Effective Field Theory for jet substructure in heavy ion collisions
MMIT–CTP 5243
Effective Field Theory for Jet substructure in heavy ioncollisions
Varun Vaidya
Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.
Abstract:
I develop an Effective Field Theory (EFT) framework to compute jet substructureobservables for heavy ion collision experiments. I consider dijet events that accompany theformation of a weakly coupled Quark Gluon Plasma(QGP) medium in a heavy ion collisionand look at the simultaneous measurement of jet mass along with the transverse momentumimbalance between the jets accounting for both vacuum and medium evolution. The jets aregroomed using a suitable grooming algorithm in order to mitigate effects of soft contaminationfrom Multi-parton interactions as well as the QGP medium. This has the great advantagethat we do not have to keep track of the evolution of the QGP medium into subsequent softhadrons since these are groomed away from the final jet. Treating the energetic jet as an openquantum system interacting with a QGP bath, I write down a factorization formula withinthe SCET(Soft Collinear Effective Theory) framework, in which the forward scattering regimeacounts for the interaction of the medium with the jet. This leads me to a Lindblad typemaster equation for the evolution of the reduced density matrix of the jet in the Markovianapproximation. The resulting solution allows a resummation of large logarithms that arisedue to the final state measurements imposed while simultaneously summing over multipleinteractions of the jet with the medium. I find that the the decoherence between the hardinteraction that creates the jet and the subsequent medium interactions leads to physical
Infra-Red(IR) collinear divergences that are otherwise absent in pure vacuum evolution. Ishow that these IR divergences are completely regulated by the medium induced gluon massand highlight the need to develop a multi-scale EFT approach in the future to resum the newlogarithms that arise from these divergences. a r X i v : . [ h e p - ph ] S e p ontents A.1 Jet functions 31A.2 Collinear-soft function 32A.3 Soft function 33A.4 Soft-collinear function 33
B Modified Quark jet function 34
B.1 Real emission diagrams 35B.2 Virtual Diagrams 37B.3 Medium induced terms 37
C Gluon interaction with medium 38
C.1 Real interaction with medium 38C.2 Virtual interaction with the medium 39– 1 –
Introduction
The natural final state of high energy hadronic or nuclear collisions are sprays of collimatedparticles consisting of hadrons and/or electrons. They are formed in an initial hard scattering,by which we mean a collision with a large transfer of momentum, followed by subsequentparton evolution know as a shower and fragmentation. Due to the large energies involvedin jet production, they can be studied via perturbative QCD since QCD is weakly coupledat high virtualities of a parton. Therefore the calculation of the initial production of jets isunder perturbative control, which makes jets powerful tools to probe the properties of thequark-gluon plasma (QGP) in heavy ion collisions.This is based on the premise, which is now widely accepted that heavy ion collisions arethe laboratory for the creation and study of the Quark Gluon Plasma medium. The highenergy collision of nuclei both at RHIC and the LHC creates sufficiently energetic partonsthat can escape confinement from color neutral hadrons and give rise to a strongly/weaklycoupled soup of quarks and gluons known as the Quark Gluon Plasma medium which, inthermal equilibrium is mainly characterized by its temperature. We can think of this plasmaas consisting of soft partons with typical energy of the order of the temperature of the mediumwhich is usually much lower than the center of mass energy of the initiating nuclear collision.These stopping collisions which create the QGP are accompanied by hard interactions whichcreate highly energetic partons which eventually form jets. These jets then have to traversethrough a region of the hot QGP as they evolve and hence they get modified in heavy ioncollision, compared with proton-proton collisions, due to the jet-medium interaction.A phenomenon that has been extensively studied in literature[1–20] is that of Jet quench-ing, which entails a suppression of particles with high transverse momenta in the medium.This has also been recently observed in experiments at both Relativistic Heavy Ion Collider(RHIC) [21–24] and Large Hadron Collider (LHC) [25–27]. The suppression mechanism hap-pens through the mechanism of energy loss when jets travel through the hot medium. Thekey to understand jet quenching and jet substructure modifications in heavy ion collisions isto understand how the jet interacts with the expanding medium. There has been tremendoustheoretical effort to study the jet energy loss mechanism (see Refs. [28–31] for recent reviews).The evolution of the jet in the medium usually depends on multiple scales such as thejet energy, the transverse momentum with respect to the jet axis, which will characterizethe collinearity of the jet and thermal scales of the QGP. In current heavy ion collisionexperiments, the temperature achieved lies in the range 150 −
500 MeV, and may not alwaysbe a perturbative scale. Thus, a fully weak coupling calculation may not be valid. A hybridmodel has been developed to address this problem [32–36], in which the initial jet productionand vacuum-like parton shower are calculated perturbatively, while the subsequent jet energyloss in the medium is calculated by mapping the field theory computation in the strongcoupling limit to a weak coupling computation in the classical gravity theory [37–42], i.e., byusing a modification of the AdS/CFT correspondence [43].In this paper, we will stick with the case of a weakly coupled QGP, i.e. the temperature– 2 –s much higher than the confinement scale Λ
QCD . The jet energy scale however, is muchhigher than the temperature. Due to the multiple scales involved in the problem, a powerfultool to deal with large hierarchies of scales is Effective Field Theory (EFT). The EFT thatis extensively used for jet studies at high energy colliders is Soft-Collinear Effective Theory(SCET). There is also a formulation of SCET (known as SCET G ) treating the Glauber gluon,which is a type of mode appearing in forward scattering, as a background field induced bythe medium interacting with an energetic jet. By making use of the collinear sector of thecorresponding EFT, this formalism has been used to address the question of jet quenchingin the medium [44–48]. I will use a complimentary approach using a new EFT for forwardscattering that has been developed recently [49] which also uses the Glauber mode to writedown contact operators between the soft and collinear momentum degrees of freedom whichis ideally suited for the situation we want to study.Another theoretical challenge in understanding the jet-medium interaction is the quantuminterference effect a phenomenon known as the Landau-Pomeranchuk-Migdal (LPM) effect.This effect has been discussed in literature extensivle [50–54].To take into account the interference effect, we can keep track of the time evolutionof the system’s density matrix. This can most easily be done by using the open quantumsystems formalism (for introductory books, see [55, 56]). For jets inside a QGP, if we onlyfocus on jet observables, the jet can be treated as an open quantum system interacting witha QGP bath. The application of the open quantum system formalism in heavy ion collisionshas been thriving in the study of color screening and regeneration of quarkonium [57–68].the understanding of quarkonium in-medium dynamics has been improved by combiningpotential nonrelativistic QCD (pNRQCD [69–71], an EFT of QCD) and the open quantumsystem formalism [72–75]. For example, a semiclassical Boltzmann transport equation ofquarkonium in the medium has been derived, under assumptions that are closely related witha hierarchy of scales [75–77].I would like to combine the tolls of EFT successfully applied for jet substructure com-putations in pp collisions with the open quantum system formalism and explore its physicalimplications on the jet-medium interaction. The long term goal is to develop a theoreticallyrobust formalism for computing jet substructure observables for both light parton and heavyquark jets. For example, the bottom quark jets have been identified as an effective probe ofthe QGP medium and will be experimentally studied at LHC, as well as by the sPHENIXcollaboration at RHIC. There has been recent work on computing jet substructure observablefor heavy quark jets in the context of proton-proton collisions [78, 79]. The objective wouldthen be to compute the same observables in heavy ion collisions and study modificationscaused by the medium.A first step was taken in [80] which looked at the transverse momentum spread of asingle energetic quark as a function of the time of propagation through the QGP medium.The multiple interactions of the quark with the medium were resummed by solving a Lindbladmaster equation for the quark jet. However for a realistic description of the system, we alsoneed to account for the initial hard interaction that creates the energetic quark which is– 3 –ressed with radiation from the subsequent parton shower which accounts for any vacuumevolution of the jet along with any medium interactions. At the same time a realistic finalstate jet produced in a heavy ion collision will usually have a large number of soft partonsthat originated from the QGP medium and are not directly associated with the evolution ofthe energetic jet. Thus any final state measurement imposed on the jet will have to accountfor these corrections which necessitates keeping track of the degrees of freedom of the QGPmedium. A way around this, which has long been used to deal with soft contamination fromMulti-Parton Interactions (MPIs) in pp collisions is that of jet grooming (For e.g. Soft-Drop[81]), which, for this paper, works by simply throwing away all the final state particles that donot pass an fractional energy cut-off ( z cut ). I maintain this cut-off to be large enough so thatall soft partons are automatically removed from the jet. This has the great advantage thatwe can once again formulate our problem in terms of the reduced density matrix of the jet,while tracing over all the degrees of freedom of the QGP medium. This also lets us avoid anyissues from the presence of non-global logarithms frequently encountered in jet substructureobservables in pp collisions.To implement this, I will borrow the tools developed in the context of pp collision forcomputing jet substructure observables. This field has progressed rapidly in recent years,both due to advances in explicit calculations, e.g. [97], as well as due to the developmentof techniques for understanding properties of substructure observables using analytic [99]approaches. Developments in jet substructure (see [100] for a recent review) have shown thatthe modified mass drop tagging algorithm (mMDT) or soft-drop grooming procedure robustlyremoves contamination from both underlying event and non-global color-correlations, see Refs.[81].The aim of this paper is to write down a factorization theorem for the reduced densitymatrix of of the jet within SCET, which subsequently allows me to resum any large logarithmsin the final state jet substructure measurements imposed on the jets. Simultaneously, I solvethe Lindblad evolution equation for the density matrix in the Markovian approximation,which also resums multiple interactions of the jet with the medium. Ultimately, I relate thisto the resummed cross section for the imposed jet substructure observable as a function ofthe propagation time of the jet through the medium. This paper sets up the theoreticalframework and we leave a detailed phenomenological study for the future.This paper is organized as follows. In Section 2 I introduce the physical system that wewish to study along with the final state measurements imposed and the relevant physical scalesthat play an important role in its description. I also describe the relevant momentum modesthat are dictated by these scales which will be a guide towards writing down a factorizationformula. The next Section 3 works out in detail, the factorization formula for the reduceddensity matrix of the jet within the framework of SCET. Section 4 deals with impact ofthe lack of coherence between the hard and medium interaction on the factorized densitymatrix. In Section 5 I present the results for the modified jet function presenting its UVand IR structure. Section 6 gives the form of the master evolution equation and solves isanalytically. Finally I conclude and discuss future directions in Section 7.– 4 – The observable
We want to a consider final state fat dijets events produced in a heavy ion collision in thebackground of a QGP medium. The jet are isolated using a suitable jet algorithm such asanti-kT with jet radius R ∼
1. We examine the scenario when the hard interaction creatingthe back to back jets happens at the periphery of the heavy ion collision, so that effectivelyonly one jet passes through the medium while the other evolves purely in vacuum. Since wedo not want to keep track of the soft partons coming from the QGP, we put a grooming onthe jets. We put in a simple energy cut-off sufficiently large to remove all partons at energyT and lower. Given a hard scale Q ∼ E J , where E J is the energy of the jet and an energycut-off, z cut E J , we work in the hierarchy Q (cid:29) z cut Q (cid:29) T (2.1)where T is the plasma temperature. The measurement we wish to impose is the transversemomentum imbalance between the two jets q T ∼ T . While this fixes the scaling of modesthat fail grooming, this does not necessarily guarantee collinear scaling inside the jet. Toensure that, we also put in a jet mass, e, measurement on both jets with Q √ e ∼ q T ∼ T .This measurement is identical to the one presented in [82] and we will refer the reader to thatpaper for a more detailed analysis of this observable.We wish to write down a factorization theorem within Soft Collinear Effective Theory(SCET)which separates out functions depending on their scaling in momentum space. This leads usto the following modes, same as ones described in [82]. p µs ∼ ( q T , q T , q T ) , Global Soft mode p µn ∼ Q (1 , λ c , λ c ) , λ c = √ e, collinear mode p µsc,n ∼ Qz cut (1 , λ sc , λ sc ) , λ sc = q T Qz cut , Soft-collinear mode p µcs,n ∼ Qz cut (1 , λ cs , λ cs ) , λ cs = r ez cut , Collinear-Soft mode (2.2)we have 3 more modes with n → ¯ n . Given the scaling we see that the Global soft modefails grooming. The Soft-collinear mode must also fail grooming since if it passes, then itwould change the scaling of jet mass that we demand. The emissions that fail grooming lieoutside the groomed jet and hence contribute to the the transverse momentum imbalance.The collinear soft mode can either pass or fail the grooming condition and contributes onlyto jet mass. The Collinear mode automatically passes grooming and also contributes to thejet mass. Notice that it has a virtuality of Q √ e ∼ q T so it can talk to the soft mode via aGlauber exchange.The Soft-collinear mode can also interact with the medium but we do not really care aboutwhat happens to that emission after it has come off the collinear particles so we ignore andsc-Glauber interactions. – 5 –he Collinear-soft mode would be thrown off-shell by a Glauber exchange with the mediumand hence we ignore any such interactions. So the only interaction with the medium that wecare about is that of the collinear mode.We assume that the particles that make up the medium also scale uniformly in temperatureand have the same scaling as the soft mode. We would like to derive a factorization formula for the system density matrix which, in thiscase is just the reduced density matrix of the jet interacting with the medium. For ease ofanalysis, we consider that the hard interaction that creates the jet is an e + e − collision. Whilethis is not a real scenario, it is an ideal playground to work out the EFT framework whichmainly deals with the final state physics. The EFT structure can then be easily carried overto the realistic case of nuclear/hadronic collisions, which we leave for future analysis as partof a detailed phenomenological application of the formalism developed in this paper. Whilea sketch of the factorization was provided in [82], we revisit the factorization in the contextof a density matrix evolution, which uses time ordered perturbation theory.The hard interaction can be encoded using an effective current operator O hard = C ( Q ) L µ J SCETµ (3.1) C ( Q ) is the Wilson co-efficient for this contact operator that depends only on the hard scaleQ. L µ is the initial state current, which in this case is just the lepton current, while J SCETµ would be the final state SCET current, which is just the gauge invariant quark current. L µ = ¯ lγ µ l, J SCETµ = ¯ χ n γ µ χ ¯ n where n = (1 , , , n = (1 , , , −
1) are light-like vectors pointing in the direction ofthe two jets. The initial state density matrix would then be ρ (0) = | e + e − ih e + e − | ⊗ ρ B (3.2)where we assume a time independent thermal density matrix for the medium. Since thepartons in the medium have the same scaling as the soft mode, we will henceforth suppressexplicitly writing out the factor ρ B till it becomes relevant for the Soft function analysis.We have also started with the assumption that the initial state participating in the hardinteraction is disentangled from the state of the QGP medium. We can follow the evolutionof this density matrix which will evolve with the effective Hamiltonian H = H QED + H IR + O hard (3.3)Since I am interested in a dijet event, I will consider only a single insertion of the hardoperator. The IR SCET Hamiltonian consists of the Hamiltonians that describe the IR modes– 6 –nd interactions between them including the Glauber. We will ignore any QED interactionshence we can drop the H QED piece hereafter. ρ ( t ) = e − iHt ρ (0) e iHt = e − iH IR t h e iH IR t e − iHt i ρ (0) h e iHt e − iH IR t i e iH IR t = e − iH IR t U ( t, ρ (0) U † ( t, e iH IR t (3.4)Our evolution operator U ( t,
0) now obeys the equation ∂ t U ( t,
0) = − i [ O hard,I ( t ) , U ( t, , with O hard,I ( t ) = e iH IR t O hard e − iH IR t (3.5)which has the solution U ( t,
0) = T n e − i R t dt O hard,I ( t ) o (3.6)Our solution for the density matrix now becomes ρ ( t ) = e − iH IR t T n e − i R t dt O hard,I ( t ) o ρ (0) ¯ T n e − i R t dt O hard,I ( t ) o e iH IR t (3.7)Since we will ultimately take a trace over the density matrix with some measurement, toleading order we insert one hard operator on each side of the cut. Then to leading order ρ ( t ) = e − iH IR t ρ (0) e iH IR t + Z t dt Z t dt e − iH IR t O hard ( t ) ρ (0) O † hard ( t ) e iH IR t (3.8)All the action happens in the second term with the hard interaction. Since we ultimatelywant to relate this to the cross section, only the second term which has connected diagramsis relevant, hence we will focus on that hereafter.We define σ ( t ) = Z t dt Z t dt e − iH IR t O hard ( t ) ρ (0) O † hard ( t ) e iH IR t = | C ( Q ) | I µν Z d x Z t dt Z d x Z t dt e − i ( x − x ) · ( p e + p ¯ e ) e − iH IR t J µSCET ( x ) | ih | J νSCET ( x ) e iH IR t (3.9)where I µν is the Lepton tensor. The IR Hamiltonian is written as a sum over theHamiltonians of all the SCET sectors( i.e., the collinear, Soft, Soft collinear et.) along withthe Glauber Hamiltonian, which in our case introduces a interaction between the Soft andCollinear sectors. H IR = H SCET + H G (3.10)– 7 –he Glauber Hamiltonian is expressed in terms of effective gauge invariant operators forquark-quark ( qq ), quark-gluon ( qg or gq ) and gluon-gluon ( gg ) interactions which have beenworked out in the Feynman gauge in Ref. [49] H G = X ij C ij O ijns O qqns = O qBn P ⊥ O q n Bs , O qgns = O qBn P ⊥ O g n Bs , O gqns = O gBn P ⊥ O q n Bs , O ggns = O gBn P ⊥ O g n Bs (3.11)where B is the color index and the subscripts n and s denote the collinear and soft operators.We will assume for the remainder of the paper that the jet which traverses the medium pointalong the n direction. C ij are the Wilson co-efficients for these contact operators and allbegin at O ( α s ). As motivated in [80], the dominant interaction of the jet with the mediumin mediated by the t channel exchange of the Glauber gluon.The Glauber operators then prevent us from factorizing the SCET current J µ in terms ofvarious momentum sectors. The way to deal with this is to expand out our result in powers ofthe Glauber Hamiltonian and establish factorization at each order in the Glauber expansion.To do this, we rearrange the the result above so as to be able to do a systematic expansionin the Glauber Hamiltonian. This assumes a weak coupling between the jet and the mediumand follows the same line of derivation as the Lindblad type equation for Open QuantumSystems.For convenience lets define Z d ˜ x = Z d x Z t dt Z d x Z t dt e − i ( x − x ) · ( p e + p ¯ e ) (3.12) σ ( t ) = | C ( Q ) | I µν Z d ˜ xe − iH IR t J µSCET ( x ) | ih | J νSCET ( x ) e iH IR t = | C ( Q ) | I µν Z d ˜ xe − iH SCET t n e iH SCET t e − iH IR ( t − t ) e − iH SCET t on e iH SCET t J µSC ( ~x , e − iH SCET t on e iH SCET t e − iH IR t o | ih | n e iH IR t e − iH SCET t on e iH SCET t J νSCET ( ~x , e − iH SCET t on e iH SCET t e iH IR ( t − t ) e − iH SCET t o e iH SCET t (3.13)Using the same process as for the hard function, we now have σ ( t ) = | C ( Q ) | I µν Z d ˜ xe − iH SCET t T n e − i R t dt H G,ISC ( t ) J µSCET,I SC ( x ) o | ih | ¯ T n e − i R t dt H G,ISC ( t ) J νSCET,I SC ( x ) o e iH SCET t (3.14)– 8 –here O I SC ( t ) = e iH SCET t Oe − iH SCET t (3.15)so that all operators are now dressed with the SCET Hamiltonian. We are now set up to doan expansion in the Glauber Hamiltonain. Ultimately, for this paper, we want to computethe trace over the reduced density matrix with an appropriate measurementΣ( t ) ≡ Tr[ σ ( t ) M ] (cid:12)(cid:12) t →∞ (3.16)We now turn towards deriving factorization formulas for this function at each order inthe Glauber expansion. The leading order term has no Glauber insertions and so should simply give us a resultproportional to the vacuum-background cross section. Since we are doing this in the contextof time ordered perturbation theory we will outline the proof for factorization of this piecehere. σ (0) ( t ) = | C ( Q ) | I µν Z d ˜ xe − iH SC ( t − t ) J µSCET ( ~x ) e − iH SCET t | ih | e iH SCET t J νSCET ( ~x ) e iH SCET ( t − t ) where, for ease of notation, we have defined Z d ˜ x = Z d x Z t dt Z d x Z t dt e − i ( x − x ) · ( p e + p ¯ e ) (3.17)Now we separate out the interactions from the free theory Hamiltonian writing H SCET = H + H int (3.18)then performing our usual trick, we can write σ (0) ( t ) = | C ( Q ) | I µν Z d ˜ xe − iH t T n e − i R t dt H int,I ( t ) J µSCET,I ( x ) o | ih | ¯ T n e − i R t dt H int,I ( t ) J νSCET,I ( x ) o e iH t where O I ( t ) = e iH t Oe − iH t (3.19)To proceed further, we put in our measurement on the dijets and take a trace over final statesand take the limit t → ∞ . – 9 – X | σ ( t → ∞ ) M| X i ≡ Σ (0) = | C ( Q ) | I µν Z d ˜ x h X |T n e − i R ∞ dt H int,I ( t ) J µSCET,I ( x ) o | ih | ¯ T n e − i R ∞ dt H int,I ( t ) J νSCET,I ( x ) o M| X i where we have M = δ (cid:16) ~q T − ~p ⊥ n, groomed jet − ~p ⊥ ¯ n, groomed jet (cid:17) δ ( e n − e n, groomed jet ) δ ( e ¯ n − e ¯ n, groomed jet )where ~q T is the transverse momentum imbalance between the two groomed jets, while e n , e ¯ n measures the jet mass for the two groomed jets. Our Hamiltonian now looks like H int = H S + n ( H n + H sc,n + H cs,n ) + n ↔ ¯ n o (3.20)while the interactions between the various sectors now appear in the form of Wilson lines inthe SCET current Without the presence of factorization violating Glauber interactions, usingstandard techniques outlined in [82], the momentum sectors can now be written as separatematrix elements. Hence the Hilbert space also factorizes into various momentum modes | X i = | X n i| X ¯ n i| X s i| X cs,n i| X cs, ¯ n i| X sc,n i| X sc, ¯ n i (3.21)Since the Hamiltonian is already factorized, we have, in principle, a factorization of all themodes at this stage. However, we still have to implement our power counting on the mea-surement functions which will ensure that only leading power corrections are retained.Acting on the final state Hilbert space, we can then pull out the co-ordinate dependence ofthe SCET current and perform all co-ordinate integralsΣ (0) = | C ( Q ) | I µν Z d x Z d x e − i ( x − x ) · ( p e + p ¯ e − p Xn − p X ¯ n − p Xs − p Xcs,n − p Xcs, ¯ n − p Xsc,n − p Xsc, ¯ n ) × h X |T n e − i R ∞ dt H int,I ( t ) J µSCET,I (0) o | ih | ¯ T n e − i R ∞ dt H int,I ( t ) J νSCET,I (0) o M| X i Performing the integrals over x and x now gives momentum conserving δ function along witha 4d volume factor V. We can then decompose the 4 momentum delta function in light-coneco-ordinates and apply power counting δ ( p e + p ¯ e − p X n − p X ¯ n − p X s − p X cs,n − p X cs, ¯ n − p X sc,n − p X sc, ¯ n ) → δ ( Q − p − X n ) δ ( Q − p + X ¯ n ) δ ( p ⊥ X groomedjet,n + p ⊥ X ¯ n,groomedjet + p ⊥ X s + p ⊥ ,failX sc,n + p ⊥ ,failX sc, ¯ n )(3.22)– 10 –here the superscript f ail indicates that the only the Sc radiation that fails the groomingcondition contributes to the transverse momentum imbalance. Our factorization now becomesΣ (0) = V × | C ( Q ) | I µν h X |T n e − i R ∞ dt H int,I ( t ) J µSCET,I (0) o | ih | ¯ T n e − i R ∞ dt H int,I ( t ) J νSCET,I (0) o | X i× δ ( Q − p − X n ) δ ( Q − p + X ¯ n ) δ ( p ⊥ X groomedjet,n + p ⊥ X ¯ n,groomedjet + p ⊥ X s + p ⊥ ,failX sc,n + p ⊥ ,failX sc, ¯ n ) × δ (cid:16) ~q T − ~p ⊥ n, groomed jet − ~p ⊥ ¯ n, groomed jet (cid:17) δ ( e n − e n, groomed jet ) δ ( e ¯ n − e ¯ n, groomed jet )Without loss of generality, we can assume that the axis of the ¯ n groomed jet is exactly alignedwith the ¯ n direction, in which case its transverse momentum becomes zero.Σ (0) = V × | C ( Q ) | I µν h X |T n e − i R ∞ dt H int,I ( t ) J µSCET,I (0) o | ih | ¯ T n e − i R ∞ dt H int,I ( t ) J νSCET,I (0) o M| X i× δ ( Q − p − X n ) δ ( Q − p + X ¯ n ) δ ( ~q T + p ⊥ X s + p ⊥ ,failX sc,n + p ⊥ ,failX sc, ¯ n ) δ P ⊥ ¯ n × δ (cid:16) ~q T − ~p ⊥ n, groomed jet (cid:17) δ ( e n − e n, groomed jet ) δ ( e ¯ n − e ¯ n, groomed jet ) δ P ⊥ ¯ n is a Kronecker delta setting the transverse momentum of the ¯ n groomed jet to 0. Wenow have a transverse momentum condition δ (cid:16) ~q T − ~p ⊥ n, groomed jet (cid:17) , which tells us that the ngroomed jet is not exactly aligned with the n axis. However, we can use RPI I invariance offrom SCET, to adjust the axis of the groomed jet without changing any physics , so that thiscondition can simply to written as δ ( ~p ⊥ n, groomed jet ) which now gets us back to the standarddefinition of the jet function. We also see that the transverse momentum imbalance receivescontributions from all the modes that fail grooming.We can convert the Kronecker delta to a Dirac delta following literature [83] δ P ⊥ ¯ n = πQ δ (cid:16) p ⊥ X ¯ n,groomedjet (cid:17) (3.23)The jet mass measurement receives contributions from the modes that pass soft-drop,which are the Collinear Soft and Collinear modes in each groomed jet. We can now write thefinal form of our factorized density matrixΣ (0) = V × H ( Q, µ ) × S ( ~q T ; µ ) ⊗ q T J ⊥ n ( e n , Q, z cut , ~q T ; µ ) ⊗ q T J ⊥ ¯ n ( e ¯ n , Q, z cut , ~q T ; µ )(3.24)– 11 –here ⊗ q T indicates a convolution in ~q T . H(Q) is the hard function which also includes theborn level term. Each J functions can be further written as J ⊥ n ( e n , Q, z cut , ~q T ; µ ) = S ⊥ sc,n ( Qz cut , ~q T ) × Z de S cs,i ( e n − e , Qz cut ) J n ( e , Q ) ≡ S ⊥ sc,n ( Qz cut , ~q T ) × S cs,i ( e n , Qz cut ) ⊗ e n J n ( e n , Q ) (3.25)with the functions defined as follows S ( ~q T ) = 1 N R tr h X S |T n e − i R ∞ dt H S ( t ) S † ¯ n S n (0) o | ih | ¯ T n e − i R ∞ dt H S ( t ) S † n S ¯ n (0) o δ ( ~q T − P ⊥ ) | X S i The trace here is a trace over color and its understood that | X S ih X S | includes a sum over softstates with their phase space integrated over. This computes the Soft function in a vacuumbackground, but as we know we actually have a background of the QGP which also scales asthe soft mode. So, in principle, we have S ( ~q T ) = 1 N R tr h X S |T n e − i R ∞ dt H S ( t ) S † ¯ n S n (0) o ρ QGP ¯ T n e − i R ∞ dt H S ( t ) S † n S ¯ n (0) o δ ( ~q T − P ⊥ ) | X S i where we have assumed a time independent QGP background. Of course, we can take intoaccount the fact that the time scales for the soft emission( which puts the collinear modeoff-shell) is much shorter than the formation time for the QGP, in which case we wouldbe justified to compute the soft function in a vacuum background, which is what we willassume for the rest of this paper. However, I stress that this assumption in no way affectsthe derivation of the factorization and is an effect which can be readily incorporated in futurecalculations. The jet function is defined as J q ( e, Q ) = (2 π ) N c tr h | ¯ T n e − i R ∞ dt H n ( t ) / ¯ n χ n o δ ( Q − P − ) δ ( P ⊥ ) δ ( e − E ) | X n ih X n |T n e − i R ∞ dt H n ( t ) ¯ χ n (0) o | i Similarly, we have S cs ( e, Qz cut ) = 1 N R tr h | ¯ T n e − i R ∞ dt H cs ( t ) U † n W t o δ ( e − (1 − Θ Groom ) E ) | X cs ih X cs ||T n e − i R ∞ dt H cs ( t ) W † t U n o | i there are two such functions, one for each jet. Finally we have the Soft Collinear function S sc ( ~q T , Qz cut ) = 1 N R tr h | ¯ T n e − i R ∞ dt H sc ( t ) U † n W t o Θ groom δ ( ~q T − Θ groom P ⊥ ) | X sc ih X sc |T n e − i R ∞ dt H n ( t ) W † t U n o | i The one loop results and the resummation of all the large logarithms was already pre-sented in [82]. For convenience we collect all the results in Appendix A.– 12 – .2 Next-to-Leading order in Glauber
We now consider the next to leading order term in the expansion of the Glauber Hamiltonianstarting from Eq.3.16. In the previous paper, the leading term we considered was the quadraticinsertion of the Glauber Hamiltonian which was the first non-trivial interaction with themedium. In the present case, we have a non-trivial soft function consisting of Soft Wilsonlines which can lead to interfering diagrams between the soft function and the medium.However, as stated in the previous section, if the time scale for Soft emissions is shorter thanthe QGP formation time, then we can factorize the QGP interactions from the explicit softradiation off the quark in which case we need to do atleast a quadratic Glauber insertion.This is what we will consider here. We can have two contributions depending on the the twoGlauber insertions on the same or opposite sides of the cut which respectively correspond tovirtual and real interactions of the medium partons. • Glauber insertion on both sides of the cut σ (1) a ( t ) = | C ( Q ) | I µν Z d ˜ xe − iH SCET t T n Z t dt H G,I SC ( t ) J µSCET,I SC ( x ) o | ih | ¯ T n Z t d ˆ tH G,I SC (ˆ t ) J νSCET,I SC ( x ) o e iH SCET t (3.26)By following the same series of steps as for the leading order term, we can write theresult in terms of the free theory interaction picture. σ (1) a ( t ) = | C ( Q ) | I µν Z d ˜ xe − iH t T n e − i R t dt H int,T ( t ) Z t dt a H G,I ( t a ) J µSCET,I ( x ) o | i× h | ¯ T n e − i R t dt H int,I ( t ) Z t dt b H G,I ( t b ) J νSCET,I ( x ) o e iH t The next step is to obtain a factorized formula in terms of our EFT modes. To do this,we explicitly put in the form of our Glauber operator, considering the case of collinearquarks interacting with the soft quarks in the medium. The rest of the operators canbe treated in exactly the same manner. H G ( x ) = C qq O An ( x ) O AS ( x ) (3.27)We can now take the trace over the density matrix inserting our measurement as before. h X |M σ (1) a ( t → ∞ ) | X i ≡ Σ (1) a = | C ( Q ) | I µν Z d ˜ x h X |T n e − i R t dt H int,T ( t ) Z t dt a H G,I ( t a ) J µSCET,I ( x ) o | i× h | ¯ T n e − i R t dt H int,I ( t ) Z t dt b H G,I ( t b ) J νSCET,I ( x ) o M| X i – 13 –e can now follow the same series of steps as for the leading order term, and apply powercounting to measurement functions as well as the momentum conserving δ functionsbased on the momentum scaling of each mode. Accordingly ,we factorize the Hilbertspace of the final states in terms of the momentum scaling of the modes and pull out theco-ordinate dependence of each mode by acting with the operators on the final state.This yields the following intermediate factors I = Z d x e − ix · ( p e + p ¯ e − p Jn, − p JS, − p ¯ n − p cs,n − p cs, ¯ n − p sc,n − p sc, ¯ n ) Z d x e − ix · ( p e + p ¯ e − p Jn, − p JS, − p ¯ n − p cs,n − p cs, ¯ n − p sc,n − p sc, ¯ n ) Z d x a e − ix a · ( p Gn, + p Gs, ) × Z d x b e − ix b · ( p Gn, + p Gs, ) (3.28)where the subscripts G, J tell us whether the momentum is coming from the action ofthe Glauber fields or the SCET current. Now performing the integrals and applyingpower counting, we have I = δ ( Q − p − Jn ) δ ( Q − p +¯ n ) δ ( p ⊥ Jn, + p ⊥ Js, + p ⊥ ¯ n + p ⊥ ,failSc,n + + p ⊥ ,failSc, ¯ n ) δ ( Q − p − Jn ) δ ( Q − p +¯ n ) δ ( p ⊥ Jn, + p ⊥ Js, + p ⊥ ¯ n + p ⊥ ,failSc,n + + p ⊥ ,failSc, ¯ n ) δ ( p − Gn, ) δ ( p + GS, + p + Gn, ) δ ( p ⊥ Gn, + p ⊥ GS, ) δ ( p − Gn, ) δ ( p + GS, + p + Gn, ) δ ( p ⊥ Gn, + p ⊥ GS, ) (3.29)where we have ignored any factors of 2 π which will be absorbed in the overall co-efficentfor Σ (1) a . We also have additional constraints since the total momentum for a particularmode must match on both sides of the cut p Jn, + p Gn, = p Jn, + p Gn, p JS, + p GS, = p JS, + p GS, (3.30)We can simplify our measurement δ functions using these set of constraints δ ( ~q T − p ⊥ ngroomed − p ⊥ ¯ ngroomed ) ≡ δ ( ~q T − p ⊥ n − p ⊥ , ¯ n )= δ ( ~q T − p ⊥ Gn, − p ⊥ Jn, − p ⊥ , ¯ n )= δ ( ~q T + [ p ⊥ GS, + p ⊥ JS, ] + p ⊥ ,failsc,n + p ⊥ ,failsc, ¯ n ) (3.31)where the term inside the square brackets is the total contribution from the Soft sector,which includes both the vacuum as well as medium effects. Then coming back to ourtransverse momentum conserving δ function, we have δ ( p ⊥ Jn, + p ⊥ Js, + p ⊥ ¯ n + p ⊥ ,failSc,n + + p ⊥ ,failSc, ¯ n ) = δ ([ p ⊥ Jn, + p ⊥ Gn, ] + p ⊥ ¯ n − ~q T ) = δ ( p ⊥ n + p ⊥ ¯ n − ~q T )(3.32)– 14 –s for the leading order term, we can set the axis of the ¯ n groomed jet to be exactlyaligned with the ¯ n axis and then using RPI I , do the same for the n jet. Using the restof the constraints then, once again we have an overall factor of V × T .Since we are ignoring interference between the Soft operators of the SCET and theGlauber insertion, we can set p GS, = p GS, ≡ p GS , so that p JS, = p JS, With all these simplifications, we are left with I = δ ( Q − p − n ) δ ( Q − p +¯ n ) δ ( p ⊥ n ) πQ δ ( p ⊥ ¯ n ) δ ( p − Gn, ) δ ( p + GS + p + Gn, ) δ ( p ⊥ Gn, + p ⊥ GS ) δ ( p − Gn, ) δ ( p + GS + p + Gn, ) δ ( p ⊥ Gn, + p ⊥ GS ) (3.33)We can now write down our factorization formula for Σ (1) a .Σ (1) a = V × | C qq | H ( Q, µ ) n Z d ~q JS S ( ~q JS ) on Z d ~q Sc, ¯ n S sc, ¯ n ( Qz cut , ~q Sc, ¯ n ) on Z d ~q Sc,n S sc,n ( Qz cut , ~q Sc,n ) on Z d ~q GS dp + GS S ABG ( q ⊥ GS , p + GS ) on Z d ~p ⊥ Gn, d ~p ⊥ Gn, dp + Gn, dp + Gn, J ABn ( e n , ~p ⊥ Gn, , ~p ⊥ Gn, , p + Gn, , p + Gn, ) o ⊗ e n CS n ( Qz cut , e n ) J ¯ n ( e ¯ n ) ⊗ e ¯ n CS ¯ n ( Qz cut , e ¯ n ) × δ ( ~q T + ~q JS + ~q GS + ~q Sc,n + ~q Sc, ¯ n ) × δ ( ~q GS + ~p ⊥ Gn, ) δ ( ~q GS + ~p ⊥ Gn, ) δ ( p + GS + p + Gn, ) δ ( p + GS + p + Gn, ) (3.34)In order to simplify notation, its easiest to express this result by rewriting some of themomentum conserving δ functions in co-ordinate space. This gives usΣ (1) a = V × | C qq | H ( Q, µ ) n Z d ~q JS S ( ~q JS ) on Z d ~q Sc, ¯ n S sc, ¯ n ( Qz cut , ~q Sc, ¯ n ) on Z d ~q Sc,n S sc,n ( Qz cut , ~q Sc,n ) o × Z d x Z d y n Z d ~q GS S ABG ( q ⊥ GS , { x ⊥ , x − } , { y ⊥ , y − } ) on J ABn ( e n , x, y ) o ⊗ e n CS n ( Qz cut , e n ) J ¯ n ( e ¯ n ) ⊗ e ¯ n CS ¯ n ( Qz cut , e ¯ n ) × δ ( ~q T + ~q JS + ~q GS + ~q Sc,n + ~q Sc, ¯ n )– 15 –hile most of the functions remain unchanged compared to their vacuum counterparts,we have two new/modified function S ABG and J ABn defined as S ABG ( q ⊥ GS , { x ⊥ , x − } , { y ⊥ , y − } ) = h X S |{ δ ( q ⊥ GS − P ⊥ ) O AS ( x ⊥ , x − ) } ρ B O BS ( y ⊥ , y − ) | X S i J ABn ( e n , x, y ) = h X n | T n ¯ χ n (0) / ¯ n O An ( x ) o | ih | ¯ T n O Bn ( y ) χ n (0) o δ ( P ⊥ ) δ ( Q − P − ) δ ( e n − E n ) | X n i (3.35)Using the translational invariance of the QGP medium we can write S ABG ( q ⊥ GS , { x ⊥ , x − } , { y ⊥ , y − } ) = Z d k (2 π ) k ⊥ e i (ˆ x − ˆ y ) · k D AB> ( k ) δ ( q ⊥ GS − ~k ⊥ ) (3.36)where ˆ x ∈ ( x + , , ~x ⊥ ) and ˆ y ∈ ( y + , , ~y ⊥ ), and as derived in [80], D AB> ( k ) is the Wight-man function in the thermal bath. We note here that this function is independent of k + .We therefore writeΣ (1) a = V × | C qq | H ( Q, µ ) n Z d ~q JS S ( ~q JS ) on Z d ~q Sc, ¯ n S sc, ¯ n ( Qz cut , ~q Sc, ¯ n ) on Z d ~q Sc,n S sc,n ( Qz cut , ~q Sc,n ) o ⊗ e n CS n ( Qz cut , e n ) ⊗ e n Z d q ⊥ GS Z d k (2 π ) k ⊥ D AB> ( k ) δ ( q ⊥ GS − ~k ⊥ ) Z d x Z d ye i (ˆ x − ˆ y ) · k n J ABn ( e n , x, y ) o × J ¯ n ( e ¯ n ) ⊗ e ¯ n CS ¯ n ( Qz cut , e ¯ n ) δ ( ~q T + ~q JS + ~q GS + ~q Sc,n + ~q Sc, ¯ n ) (3.37) • Glauber insertion on the same side of the cut
We can now look the piece we get by doing two Glauber insertions on the same side ofthe cut starting from Eq. 3.16. Here, I will consider inserting on the bra side and dealwith the ket in the same manner later on. σ (1) b ( t ) = ( − i ) I µν | C ( Q ) | Z d ˜ xe − iH SCET t T n Z t dt H G,I SC ( t ) Z t d ˆ tH G,I SC (ˆ t ) J µSCET,I SC ( x ) o | ih | ¯ T n J νSCET,I SC ( x ) o e iH SCET t This can again be written in terms of the free theory interaction picture following thesame steps as in the previous section – 16 – (1) b ( t ) = ( − i ) I µν | C ( Q ) | Z d ˜ xe − iH t T n e − i R t dt i H int,I ( t i ) Z t dt H G,I ( t ) Z t d ˆ tH G,I (ˆ t ) J µSCET,I ( x ) o | i× h | ¯ T n e − i R t dt H int,I ( t ) J νSCET,I ( x ) o e iH t As before, we can pull out the co-ordinate dependence explicitly from each momentumsector in order to get momentum conserving δ functions. In this case we obtain thefollowing co-ordinate integrals I = Z d x e − ix · ( p e + p ¯ e − p Jn − p JS − p ¯ n − p cs,n − p cs, ¯ n − p sc,n − p sc, ¯ n ) Z d x e − ix · ( p e + p ¯ e − p n − p S − p ¯ n − p cs,n − p cs, ¯ n − p sc,n − p sc, ¯ n ) Z d x a e − ix a · ( p Gn,a + p GS,a ) × Z d x b e − ix b · ( p Gn,b + p GS,b ) (3.38)along with the constraints matching the momentum in a particular sector on both sidesof the cut. p Jn + p Gn,a + p Gn,b = p n p JS + p GS,a + p GS,b = p S (3.39)which after applying power counting leads to I = δ ( Q − p − Jn ) δ ( Q − p +¯ n ) δ ( p ⊥ Jn + p ⊥ Js + p ⊥ ¯ n + p ⊥ ,failSc,n + + p ⊥ ,failSc, ¯ n ) δ ( Q − p − n ) δ ( Q − p +¯ n ) δ ( p ⊥ n + p ⊥ s + p ⊥ ¯ n + p ⊥ ,failSc,n + + p ⊥ ,failSc, ¯ n ) δ ( p − Gn,a ) δ ( p + GS,a + p + Gn,a ) δ ( p ⊥ Gn,a + p ⊥ GS,a ) δ ( p − Gn,b ) δ ( p + GS,b + p + Gn,b ) δ ( p ⊥ Gn,b + p ⊥ GS,b )The measurement of the transverse momentum imbalance now reads δ ( ~q T − p n, groomed jet − p ¯ n, groomed jet ) = δ ( ~q T − p ⊥ n − p ⊥ ¯ n )= δ ( ~q T + p ⊥ s + p ⊥ ,failSc,n + p ⊥ ,failSc, ¯ n ) = δ ( ~q T + p ⊥ Js + p ⊥ GS,a + p ⊥ GS,b + p ⊥ ,failSc,n + p ⊥ ,failSc, ¯ n )So we see that with slight modifications, everything works out in the same manner asfor the previous case. Since we are not including any interference between the SCETand the Glauber Soft functions, we therefor have p GS,a + p GS,b = 0 (3.40)– 17 –o, in this case all the qT imbalance only comes from the SCET vacuum correctionswhich makes sense since this piece only has virtual interactions with the medium. Ourconstraints now become I = V πQ δ ( Q − p − Jn ) δ ( Q − p +¯ n ) δ ( p ⊥ n ) δ ( p ⊥ ¯ n ) δ ( p − Gn,a ) δ ( p + GS,a + p + Gn,a ) δ ( p ⊥ Gn,a + p ⊥ GS,a ) δ ( p − Gn,b ) δ ( p + GS,b + p + Gn,b ) δ ( p ⊥ Gn,b + p ⊥ GS,b )The form of the factorization theorem remains unchanged although the definitions ofthe various functions now changeΣ (1) b = − V × | C qq | H ( Q, µ ) n Z d ~q JS S ( ~q JS ) on Z d ~q Sc, ¯ n S sc, ¯ n ( Qz cut , ~q Sc, ¯ n ) on Z d ~q Sc,n S sc,n ( Qz cut , ~q Sc,n ) on Z d ~q GS dp + GS S AG ( q ⊥ GS , p + GS ) on Z d ~p ⊥ Gn, d ~p ⊥ Gn, dp + Gn,a dp + Gn,b J An ( e n , ~p ⊥ Gn,a , ~p ⊥ Gn,b , p + Gn,a , p + Gn,b ) o ⊗ e n CS n ( Qz cut , e n ) J ¯ n ( e ¯ n ) ⊗ e ¯ n CS ¯ n ( Qz cut , e ¯ n ) × δ ( ~q T + ~q JS + ~q Sc,n + ~q Sc, ¯ n ) × δ ( ~q GS + ~p ⊥ Gn,a ) δ ( − ~q GS + ~p ⊥ Gn,b ) δ ( p + GS + p + Gn,a ) δ ( − p + GS + p + Gn,b )As before , for compactness we write this asΣ (1) b = − V × | C qq ( Q ) | H ( Q, µ ) n Z d ~q JS S ( ~q JS ) on Z d ~q Sc, ¯ n S sc, ¯ n ( Qz cut , ~q Sc, ¯ n ) on Z d ~q Sc,n S sc,n ( Qz cut , ~q Sc,n ) o Z d x Z d y n S ABG ( { x − , x ⊥ } , { y − , y ⊥ } ) on J ABn ( e n , x, y ) o ⊗ e n CS n ( Qz cut , e n ) J ¯ n ( e ¯ n ) ⊗ e ¯ n CS ¯ n ( Qz cut , e ¯ n ) × δ ( ~q T + ~q JS + ~q Sc,n + ~q Sc, ¯ n ) (3.41)with the following definitions S ABG ( { x − , x ⊥ } , { y − , y ⊥ } ) = h X S | T n O AS ( x ⊥ , x − ) O BS ( y ⊥ , y − ) o ρ B | X S i J ABn ( e n , x, y ) = h X n | T n ¯ χ n (0) / ¯ n O An ( x ) O Bn ( y ) o | ih | ¯ T n χ n (0) o δ ( P ⊥ ) δ ( Q − P − ) δ ( e n − E n ) | X n i (3.42)The term with Glauber insertions on the other side of the cut can now be triviallyobtained from this result. – 18 – Decoherence of the hard interaction from the medium
One aspect of this factorization which is different compared to a vacuum factorization resultis the presence of the environment. Since the QGP medium in not coherently connected withthe hard interaction that produces the jet, the phase space of the jet allows the partons to goon-shell before they interact with the medium. In fact, the most dominant contribution tothe cross section comes from this region of phase space. To see this explicitly we can look atthe tree level result for our modified jet function defined in Eq.3.35 that appears in the thefactorized formula for Σ (1) a in Eq.3.37 Z d x Z d ye − ik · (ˆ x − ˆ y ) J ABn ( e n , x, y )= Z d x Z d ye − ik · (ˆ x − ˆ y ) h X n | T n O An ( x ) ¯ χ n (0) / ¯ n o | ih | ¯ T n χ n (0) O Bn ( y ) o δ ( P ⊥ ) δ ( Q − P − ) | X n i = Z d x Z d ye − ik · (ˆ x − ˆ y ) e ip · ( x − y ) Z ˜ dp Tr h ¯ u ( p ) T A / ¯ n D ( x ) / ¯ n D † ( y ) / ¯ n T B u ( p ) i δ ( p ⊥ ) δ ( Q − p − )= Z d x Z d ye − ik · (ˆ x − ˆ y ) e ip · ( x − y ) Z ˜ dp Tr h ¯ u ( p ) T A / ¯ n Z d q /n q − e − iq · x q − i(cid:15) / ¯ n Z d q /n q ) − e q · y ( q ) + i(cid:15) / ¯ n T B u ( p ) i δ ( p ⊥ ) δ ( Q − p − ) (4.1)where henceforth we will reserve the notation ˜ dp to mean the integral over the phase spaceof the on-shell massless particle p ˜ dp = d p E p ≡ d pδ + ( p ) (4.2)The integrals over the co-ordinates x and y now sets q = q = p + k . Since we are integratingover k, we see that there is a pinch singularity in the integral over k + and the dominant con-tribution comes from the region when the intermediate propagators(in q = q ) go on-shell. Sowe can replace the propagators with their on-shell condition ignoring for the rest of this paperany contribution from the residual principal value. We leave for the future an investigationof the corrections from the principal value terms.We can therefore revisit our factorized functions and simplify them by looking at this domi-nant contribution where the intermediate partons produced in the hard interaction go on-shellbefore interacting with the medium. • Σ (1) a For convenience we give our factorized formula for the Glauber insertion on opposite– 19 –ides of the cut Eq. 3.37Σ (1) a = V × H ( Q, µ ) | C qq ( Q ) | n Z d ~q JS S ( ~q JS ) on Z d ~q Sc, ¯ n S sc, ¯ n ( Qz cut , ~q Sc, ¯ n ) on Z d ~q Sc,n S sc,n ( Qz cut , ~q Sc,n ) o ⊗ e n CS n ( Qz cut , e n ) ⊗ e n Z d k (2 π ) k ⊥ D AB> ( k ) Z d x Z d ye i (ˆ x − ˆ y ) · k n J ABn ( e n , x, y ) o × J ¯ n ( e ¯ n ) ⊗ e ¯ n CS ¯ n ( Qz cut , e ¯ n ) δ ( ~q T + ~q JS + ~k ⊥ + ~q Sc,n + ~q Sc, ¯ n ) (4.3)Since we are working in a regime where the intermediate partons are going on-shell,we can do a a further factorization of the jet function to separate out the Glauber andvacuum terms explicitly. What this will do is separate out a piece which is now lookslike the standard jet function in the SCET piece, albeit, this is now convoluted withthe Glauber jet function due to the jet mass measurement which happens on the finalstate. While both the hard current as well as the Glauber vertex receives higher ordercorrections from the SCET Hamiltonian, the large logarithms that dominate the crosssection come from the SCET corrections to the hard vertex. Hence for this paper, wewill only evaluate the Glauber vertex at tree level while allowing for all order correctionsto the hard interaction, although it is straightforward to compute higher order SCETcorrections to the Glauber vertex given our factorized formula. To proceed, we caninsert a complete set of on-shell states separating the hard and Glauber operators,which is allowed since we are going to place the intermediate partons on-shell. J AB ( e n , k ) = Z d x Z d ye i (ˆ x − ˆ y ) · k n J ABn ( e n , x, y ) o = Z d x Z d ye − ik · (ˆ x − ˆ y ) h X n | O An ( x ) | Y n ih Y n | δ ( P ⊥ ) δ ( Q − p − ) T n e − i R dt H n ( t ) χ n (0) ¯ χ / ¯ n o | i× h | ¯ T n e − i R dt H n ( t ) χ n (0) | ˜ Y n ih ˜ Y n | O Bn ( y ) δ ( e n − E ) | X n i (4.4)which can be rearranged to give J AB ( e n , k ) = Z d x Z d ye − ik · (ˆ x − ˆ y ) h X n | O An ( x ) | Y n ih ˜ Y n | O Bn ( y ) δ ( e n − E ) | X n i× h Y n | δ ( P ⊥ ) δ ( Q − p − ) T n e − i R dt H n ( t ) ¯ χ n (0) / ¯ n o | ih | ¯ T n e − i R dt H n ( t ) χ n (0) | ˜ Y n i≡ G AB J (4.5)– 20 –here its understood that there is an integral over the phase space of all the insertedstates Y n , ˜ Y n . The Glauber piece can now be evaluated at tree level. G AB = Z d x Z d ye − ik · (ˆ x − ˆ y ) h X n | O An ( x ) | Y n ih ˜ Y n | O Bn ( y ) δ ( e n − E ) | X n i = Z d x Z d ye − ik · (ˆ x − ˆ y ) e − ix · ( p Y − p X ) e iy · ( p ˜ Y − p X ) h X n | O An (0) | Y n ih ˜ Y n | O Bn (0) δ ( e n − E ) | X n i = δ ( P − X − p − Y ) δ ( k + + p + Y − p + X ) δ ( k ⊥ + p ⊥ Y + p ⊥ X ) δ ( P − X − p − ˜ Y ) δ ( k + + p +˜ Y − p + X ) δ ( k ⊥ + p ⊥ ˜ Y + p ⊥ X ) h X n | O An (0) | Y n ih ˜ Y n | O Bn (0) δ ( e n − E ) | X n i (4.6)The δ functions then set the total momentum p Y = p ˜ Y . For now, since we only have aquark operator, this will create and annihilate a single quark which we choose to havemomenta p X and p Y . So we have G AB = Π ∞ i =1 Z ˜ dp Xi ˜ dp Y ˜ dp ˜ Y δ ( P − X − p − Y ) δ ( k + + p + Y − p + X ) δ ( k ⊥ + p ⊥ Y + p ⊥ X ) × δ ( P − X − p − ˜ Y ) δ ( k + + p +˜ Y − p + X ) δ ( k ⊥ + p ⊥ ˜ Y + p ⊥ X )Tr[¯ u ( p X ) T A / ¯ n u ( p Y )¯ u ( p ˜ Y ) T B u ( p X )] × Π ∞ i =2 Z ˜ dp Y i Z ˜ dp ˜ Y i [2 E p Y i δ ( p Xi − p Y i )][2 E p ˜ Y i δ ( p Xi − p ˜ Y i )] δ ( e n − Q X i,j p Xi · p Xj )= Π ∞ i =1 Z ˜ dp Xi ˜ dp Y ˜ dp ˜ Y δ ( P − X − p − Y ) δ ( k + + p + Y − p + X ) δ ( k ⊥ + p ⊥ Y + p ⊥ X ) × δ ( p − X − p − ˜ Y ) δ ( k + + p +˜ Y − p + X ) δ ( k ⊥ + p ⊥ ˜ Y + p ⊥ X )Tr[¯ u ( p X ) T A / ¯ n u ( p Y )¯ u ( p ˜ Y ) T B u ( p X )] δ ( e n − Q X i,j p Xi · p Xj ) (4.7)We can now eliminate the phase space integrals over p Y and p ˜ Y . This will, as before,lead to a redundant δ function in energy which we will interpret as a factor of t. G AB = t Π ∞ i =1 Z ˜ dp Xi Tr[ T A T B ] δ ( e n − Q X i,j p Xi · p Xj ) δ p + X − k + + ( p ⊥ X − k ⊥ ) p − X ! (4.8)– 21 –ow putting everything together we have J AB ( e n , k ) = t Π ∞ i =1 Z ˜ dp Xi Tr[ T A T B ] δ ( e n − Q X i,j p Xi · p Xj ) δ p + X − k + − ( p ⊥ X − k ⊥ ) p − X ! h P X − k, P Xi | δ ( P ⊥ ) δ ( Q − p − ) T n e − i R dt H n ( t ) ¯ χ n (0) / ¯ n o | ih | ¯ T n e − i R dt H n ( t ) χ n (0) | P X − k, P Xi i (4.9)we can make a change of variables taking p X − k → p X . We can rearrange themeasurement function as M = X i,j p Xi · p Xj = X j =1 p X · p Xj + X i,j =1 p Xi · p Xj → X i,j p Xi · p Xj + X j k · p Xj − k · p X Now using the fact that P j p − Xj = Q , P j ~p ⊥ Xj = 0 and using the power counting of ourGlauber mode, M = X i,j p Xi · p Xj + 12 k + Q − k + p − X + ~k ⊥ · p ⊥ X (4.10)Given the on-shell condition on p − k from Eq. 4.9, k + p − X = 2 ~p ⊥ X · ~k ⊥ + ~k ⊥ (4.11) M = X i,j p Xi · p Xj + Q p − X ~p ⊥ X · ~k ⊥ + ~k ⊥ p − X (cid:16) Q − p − X (cid:17) (4.12)so that our jet function becomes J AB ( e n , k ) = t Π ∞ i =1 Z d ˜ p Xi Tr[ T A T B ] δ ( e n − ~k ⊥ · p ⊥ X Qp − X − k ⊥ p − X Q ( Q − p − X ) − Q X i,j p Xi · p Xj ) δ p + X − k + − ( p ⊥ X − k ⊥ ) p − X ! h P Xi | δ ( P ⊥ ) δ ( Q − p − ) T n e − i R dt H n ( t ) ¯ χ n (0) / ¯ n o | ih | ¯ T n e − i R dt H n ( t ) χ n (0) o | P Xi i (4.13)So the definition of the jet mass measurement is altered from its standard definitionby a function of k ⊥ due to the interaction with the medium. Even though we haveworked this out for the case of jet mass measurement, this is a general property for alljet substructure observables. p X here is the momentum of a quark so that when we– 22 –ompute the jet function, it will automatically sum over all the quarks in the jet asit should. When we include the gluon-medium interaction, we will have a similar sumover gluon states.To write the final form for Σ (1) a , we use the k + independence of D AB> to remove theon-shell condition for p − k from Eq. 4.13Σ (1) a = V × t ( H ( Q, µ ) | C qq ( Q ) | n R d ~q JS S ( ~q JS ) on R d ~q Sc, ¯ n S sc, ¯ n ( Qz cut , ~q Sc, ¯ n ) on R d ~q Sc,n S sc,n ( Qz cut , ~q Sc,n ) o ⊗ e n CS n ( Qz cut , e n ) ⊗ e n R d k ⊥ dk − (2 π ) k ⊥ D AA> ( k ) ˜ J n ( e n , ~k ⊥ ) × J ¯ n ( e ¯ n ) ⊗ e ¯ n CS ¯ n ( Qz cut , e ¯ n ) δ ( ~q T + ~q JS + ~k ⊥ + ~q Sc,n + ~q Sc, ¯ n ) ) Where we have absorbed all factors of 2 π resulting from the δ functions inside C qq .Where we have now defined a modified jet function˜ J n ( e n , ~k ⊥ ) = Π ∞ i =1 Z d ˜ p Xi δ ( e n − ~k ⊥ · p ⊥ X Qp − X − k ⊥ p − X Q ( Q − p − X ) − Q X i,j p Xi · p Xj ) h P Xi | δ ( P ⊥ ) δ ( Q − p − ) T n e − i R dt H n ( t ) ¯ χ n (0) / ¯ n o | ih | ¯ T n e − i R dt H n ( t ) χ n (0) o | P Xi i (4.14)The extra piece in the measurement, however, is not readily expressed as a convolution,so the master equation will be more complicated than before. • Σ (1) b The factorized form for this piece was derived in Eq. 3.41 in terms of the functions S G convoluted with a a jet function with the following definitions S ABG ( { x − , x ⊥ } , { y − , y ⊥ } ) = h X S | T n O AS ( x ⊥ , x − ) O BS ( y ⊥ , y − ) o ρ B | X S i J ABn ( e n , x, y ) = h X n | T n χ n (0) / ¯ n O An ( x ) O Bn ( y ) o | ih | ¯ T n ¯ χ n (0) o δ ( P ⊥ ) δ ( Q − P − ) δ ( e n − E n ) | X n i (4.15)Once again we are going to put the intermediate particles on-shell before they interactwith the medium. This allows us to simplify our jet function by inserting a complete– 23 –asis on on-shell states J ABn ( e n , x, y ) = h X n | T n O An ( x ) O Bn ( y ) o | Y n ih Y n | T n χ n (0) / ¯ n o | ih | ¯ T n ¯ χ n (0) o δ ( P ⊥ ) δ ( Q − P − ) δ ( e n − E n ) | X n i = ˆ J ABn ( x, y ) J n (4.16)We can now completely separate out the medium dependent piece G = Z d x Z d yS ABG ( { x − , x ⊥ } , { y − , y ⊥ } ) ˆ J ABn ( x, y )= 2 Z d x Z d y Θ( x − y ) h X S | O AS ( x ⊥ , x − ) O BS ( y ⊥ , y − ) ρ B | X S ih X n | O An ( x ) O Bn ( y ) | Y n i We can now write Θ( t ) = 1 / t )) separating out the terms corresponding toUnitary evolution from the dissipative. G = G U + G D (4.17)The piece corresponding to unitary evolution reads G U = 2 Z d x Z d y sgn( x − y ) h X S | O AS ( x ⊥ , x − ) O BS ( y ⊥ , y − ) ρ B | X S ih X n | O An ( x ) O Bn ( y ) | Y n h Y n | T n χ n (0) / ¯ n o | ih | ¯ T n ¯ χ n (0) o δ ( P ⊥ ) δ ( Q − P − ) δ ( e n − E n ) | X n i This piece cancels out with the corresponding Unitary evolution term from doubleGlauber insertions on the other side of the cut.The Dissipative piece, on the other hand no longer contains any time ordering. Then,as before, using the translational invariance of the QGP medium, h X S | O AS ( x ⊥ , x − ) O BS ( y ⊥ , y − ) ρ B | X S i = Z d k (2 π ) k ⊥ e i (ˆ x − ˆ y ) · k D AB> ( k )where ˆ x ∈ ( x + , , ~x ⊥ ) and ˆ y ∈ ( y + , , ~y ⊥ ), D AB> ( k ) is again the Wightman function inthe thermal bath.Following the same series of steps as for Σ (1) a , evaluating the Glauber vertex at treelevel brings us to Z d x Z d yS ABG ( { x − , x ⊥ } , { y − , y ⊥ } ) J ABn ( x, y ) = t Z d k ⊥ dk − (2 π ) k ⊥ D AA> ( k ) J n ( e n )– 24 –e see that the jet function J n ( e n ) is independent of k ⊥ and is in fact the standardvacuum jet function. This allows us to write the result for Σ (1) b in terms of the vacuumresult Σ (0) from Eq. 3.24.Σ (1) b = −
12 Σ (0) × t | C qq | Z d k ⊥ dk − (2 π ) k ⊥ D AA> ( k ) (4.18)The result for our reduced density matrix from Eq. 3.16 expanded to quadratic order in theGlauber vertex now becomesΣ( t ) = Σ (0) + Σ (1) a + n Σ (1) b + c.c o + O ( H G ) (4.19)which can be rewritten as Σ( t ) = Σ (0) (1 − tR ) + Σ (1) a (4.20)where R = | C qq | Z d k ⊥ dk − (2 π ) k ⊥ D AA> ( k ) (4.21) Given the results of the previous sections, we see that the only new term that needs to becomputed is modified jet function from Eq. 4.14˜ J n ( e n , ~k ⊥ ) = Π ∞ i =1 Z d ˜ p Xi δ ( e n − ~k ⊥ · p ⊥ X Qp − X − k ⊥ p − X Q ( Q − p − X ) − Q X i,j p Xi · p Xj ) h P Xi | δ ( P ⊥ ) δ ( Q − p − ) T n e − i R dt H n ( t ) ¯ χ n (0) / ¯ n o | ih | ¯ T n e − i R dt H n ( t ) χ n (0) o | P Xi i (5.1)The one loop calculation is given in Appendix B. As expected, the anomalous dimensionof the modified jet function remains unaltered as is required by RG consistency of the fac-torization. The effect of the medium then is to alter the IR behavior of the jet function. Inparticular, the medium interaction induces new IR divergences which are fully regulated bythe medium induced gluon mass. The one loop result can therefore be written as˜ J (1) n ( e n , ~k ⊥ ) = J (1) n ( e n ) + J Mn ( e n , k ⊥ , m g ) (5.2)where J (1) n ( e n ) is the one loop vacuum jet function. The function J Mn ( e n , k ⊥ , m g ) is UV andIR finite with the IR divergences regulated by the gluon mass m g . J Mn = α s C F π ( e n + y ) ( − B A √ A + B + ( A + A −
4) ln 4 AB ) (5.3)– 25 –here we have defined A = e n / ( e n + y ) and B = 4 M y/ ( e n + y ) with M = 4 m g Q , y = 4 k ⊥ Q (5.4)This result goes to 0 as k ⊥ goes to 0 so that it is a purely medium induced result. There aretwo effects to note here • Unlike the vacuum result, there is no corresponding virtual diagram which regulates J Mn in the limit e n →
0. We have to rely on the gluon mass once again to keep theresult finite in this limit. Hence, if we treat this piece as per of the fixed order term inthe jet function, we need to go to Laplace space to do the resummation using the RGrunning. But unlike the vacuum jet function, the medium induced piece we will get acontribution from values of e n all the way down to m g which is contrary to the powercounting of our EFT which assumes e n (cid:29) M . This breakdown is again a result of thedecoherence between the hard interaction and the medium interaction, which signalsthe need for a multi-scale EFT at higher order ( O ( α s )) which will work in the full range M ≤ e n ≤ T /Q . • We also have a logarithm of ln M /e n in this result. This appears due to the IR polewhen the gluon become collinear to the quark. Unlike the collinear pole in the vacuumresult which always appears at e n → e n . A possible resummationof this logarithm would require us to separate the scale e n from M , which again requiresa multi-stage EFT.For the purpose of this paper, therefore, we will treat this new term as a fixed ordercorrection, only keeping its effect to O( α s ) in the jet function and hence O ( α s ) in the fullcross section. This means than we will evaluate all the other functions in the factorized crosssection at tree level.We can then writeΣ (1) a = t | C qq | Z d k ⊥ dk − (2 π ) k ⊥ D AA> ( k )Σ (0) ( ~q T + ~k ⊥ )+ V × tH ( Q ) | C q q | Z d k ⊥ dk − (2 π ) k ⊥ D AA> ( k ) δ ( ~q T + ~k ⊥ ) J Mn ( e n , k ⊥ , m g ) (5.5)While this result has been obtained by considering the quark interaction with the medium,we also have a contribution from the gluon interacting with the medium. The one loop resultfor this is evaluated in Appendix C. These contributions are again both IR and UV finite.Once again we treat them as fixed order contributions at one loop and ignore any higherorder resummation associated with these terms, leaving that analysis for the future. Theresult gives us two contributions, one from real interaction Eq.C.4 with the medium andanother from virtual Eq.C.6 which we now dress with the medium correlators.– 26 – (1) a,g + n Σ (1) b,g + c.c o = V × tH ( Q ) | C gq | Z d k ⊥ dk − (2 π ) k ⊥ D AA> ( k ) × ( δ ( ~q T + ~k ⊥ ) G Mn ( e n , k ⊥ , m g ) − δ ( ~q T ) G Mn ( e n , k ⊥ = 0 , m g ) ) (5.6)with G Mn ( e n , k ⊥ , m g ) = α s C F π ( e n + y ) ( A − A + 2) ln 4(1 − A )2( M − A ) − B + 2 p ( M − A ) + B (5.7)where A = y/ ( e n + y ), B = 4 M y/ ( e n + y ) We can now gather all the pieces and write down the Master equation for the density matrixevolution.In our case, we are now measuring the jet mass and the transverse momentum imbalancebetween the jets on the final state. In this case, the matrix element now depends on the jetmass and the transverse imbalance ~q T . The results till now have been expressed in terms ofa measurement operator imposed of the final state density matrix.Combining all the pieces from the previous section and using them in Eq. 4.20, we canwriteTr[ ρ ( t ) M ] ≡ Σ( e n , ~q T , t )= V × " Σ (0) ( e n , ~q T ) − Rt Σ (0) ( e n , ~q T ) + t | C qq | Z d k ⊥ dk − (2 π ) ( k ⊥ + m g ) D AA> ( k )Σ (0) ( ~q T + ~k ⊥ )+ tH ( Q ) Z d k ⊥ dk − (2 π ) ( k ⊥ + m g ) D AA> ( k ) ( δ ( ~q T + ~k ⊥ ) h | C qq | J Mn ( e n , k ⊥ , m g ) + | C gq | G Mn ( e n , k ⊥ , m g ) i − δ ( ~q T ) | C gq | G Mn ( e n , k ⊥ = 0 , m g ) ) (6.1)where we have explicitly pulled out a factor of V. The Wightman function has been ccaluclatedin [80] – 27 – AA> ( k − , k ⊥ ) = N f (8 πα s ) (2 π ) πT F × I ( k − , k ⊥ ) , with I ( k − , k ⊥ ) = Z | p ⊥ | d | p ⊥ | dp − dp + dφδ ( p − p + − | p ⊥ | ) δ (cid:16) ( p − + k − ) p + − | p ⊥ + k ⊥ | (cid:17) ( p + ) × Θ( p − + p + )Θ( p − + k − + p + ) n F (cid:16) p − + p + (cid:17)" − n F (cid:16) p − + k − + p + (cid:17) (6.2)where N f is the number of quark flavors in the QGP and we are ignoring any contribution fromthe gluons in the medium for simplicity. n F is the Fermi distribution function at temperatureT.We can now relate the trace over the density matrix to the scattering cross section, notingthat dσde n d ~q T ( t ) = N Tr[ ρ ( t ) M ] V (6.3)where N is a normalization factor that depends on the initial state kinematics which we canabsorb in the born level cross section. Notice here that there is still a time dependence inthe cross section which is unusual but we interpret this as the time of propagation throughthe quark gluon plasma which in turn will depend on the length over which the jet traversesthrough the QGP medium. Hence, this should be treated as a length scale.Defining P ( e n , ~q T )( t ) = dσde n d ~q T ( t ) K ( ~k ⊥ ) = | C qq | Z dk − (2 π ) ( k ⊥ + m g ) D AA> ( k ) F ( e n , k ⊥ ) = N H ( Q ) J Mn ( e n , k ⊥ , m g ) + | C gq | | C qq | G Mn ( e n , k ⊥ , m g ) ! (6.4)and taking the limit t →
0, which is justified in the Markovian approximation as explainedin [80], we can write an evolution equation for our observable as a function of the time ofpropagation in the QGP medium. ∂ t P ( e n , ~q T )( t ) = − RP ( e n , ~q T ) + Z d k ⊥ K ( k ⊥ ) P ( e n , ~q T + ~k ⊥ )( t )+ Z d k ⊥ K ( k ⊥ ) " δ ( ~q T + ~k ⊥ ) F ( e n , k ⊥ ) − δ ( ~q T ) F ( e n , = − RP ( e n , ~q T ) + Z d k ⊥ K ( k ⊥ ) P ( e n , ~q T + ~k ⊥ , t ) + K ( ~q T ) F ( e n , q T ) − δ ( ~q T ) Z d k ⊥ K ( k ⊥ ) F ( e n , − RP ( e n , ~q T ) + Z d k ⊥ K ( k ⊥ ) P ( e n , ~q T + ~k ⊥ , t ) + K ( ~q T ) F ( e n , q T ) − δ ( ~q T ) F ( e n , R (6.5)The new term then explicitly disappears as k ⊥ →
0. Notice that this term does not have asimple convolution in transverse momentum space. To Solve this we go once again to impactparameter space,˜ P ( e n , ~r ⊥ ) = Z d ~q T P ( e n , ~q T ) e − i~q T · ~r ⊥ ˜ K ( ~r ⊥ ) = Z d ~k ⊥ K ( ~k ⊥ ) e − i~k ⊥ · ~r ⊥ ˜ G ( e n , ~r ⊥ ) = Z d ~q T e − i~q T · ~r ⊥ n K ( ~q T ) F ( e n , q T ) − δ ( ~q T ) F ( e n , R o (6.6)so that the convolution turns into a product, ∂ t ˜ P ( e n , ~r ⊥ ) = ( − R + ˜ K ( ~r ⊥ )) ˜ P ( e n , ~r ⊥ ) + ˜ G ( e n , ~r ⊥ ) d ˜ P ( e n , ~r ⊥ )˜ P ( e n , ~r ⊥ ) + ˜ g ( e n , ~r ⊥ ) = ( − R + ˜ K ( ~r ⊥ )) dt (6.7)with ˜ g ( e n , ~r ⊥ ) = ˜ G ( e n , ~r ⊥ ) − R + ˜ K ( ~r ⊥ ) (6.8)which leads us to the solutionln h ˜ P ( e n , ~r ⊥ ) + ˜ g ( e n , ~r ⊥ ) i = ( − R + ˜ K ( ~r ⊥ )) t + C ˜ P ( e n , ~r ⊥ )( t ) = P e ( − R + ˜ K ( ~r ⊥ )) t − ˜ g ( e n , ~r ⊥ ) (6.9)where C is the constant of integration and P = e C . At t=0 we need the result to be thevacuum result , V( e n , ~r ⊥ ) which is just the Fourier transform of the resummed vacuum crosssection V ( e n , ~r ⊥ ) = Z d ~q T e − i~q T · ~r ⊥ dσde n d ~q T (cid:12)(cid:12) vacuum = N V Z d ~q T e − i~q T · ~r ⊥ Σ (0) ( e n , ~q T ) (6.10)The factorization for the vacuum cross section is given by Eq. 3.24 in terms of functionswhose one loop results are collected in Appendix A for completeness. The resummation ofthe vacuum cross section was accomplished in [82] to NNLL accuracy and we refer the readerto that paper for more details. P = V ( e n , ~r ⊥ ) + ˜ g ( e n , ~r ⊥ ) (6.11)which now leads us to the solution˜ P ( e n , ~r ⊥ ) = [ V ( e n , ~r ⊥ ) + ˜ g ( e n , ~r ⊥ )] e ( − R + ˜ K ( ~r ⊥ )) t − ˜ g ( e n , ~r ⊥ ) (6.12)– 29 –o that dσde n d~q T ( t ) = Z d ~r ⊥ e i~r ⊥ · ~q T (h V ( e n , ~r ⊥ ) + ˜ g ( e n , ~r ⊥ ) i e ( − R + ˜ K ( ~r ⊥ )) t − ˜ g ( e n , ~r ⊥ ) ) (6.13)which is the main result of this paper.While the result gives us a closed form expression, we still to do the integrals numericallyfrom this point onwards. A realistic comparison with data will need us to include the effectsof nuclear/ hardonic pdfs and we leave a detailed phenomenological study based on thisframework for the future. In this paper, I develop an Effective Field Theory (EFT) framework to compute jet substruc-ture observables for heavy ion collision experiments. I consider dijet events that happen atthe periphery of the collision so that one of the jets evolves through vacuum while the othertravels through the Quark Gluon Plasma medium that is created in the background. The jetsare groomed using a grooming algorithm in order to mitigate effects of soft contaminationfrom Multi-parton interactions as well as the QGP medium. This means that the finals statemeasurements do not include any soft hadrons from the cooling QGP medium. We can thenonly follow the evolution of the reduced density matrix of the jet tracing over the QGP bath.This effectively treats the jet as an open quantum system interacting with a thermal bathand allows us to derive a Lindblad type master equation for the density matrix evolution.I measure two quantities on the final state di-jet configuration: The transverse momentumimbalance between the jets as well as a jet mass constraint imposed on each jet which re-stricts the radiation inside the large radius jets to a collinear core. I note that the dominantcontribution to the evolution of the jet in the medium comes from the regime where the par-tons created in the hard interaction and subsequent vacuum shower go on-shell before theyinteract with the medium. This happens due to the lack of quantum coherence between thehard and medium interactions.The physical scales that arise from the presence of the medium, the initial state kinematicsas well as the final state measurements create a hierarchy of scales which are reflected in theform of large logarithms in the cross section. These large logarithms lead to a breakdown ofperturbation theory and it becomes necessary to resum them in order to have convergence.To do this, I write down a factorization formula within SCET for each piece of the Lind-blad equation. This formula is valid to leading (quadratic) order in the Glauber vertex thatencodes the forward scattering interaction of the jet with the medium and to leading powerin the expansion parameters of our EFT. All the elements except the jet function remainidentical compared to the corresponding factorized formula for a pure vacuum evolution ofthe jet. For consistency of Renormalization group equations, we therefore require that theanomalous dimension of the medium modified jet function remain the same as its vacuumcounterpart. We explicitly show this by doing a one loop calculation and recovering the cusp– 30 –nd non-cusp anomalous dimension. The interaction with medium however, changes the IRstructure of the jet function inducing new Infra-Red divergences in the cross section whichwould otherwise be absent in a pure vacuum evolution of the jet. These IR divergences arefully regulated by the medium induced gluon mass and more significantly, lead to a breakdownof the power counting of the EFT and higher orders in the strong coupling, which preventsus from resumming them systematically within the constructed EFT formalism. This can beremedied by utilizing a multi-scale EFT approach at higher orders in the coupling, somethingwhich we leave for the future. In this paper, I treat these new IR corrections as fixed orderterms and derive a master equation for the evolution of the jet cross section as a function ofthe propagation time in the medium. This equation can be solved analytically in impact pa-rameter space and the solution resums multiple scattering events of the jet with the medium.The purpose of this paper is to set up an EFT framework for systematic computation of jetsubstructure in heavy ion collisions. While I have treated the initial hard interaction as an e + e − collision for ease of analysis, this can be easily extended to the realistic case of hadroniccollisions which will also require us to input nuclear pdfs for comparison with data. We leavethe detailed phenomenological analysis based on this framework for future work.While I have accounted for the vacuum evolution, the factorization formula is also setup to systematically compute the corrections to the Glauber vertex as well which we hope toinclude in the future. At the same time, this work now paves the way to apply this formalismfor the case of jets initiated by heavy quarks. Acknowledgments
I thank Xiaojun Yao for useful discussions during the intial stages of this project. This work issupported by the Office of Nuclear Physics of the U.S. Department of Energy under ContractDE-SC0011090 and Department of Physics, Massachusetts Institute of Technology.
A Operator definitions and one loop results for vacuum evolution
In this appendix we give the operator definitions of the factorization elements that appearin the vacuum and their NLO expansions. From those we determine the renormalizationfunctions, group equations, and corresponding anomalous dimensions. Part of these resultswere derived in [82] while the rest were already available in literature. Hence we will onlyquote the final results here for the sake of completeness without going into any detailedderivations.
A.1 Jet functions
The quark and gluon jet function definitions, one loop calculation, and the correspondingLaplace transforms can be found in ref. [101]. Here we summarize their results. The quark– 31 –et function is given by, J q ( e, Q ) = (2 π ) N c tr h / ¯ n χ n (0) δ ( Q − P − ) δ (2) ( PPP ⊥ ) δ ( e − E ) ¯ χ n i , (A.1)It is useful to work in Laplace space. The renormalized groomed jet function up to NLOcontributions in Laplace space is given by J i ( s, Q ; µ ) = 1 + α s C i π n L J + ¯ γ i L J − π c i o + O ( α s ) , (A.2)where for quark initiated jets we have C q = C F = N c − N c , ¯ γ q = 32 , c q = 72 , (A.3)The logarithms, L J that appear in eq. (A.2) and the corresponding one loop anomalousdimensions are L J = ln (cid:16) µ ˜ sQ (cid:17) , γ J = α s C i π (cid:16) L J + ¯ γ i (cid:17) + O ( α s ) . (A.4)where ˜ s is given, by 4 se γ E , s being the Laplace variable conjugate to the normalized jet masse. The anomalous dimension is defined through the RG equation satisfied by renormalizedjet functions. In Laplace space this is dd ln µ J i ( s, Q ; µ ) = γ J ( s, Q ; µ ) J i ( s, Q ; µ ) . (A.5)In momentum space the above equation is written as convolution (in the invariant massvariable e ), of the anomalous dimension and the renormalized jet function. A.2 Collinear-soft function
The operator definition of the invariant mass measurement collinear soft function is given by S cs ( e, Qz cut ) = 1 N R tr h T (cid:16) U † n W t (cid:17) M SDe ¯ T (cid:16) W † t U n (cid:17) i , (A.6)where M SDe is the invariant measurement function, M SDe = δ ( e − (1 − Θ SD ) E ) . (A.7)Here we dropped the jet flavor (quark/anti-quark or gluon) for simplicity of notation and thenormalization constant N R is simply the size of the representation for SU( N c ) of the W t and U n Wilson lines. For quark jets (fundamental representation) we have N R = N c . Defining ξ ≡ µ Q z cut . (A.8)– 32 –he renormalized collinear-soft function in Laplace space is given as, S cs ( s, Qz cut ; µ ) = 1 − α s C i π L cs + O ( α s ) , (A.9)which satisfies the following RGE dd ln µ S cs ( s, Qz cut ; µ ) = γ cs ( s, µ ) S cs ( s, Qz cut ; µ ) . (A.10)The logarithm L cs and the corresponding anomalous dimension are L cs = ln( ξ ˜ s ) , γ cs ( s, µ ) = − α s C i π L cs + O ( α s ) . (A.11) A.3 Soft function
The soft function that appears in the factorization theorems in eq. (3.24) is defined in eq. (3.26)and it has been calculated in several schemes at higher orders in QCD, as in [102]. and satisfiesthe following renormalization group equations dd ln µ S ( µ, ν ) = γ s ( µ, ν ) S ( µ, ν ) , dd ln ν S ( µ, ν ) = γ sν ( µ, ν ) ⊗ S ( µ, ν ) . (A.12)Therefore we find for the one-loop corresponding impact parameter space quantities S ( µ, ν ) = 1 + α s ( µ ) C i π n (cid:16) µ E µ (cid:17) ln (cid:16) νµ (cid:17) − (cid:16) µ E µ (cid:17) − π o + O ( α s ) , (A.13)with γ s ( µ, ν ) = − α s ( µ ) C i π ln (cid:16) νµ (cid:17) + O ( α s ) , γ sν ( µ, ν ) = 4 α s ( µ ) C i π ln (cid:16) µ E µ (cid:17) + O ( α s ) . (A.14)where µ − E = be γ E with b = | ~b | is the impact parameter variable conjugate to ~q T . A.4 Soft-collinear function
The soft-collinear function is defined by the matrix element S ⊥ sc ( Qz cut ) = 1 N R tr h T (cid:16) U † n W t (cid:17) M SD ⊥ ¯ T (cid:16) W † t U n (cid:17) i , (A.15)and the groomed jet measurement function, M SD ⊥ is given in terms of the label momentumoperator, P , M SD ⊥ = Θ SD × δ ( qqq T − Θ SD PPP ⊥ ) , (A.16)where Θ SD denotes the soft drop groomer. The collinear-soft modes only contribute to theinvariant mass measurement if they pass soft-drop, which is implemented by the Θ SD term.and satisfies the following renormalization group equations dd ln ν S ⊥ sc ( µ, ν ) = γ scν ( µ, ν ) ⊗ S ⊥ sc ( µ, ν ) , dd ln µ S ⊥ sc ( µ, ν ) = γ sc ( µ, ν ) S ⊥ sc ( µ, ν ) , (A.17)– 33 –here the Qz cut dependence is suppressed to improve readability. In MS scheme the corre-sponding Fourier transform is given by˜ S ⊥ sc ( Qz cut ; µ, ν ) = 1 + α s C i π n − (cid:16) νQz cut (cid:17) ln (cid:16) µ E µ (cid:17)o + O ( α s ) , (A.18)and thus for the one-one-loop anomalous dimensions we get γ scν ( µ, ν ) = − α s ( µ ) C i π ln (cid:16) µ E µ (cid:17) + O ( α s ) γ sc ( µ, ν ) = 2 α s ( µ ) C F π ln (cid:16) νQz cut (cid:17) + O ( α s ) . (A.19) B Modified Quark jet function
In this section we present the one loop results for the modified jet function defined as follows(Eq.4.14)˜ J n ( e n , ~k ⊥ ) = (2 π ) N c Π ∞ i =1 Z ˜ dp Xi δ ( e n − ~k ⊥ · p ⊥ X Qp − X − k ⊥ p − X Q ( Q − p − X ) − Q X i,j p Xi · p Xj ) h P Xi | δ ( P ⊥ ) δ ( Q − p − ) T n e − i R dt H n ( t ) ¯ χ n (0) / ¯ n o | ih | ¯ T n e − i R dt H n ( t ) χ n (0) o | P Xi i (B.1)At tree level, we have a single quark which leads to˜ J (0) n ( e n , ~k ⊥ ) = δ ( e n ) Z ˜ dp q δ ( p ⊥ q ) δ ( Q − p − q ) p − Tr " /n / ¯ n = δ ( e n ) (B.2)and is independent of ~k ⊥ . I will give all results in Laplace space where I will use s as theconjugate variable to e n , so that at tree level˜ J (0) n ( s, ~k ⊥ ) = 1 (B.3)At one loop, I have both real and virtual diagrams. The gluon emitted as a radiative correctioninteracts with the medium and acquires a mass m g with the hierarchy m g (cid:28) e n since we areworking in a weak coupling regime. If the jet function is IR finite, then the m g scale isirrelevant and we can use dimensional regularization which does not distinguish between UVand IR divergences. However, we expect that the medium will induce non-trivial in the Infra-Red physics of this function, it is no longer guaranteed that this function is IR finite, in thesense that it can be sensitive to scale m g . With this is mind, the strategy I will adopt isto explicitly separate out terms that correspond to the vacuum jet function and use the factthat it its IR finite to complete the calculation.– 34 – a) (b) Figure 1 : Real emission diagrams
B.1 Real emission diagrams
These are the diagrams that arise from the insertion of the Glauber interaction on oppositesides of the cut. There are two Feynman diagram that contribute as shown in Fig. 1 (alongwith a zero bin term). Only the zero bin terms has UV divergences while the rest of the realpieces only have IR divergences which are fully regulated by the gluon mass. The zero-bin isthere to cancel out the overlap between the jet and the Collinear-Soft function. R = R a + R b − R Z (B.4)Our real diagram with a gluon mass now becomes R a = 8 g C F Z d pδ + ( p ) Z d q (2 π ) δ + ( q − m g ) δ ( Q − p − − q − ) δ ( ~q ⊥ + ~p ⊥ ) × δ e n − ~k ⊥ · ~p ⊥ Qp − − k ⊥ ( Q − p − )2 p − Q + 82 Q (cid:16) p + + q + (cid:17)! p − ( p − + q − ) q − ( p + q ) (B.5)The on-shell condition also requires q − ≥ m which will cut-off any soft divergence. Wealso have the constraint e n ≥ m g Q (B.6)After doing the integrals over q + , ~q ⊥ , we can rescale q − by Q and define M = 2 m g /Q, y =4 k ⊥ /Q , R a = g C F Z M dq − (2 π ) (1 − q − ) q − (cid:16) ( e n (1 − q − ) − q − y ) + 4 M (1 − q − ) y (cid:17) / (B.7)– 35 –e will rewrite our result explicitly separating out the piece corresponding to the vacuum jetfunction R a = g C F (2 π ) " Z M dq − (1 − q − ) q − (cid:16) ( e n (1 − q − ) − q − y ) + 4 M (1 − q − ) y (cid:17) / ≡ g C F (2 π ) " Z M dq − (1 − q − ) q − (cid:16) ( e n (1 − q − ) − q − y ) + 4 M (1 − q − ) y (cid:17) / − Z M dq − (1 − q − ) q − e n + ( Z M dq − q − e n − g C F (2 π ) Z M dq − e n ) (B.8)Due to the presence of y, this will now induce a non-trivial y dependent IR term which ison top of the SCET vacuum result. But it is important to see that this does not have anyUV divergences and all IR divergences are fully regulated by the physical gluon mass. Wehave written this result so that the first term explicitly goes to 0 in the limit y → R b = 4 g C F Z d pδ + ( p ) Z d q (2 π ) δ + ( q − m g ) δ ( Q − p − − q − ) δ ( ~p ⊥ + ~q ⊥ ) × δ e n − ~k ⊥ · ~p ⊥ Qp − − k ⊥ ( Q − p − )2 p − Q + 82 Q (cid:16) p + + q + (cid:17)! ( p − + q − ) [( p + q ) ] p ⊥ p − (B.9)which gives us the IR and UV finite result R b = 12 g C F (2 π ) Z M dq − q − (1 − q − ) (cid:16) ( e n (1 − q − ) − q − y ) + 4 M (1 − q − ) y (cid:17) / (B.10)We can rewrite this term, adding and subtracting a y independent piece R b = ( g C F π ) Z M dq − q − (1 − q − ) (cid:16) ( e n (1 − q − ) − q − y ) + 4 M (1 − q − ) y (cid:17) / − g C F π ) Z M dq − q − e n ) + g C F π ) Z M dq − q − e n (B.11)The first line here explicitly goes to 0 as y → igure 2 : Virtual emission with real medium interaction B.2 Virtual Diagrams
Due to the presence of the mass scales, we also need to consider the virtual diagrams. One ofthe diagrams which we will now evaluate in shown in Fig.2. Once again we have a zero-bincontribution V = V a + V b − V Z (B.12)However, we see immediately that the medium interaction does not affect the virtualdiagrams at all, except for a contribution from the medium correlator which only appears asoverall multiplicative factor. Hence the virtual diagrams remain identical to the case of thevacuum jet function. B.3 Medium induced terms
Combining all the terms computed in the previous two sections, we can write our result forthe modified jet function as ˜ J (1) n = J (1) n + J ( M ) n (B.13)where J (1) n is identical to the vacuum jet function result presented in Eq.A.2 and is independentof the gluon mass m g , since it is IR finite. J Mn is the medium induced term which we collecthere from Eqn. B.8 and Eq.B.11. J Mn = g C F (2 π ) " Z M dq − (1 − q − ) q − (cid:16) ( e n (1 − q − ) − q − y ) + 4 M (1 − q − ) y (cid:17) / − Z M dq − (1 − q − ) q − e n + ( g C F π ) Z M dq − q − (1 − q − ) (cid:16) ( e n (1 − q − ) − q − y ) + 4 M (1 − q − ) y (cid:17) / − g C F π ) Z M dq − q − e n ) (B.14)The crucial point to note here is that there is no corresponding virtual diagram so thatthere is a real IR divergence, albeit regulated by the gluon mass as e n →
0. This is counter to– 37 – igure 3 : Gluon real interaction with the mediumour scaling argument for e n at higher orders where we now seem to get a contribution fromthis region e n → e n all the way down to the IR scale m /Q . Weleave this for the future.This also suggests that we can only truly trust this new term at leading order where ourpower counting holds. So we will treat this piece as a fixed order correction to our eventualmaster equation and not include it as part of our resummed jet function.Since we are treating this term as a fixed order correction, the e n that appears here isthe final e n that scales as y,T. Defining A = e n / ( e n + y ) and B = 4 M y/ ( e n + y ) , we nowkeep only the singular, logarithmic terms as M → J Mn = α s C F π ( e n + y ) ( − B A √ A + B + ( A + A −
4) ln 4 AB ) (B.15)In the limit y → A → B →
0, so that the full result goes to 0 as y → C Gluon interaction with medium
C.1 Real interaction with medium
We have another real diagram which is induced by the medium and is otherwise absent inthe vacuum. Fig. 3. This involves the interaction of the gluon with the medium. R ga = 8 g C F Z d pδ + ( p ) Z d q (2 π ) d − δ ( Q − p − − q − ) δ ( ~q ⊥ + ~p ⊥ ) × δ e n − Q ( p + + q + ) − Q ~q ⊥ · ~k ⊥ + 8 ~k ⊥ ( Q − q − ) /Qq − ! p − ( p − + q − ) q − ( p + q ) (C.1)– 38 – igure 4 : Gluon virtual interaction with the mediumThis is again UV finite so that we do not need to use dim.reg. At the same time all the IRdivergences are regulated by the gluon mass. Rescaling q − by Q and defining y = k ⊥ /Q , M = m/Q R ga = g C F (2 π ) Z dq − (2 π ) d − (1 − q − ) q ( q − e n − y (1 − q − )) + 4 M (1 − q − ) y (C.2)which only has a collinear singularity, We also have R gb = g C F π ) Z dq − (2 π ) d − ( q − ) q ( q − e n − y (1 − q − )) + 4 M (1 − q − ) y (C.3)which now gives us the result, keeping only the dominant logarithmic pieces R g = α s C F π ( e n + y ) ( A − A + 2) ln 4(1 − A )2( M − A ) − B + 2 p ( M − A ) + B (C.4)where A = y/ ( e n + y ), B = 4 M y/ ( e n + y ) . As we can see, in the limit y →
0, this resultreduces to R g (cid:12)(cid:12) y → = − α s C F e n π ln M (C.5)These terms do not go to 0 as y →
0. This is because we still have another diagram witha virtual interaction with the medium which needs to be added and which we now turn to.
C.2 Virtual interaction with the medium
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