereas in the CMS preliminary measurements thecut ∆ R > . R > . z g distribution with widersplitting angles. We find that the modification increases(decreases) with ∆ R for low (high) p T jets, renderinga stronger p T dependence in the modification pattern.This can be understood because the angular scale set inthe medium-induced splitting function should be propor-tional to the ratio between the medium temperature andthe jet transverse momentum. By probing radiation withwider angles we should see that the medium modificationdecreases faster with the jet p T .An important new observable that we propose to studyin heavy ion collisions is the angular separation distribu-tion r g ≡ ∆ R of the leading subjets inside a groomedjet. At leading order, p i ( r g ) = R / z cut dx p T x (1 − x ) P i ( x, k ⊥ ( r g , x )) R / z cut dx R k R k ∆ dk ⊥ P i ( x, k ⊥ ) , (13)and k ⊥ ( r g , x ) = ωx (1 − x ) tan r g . The power of thisobservable is that it is sensitive to the medium modifica-tion of the hardest branching inside jets, rather than thesoft radiation which can be transported to larger anglesthrough different mechanisms such QGP excitations. InFIG. 5 we predict the angular separation modification forthe leading subjets in the SCET G framework. The samejet selection cuts and soft drop parameters are used as inthe preliminary CMS momentum sharing measurements.We examine the p T dependence of the angular regionwhere the distribution is enhanced shifts to smaller valueswhen the jet p T increases. The peak of this distributioncorresponds to the characteristic r g where the medium s NN = = È Η È < - % Β= z cut = < p T <
80 GeV100 GeV < p T <
120 GeV140 GeV < p T <
160 GeV250 GeV < p T <
300 GeVg = H ± L r gp ( r g ) PbPb p ( r g ) pp FIG. 5: Theoretical calculations for the groomed jet radiusmodification of inclusive jets in proton-proton and centrallead-lead collisions at √ s NN = 5 .
02 TeV. The soft drop pa-rameters β = 0, z cut = 0 . R > . p T bins with 60 GeV < p T <
80 GeV (redband), 100 GeV < p T <
120 GeV (green band), 140 GeV
160 GeV (blue band) and 250 GeV < p T <
300 GeV(purple band). enhancement of large-angle splitting for hard branchingprocesses is most significant.To conclude, we presented the first calculation of themomentum sharing distribution p ( z g ), defined in the softdrop jet grooming procedure, and examined its modifica-tion in heavy ion collisions. This observable probes thehard branching at the early stage of parton shower for-mation and is a new powerful way to investigate the jetformation mechanism. In heavy ion collisions, the mo-mentum sharing distribution of the two leading subjetsin a reconstructed jets allows us to probe the early stagesof the QGP evolution. We found that the z g distributionis significantly modified in the medium, as shown in ourtheory calculation and the preliminary CMS data. Thissuggests that the parton shower modification in the QGPstarts early with the first hard splittings, and the in-medium splitting functions have a qualitatively differentbehavior from the ones in the vacuum. We also proposeda new measurement of the angular separation distribu-tion between the leading subjets inside a groomed jetwhich encodes the angular distribution of the hard split-ting, and we present theoretical predictions for its behav-ior. Future studies of jet substructure observables moresensitive to the soft radiation, for example the jet mass[49–53], will allow us to map out the whole jet formationhistory.Y.-T. C. would like to thank Andrew Larkoski, Yen-JieLee, Yacine Mehtar-Tani, Felix Ringer, Jesse Thaler andMarta Verweij for very helpful discussions, and the ESIat Universit¨at Wien for hospitality and support. Thisresearch was supported by the US Department of En-ergy, Office of Science under Contract No. DE-AC52-06NA25396 and the DOE Early Career Program. [1] K. Adcox et al. (PHENIX), Phys. Rev. Lett. , 022301(2002), nucl-ex/0109003.[2] C. Adler et al. (STAR), Phys. Rev. Lett. , 202301(2002), nucl-ex/0206011.[3] K. Adcox et al. (PHENIX), Nucl. Phys. A757 , 184(2005), nucl-ex/0410003.[4] I. Arsene et al. (BRAHMS), Nucl. Phys.
A757 , 1 (2005),nucl-ex/0410020.[5] B. B. Back et al., Nucl. Phys.
A757 , 28 (2005), nucl-ex/0410022.[6] J. Adams et al. (STAR), Nucl. Phys.
A757 , 102 (2005),nucl-ex/0501009.[7] G. Aad et al. (ATLAS), Phys.Rev.Lett. , 252303(2010), 1011.6182.[8] K. Aamodt et al. (ALICE), Phys. Lett.
B696 , 30 (2011),1012.1004.[9] S. Chatrchyan et al. (CMS), Phys.Rev.
C84 , 024906(2011), 1102.1957.[10] S. Chatrchyan et al. (CMS), Eur. Phys. J.
C72 , 1945(2012), 1202.2554.[11] G. Aad et al. (ATLAS), Phys.Lett.
B719 , 220 (2013),1208.1967.[12] G. Aad et al. (ATLAS), Phys. Rev. Lett. , 072302(2015), 1411.2357.[13] S. Chatrchyan et al. (CMS) (2012), CMS-HIN-12-004.[14] J. Adam et al. (ALICE), Phys. Lett.
B746 , 1 (2015),1502.01689.[15] S. Chatrchyan et al. (CMS), Phys.Lett.
B730 , 243(2014), 1310.0878.[16] S. Chatrchyan et al. (CMS), Phys. Rev.
C90 , 024908(2014), 1406.0932.[17] G. Aad et al. (ATLAS), Phys. Lett.
B739 , 320 (2014),1406.2979.[18] C. Collaboration (CMS) (2016).[19] S. D. Ellis, Z. Kunszt, and D. E. Soper, Phys.Rev.Lett. , 3615 (1992), hep-ph/9208249.[20] M. Procura and I. W. Stewart, Phys. Rev. D81 ,074009 (2010), [Erratum: Phys. Rev.D83,039902(2011)],0911.4980.[21] Y.-T. Chien and I. Vitev, JHEP , 061 (2014),1405.4293.[22] Y.-T. Chien and I. Vitev, JHEP , 023 (2016),1509.07257.[23] Y.-T. Chien, Z.-B. Kang, F. Ringer, I. Vitev, andH. Xing, JHEP , 125 (2016), 1512.06851.[24] Y.-T. Chien, Z.-B. Kang, F. Ringer, I. Vitev, and H. Xing(2016), in preparation.[25] A. J. Larkoski, S. Marzani, G. Soyez, and J. Thaler,JHEP , 146 (2014), 1402.2657.[26] A. J. Larkoski, S. Marzani, and J. Thaler, Phys. Rev. D91 , 111501 (2015), 1502.01719.[27] G. Altarelli and G. Parisi, Nucl. Phys.
B126 , 298 (1977).[28] M. Cacciari, G. P. Salam, and G. Soyez, JHEP , 063(2008), 0802.1189. [29] Y. L. Dokshitzer, G. D. Leder, S. Moretti, and B. R.Webber, JHEP , 001 (1997), hep-ph/9707323.[30] M. Wobisch and T. Wengler(1998), hep-ph/9907280, URL https://inspirehep.net/record/484872/files/arXiv:hep-ph_990728 [31] M. Dasgupta, F. Dreyer, G. P. Salam, and G. Soyez,JHEP , 039 (2015), 1411.5182.[32] Y.-T. Chien, A. Hornig, and C. Lee, Phys. Rev. D93 ,014033 (2016), 1509.04287.[33] T. Becher, M. Neubert, L. Rothen, and D. Y. Shao, Phys.Rev. Lett. , 192001 (2016), 1508.06645.[34] Z.-B. Kang, F. Ringer, and I. Vitev (2016), 1606.06732.[35] C. W. Bauer, S. Fleming, and M. E. Luke, Phys.Rev.
D63 , 014006 (2000), hep-ph/0005275.[36] C. W. Bauer, S. Fleming, D. Pirjol, and I. W. Stewart,Phys.Rev.
D63 , 114020 (2001), hep-ph/0011336.[37] C. W. Bauer and I. W. Stewart, Phys.Lett.
B516 , 134(2001), hep-ph/0107001.[38] C. W. Bauer, D. Pirjol, and I. W. Stewart, Phys.Rev.
D65 , 054022 (2002), hep-ph/0109045.[39] C. W. Bauer, S. Fleming, D. Pirjol, I. Z. Rothstein,and I. W. Stewart, Phys.Rev.
D66 , 014017 (2002), hep-ph/0202088.[40] A. Idilbi and A. Majumder, Phys.Rev.
D80 , 054022(2009), 0808.1087.[41] G. Ovanesyan and I. Vitev, JHEP , 080 (2011),1103.1074.[42] G. Ovanesyan and I. Vitev, Phys.Lett.
B706 , 371 (2012),1109.5619.[43] M. Fickinger, G. Ovanesyan, and I. Vitev, JHEP ,059 (2013), 1304.3497.[44] J. D. Bjorken, pp. FERMILAB–PUB–82–059–THY,FERMILAB–PUB–82–059–T (1982).[45] Z.-B. Kang, R. Lashof-Regas, G. Ovanesyan, P. Saad,and I. Vitev (2014), 1405.2612.[46] Y.-T. Chien, A. Emerman, Z.-B. Kang, G. Ovanesyan,and I. Vitev, Phys. Rev.
D93 , 074030 (2016), 1509.02936.[47] I. Vitev, Phys. Rev.
C75 , 064906 (2007), hep-ph/0703002.[48] W. Tung, H. Lai, A. Belyaev, J. Pumplin, D. Stump,et al., JHEP , 053 (2007), hep-ph/0611254.[49] Y.-T. Chien and M. D. Schwartz, JHEP , 058(2010), 1005.1644.[50] Y.-T. Chien, R. Kelley, M. D. Schwartz, and H. X. Zhu,Phys. Rev.