Non-Abelian Bremsstrahlung and Azimuthal Asymmetries in High Energy p+A Reactions

Non-Abelian Bremsstrahlung and Azimuthal Asymmetries
in High Energy p+A Reactions

M. Gyulassy gyulassy@phys.columbia.edu MTA WIGNER Research Centre for Physics, RMI, Budapest, Hungary Department of Physics, Columbia University, New York, NY 10027, USA    P. Levai MTA WIGNER Research Centre for Physics, RMI, Budapest, Hungary    I. Vitev Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA    T. Biro MTA WIGNER Research Centre for Physics, RMI, Budapest, Hungary
May 30, 2014
Abstract

We apply the GLV reaction operator solution to the Vitev-Gunion-Bertsch (VGB) boundary conditions to compute the all-order in nuclear opacity non-abelian gluon bremsstrahlung of event-by-event fluctuating beam jets in nuclear collisions. We evaluate analytically azimuthal Fourier moments of single gluon, , and even number gluon, inclusive distributions in high energy p+A reactions as a function of harmonic , target recoil cluster number, , and gluon number, , at RHIC and LHC. Multiple resolved clusters of recoiling target beam jets together with the projectile beam jet form Color Scintillation Antenna (CSA) arrays that lead to characteristic boost non-invariant trapezoidal rapidity distributions in asymmetric nuclear collisions. The scaling of intrinsically azimuthally anisotropic and long range in nature of the non-abelian bremsstrahlung leads to moments that are similar to results from hydrodynamic models, but due entirely to non-abelian wave interference phenomena sourced by the fluctuating CSA. Our analytic non-flow solutions are similar to recent numerical saturation model predictions but differ by predicting a simple power-law hierarchy of both even and odd without invoking factorization. A test of CSA mechanism is the predicted nearly linear rapidity dependence of the . Non-abelian beam jet bremsstrahlung may thus provide a simple analytic solution to Beam Energy Scan (BES) puzzle of the near independence of moments observed down to 10 AGeV where large- valence quark beam jets dominate inelastic dynamics. Recoil bremsstrahlung from multiple independent CSA clusters could also provide a partial explanation for the unexpected similarity of in and non-central at same multiplicity as observed at RHIC and LHC.

pacs:
24.85.+p; 12.38.Cy; 25.75.-q

I Introduction

An unexpected discovery at RHIC/BNL in reactions at AGeV Adare:2013piz () and at LHC/CERN in ATeV reactions CMS:2012qk (); Abelev:2012ola (); Aad:2012gla () is the large magnitude of mid-rapidity azimuthal anisotropy moments, , that are remarkably similar to those observed previously in non-central  Adams:2005dq (); Adcox:2004mh (); Adare:2010ux () and  Aamodt:2011by (); Aamodt:2010pa (); ALICE:2011ab (); Chatrchyan:2012wg (); ATLAS:2012at () reactions. See preliminary data in Fig. 1 taken from ATLAS ATLASvn () Fig. 24 that also shows a large rapidity-even dipole harmonicATLASpAv1 ().

In addition, the Beam energy Scan (BES) at RHIC Adamczyk:2013gw () revealed a near independence from 8 AGeV to 2.76 ATeV of the in at fixed centrality that was also unexpected.

Figure 1: (Color online) Reproduced from ATLAS ATLASvn () p+Pb Figure 24 with n=2 to 5 obtained for and the range of 1-3 GeV. An overlay sketch of preliminary rapidity-even data shown at QM14 ATLASpAv1 () is also indicated. The error bars and shaded boxes represent the statistical and systematic uncertainties, respectively. Results in are compared to the CMS data CMS:2012qk () obtained by subtracting the peripheral events (the number of offline tracks ), shown by the solid and dashed lines.

In high energy , the moments have been interpreted as possible evidence for the near “perfect fluidity” of the strongly-coupled Quark Gluon Plasmas (sQGP) produced in such reactions Romatschke:2007mq (); Luzum:2008cw (); Alver:2010dn (); Gale:2012rq (); Heinz:2013th (). However, the recent observation of similar in much smaller systems and also the near beam energy independence of the moments observed in the Beam Energy Scan (BES) Adamczyk:2013gw () at RHIC together with LHC, from 7.7 AGeV to 2.76 ATeV have posed a problem for the perfect fluid interpretation because near inviscid hydrodynamics is not expected to apply in space-time regions where the local temperature falls below the confinement temperature, MeV. In that Hadron Resonance Gas (HRG) “corona” region, the viscosity to entropy ratio is predicted to grow rapidly with decreasing temperature Danielewicz:1984ww () and the corona volume fraction must increase relative to the ever shrinking volume of the perfect fluid “core” with when either the projectile atomic number and size fm or the center-of-mass (CM) energy decrease .

While hydrodynamic equations have been shown to be sufficient to describe data with particular assumptions about initial and freeze-out conditions Bozek:2011if (), its necessity as a unique interpretation of the data is not guaranteed. This point was underlined recently using a specific initial state saturation model Dusling:2013oia () that was shown to be able to fit correlation even moments data without final state interactions. That saturation model has also been used Gale:2012rq () to specify initial conditions for perfect fluid hydrodynamics in . However, in such initial conditions for hydrodynamics are not as well-controlled because the gluon saturation scale scale, GeV, is small and its fluctuations in the transverse plane on sub-nucleon scales are not reliably predicted.

The near independence of moments on beam energy observed in BES Adamczyk:2013gw () at RHIC from 7.7 AGeV to 2760 AGeV pose further serious challenges to the uniqueness of the perfect fluid interpretations of the data because of previous predictions Teaney:2000cw () for systematic reduction of the moments due to the increasing HRG corona. Those predictions appeared to be confirmed by SPS  AGeV data Agakichiev:2003gg (). The most recent BES measurements, however, appear to contradict the diluting role of the HRG corona. The HRG corona fraction should dilute perfect fluid QGP core flow signatures at lower energies unless additional dynamical mechanisms possibly associated with increasing baryon density accidentally conspire to compensate for growing HRG corona fraction. Such combination of canceling effects with was demonstrated to be possible using a specific hybrid hydro+URQMD model Auvinen:2013sba () or three fluid models Ivanov:2014zqa (). While such hybrid models are sufficient to explain the BES independence of data in , the necessity and, hence, uniqueness of such hybrid descriptions are not guaranteed.

The BES Adamczyk:2013gw () data also pose a challenge to color glass condensate (CGC) gluon saturation model Gale:2012rq () used to specify initial conditions for hydrodynamic flow predictions in . This is because is predicted to decrease with , and thus gluon saturation-dominated high energy gluon fusion models of initial-state dynamics should switch over into valence quark-diquark dominated inelastic dynamics when partons with fractional energy play the dominant role. At RHIC and lower energies valence quark and diquark QCD string phenomenology based on the LUND Andersson:1986gw () model with (diquark-quark) beam jets and its nuclear collision generalization via HIJING Gyulassy:2003mc () can smoothly interpolate between AGS and RHIC energies. Such multiple beam jet based approach to naturally accounts, for example, for the striking long range triangular, boost non-invariant, form of nuclear enhancement of the final hadron rapidity density in observed at all CM energies up to LHC Debbeqm14 (). By including multiple mini and hard jet production it can account for the growth of though at top AGeV RHIC and at LHC energies there is strong evidence for the onset for gluon saturation Gyulassy:2004zy () that limits minijet processes to that grows with and .

The importance of multiple beam jets with rapidity kinematics controlled by valence quarks and diquarks was first proposed within the Brodsky-Gunion-Kuhn (BGK) model Brodsky:1977de () which is reproduced also in the HIJING Adil:2005qn () model. The trapezoidal boost non-invariant dependence of the local density, , predicted in Adil:2005qn () as a function of the transverse coordinate even in symmetric , may also play an important role in in the triangular long range dependence of ) as observed in by PHOBOS Busza:2004mc ().

In this paper we explore the possibility that a dynamical source that could partially account for the above puzzling azimuthal moment systematics may be traced to a basic perturbative QCD (pQCD) feature. The pQCD based model here extends the opacity Gunion-Bertsch Gunion:1981qs () (GB) perturbative QCD bremsstrahlung used to model for to all orders in opacity, , Vitev-Gunion-Bertsch (VGA) multiple interaction pQCD bremsstrahlung for applications to nuclear collisions. We show that VGA bremsstrahlung naturally leads on an event by event basis to a hierarchy of non trivial azimuthal asymmetry moments similar to those observed in (see fig.1) and peripheral at fixed  Chatrchyan:2012wg (); Aamodt:2010pa (); ATLAS:2012at () .

A particularly important feature of beam jet non-abelian bremsstrahlung is that it automatically leads to long range rapidity “ridge” correlations and to all even and odd azimuthal harmonics with . Conventional Lund string beam jet models Andersson:1986gw (), as encoded e.g. in HIJING, on the other hand neglect recoil induced moderate color bremsstrahlung azimuthal asymmetries. From the pQCD perspective, beam jets are simply arrays of parallel color antennas that radiate due to multiple soft transverse momentum transfers GeV between participant projectile and target nucleons. Many event generators include averaged (azimuthally randomized) bremsstrahlung effects via up to the minijet scale . In HIJING the ARIADNE Ariadne () code is used in conjunction with the non-perturbative Lund string fragmentation code JETSET Sjostrand:1993yb () to incorporate this effect, while highly azimuthally asymmetric hard pQCD jets with are included via the PYTHIA Sjostrand:1993yb ()) code. In Andersson:1986gw () it was emphasized that the high string tension of color strings reduces greatly the sensitivity of Lund string fragmentation to QCD bremsstrahlung , and that this is an important infrared safety feature of that non-perturbative hadronization phenomenology.

In multiple collisions, however, the projectile accumulates multiple transverse momentum kicks (the Cronin effect) from scattering with cold nuclear participants Gyulassy:2002yv (); Ovanesyan:2011xy () that enhances the bremsstrahlung mean square via random walk in the target frame. In the CGC approach this growth is built into in the infinite momentum frame.

At the minijet scale the underlying azimuthal asymmetry of non-abelian bremsstrahlung will tend to focus gluons toward the azimuthal directions of exchanged momenta. At present, this basic azimuthal dependence is not taken into account in HIJING.

As we show below, there is a very important aspect to the multiple color antenna arrays in high energy due to the longitudinal coherence of clusters of participant target beam jets separated by small transverse coordinates too small to be resolved by the transverse momenta involved. While the total average number of Glauber participant nucleons that interact with a projectile at impact parameter is determined by the area of the inelastic cross section few fm as , for moderate momentum transfers with 1-2 GeV bremsstrahlung the target participant antennas naturally group event by event into resolved clusters separated in the transverse plane by sub-nucleon distances fm, similar to the CGC model Lappi:2006fp () and in AdS/CFT shock modeling Noronha:2014vva () of , but here simply to transverse resolution scale of multiple scattering recoil kinematics in the target frame versus the infinite momentum frame.

This partial decoherence of the participating target dipoles creates non-isotropic spatial distributions of color antennas that radiate according to the fluctuating spatial asymmetries from even to event. Each cluster is characterized by the number of target participant dipole antennas that exchange coherently with the projectile at a specific azimuthal angle controlled by the transverse geometrical distribution of the clusters.

Each recoil cluster radiates coherently into a broad range of rapidities that appears in two particle correlations as “ridge” components with near the cluster accumulated recoil transverse momenta and with near the projectile dipole (cluster) radiates near the total momentum transfer received . On an event-by-event basis and the color antenna geometry fluctuate producing naturally and other azimuthal harmonics in two gluon .

Our goal here is to estimate analytically the magnitude of the color bremsstrahlung source of pQCD dynamical azimuthal two particle correlations and its dependence of . We illustrate the results with specific analytic cluster geometric limits, including symmetric and Gaussian random CSA. We propose a future generalization of HIJING that could enable more realistic testing the influence of anisotropic VGA bremsstrahlung on the final hadron flavor dependent azimuthal moments and competing minijet and hard jet sources of anisotropies.

Ii First order in opacity (GB) bremsstrahlung and azimuthal asymmetries

The above puzzles with BES Adamczyk:2013gw (), at RHIC, and with at LHC motivate us to consider an alternative, more basic, perturbative QCD source of azimuthal asymmetries.The well known non-abelian bremsstrahlung Gunion-Bertsch (GB) formula Gunion:1981qs () for the soft gluon radiation single inclusive distribution is

(1)

where we characterize the parton scattering elastically with the cross section off color neutral target participants with a momentum transfer in terms of a characteristic cold nuclear matter scale GeV taken from fits to forward dihadron correlations in Qiu:2004da (); Neufeld:2010dz (); Kang:2012kc (). Here and the produced gluon has rapidity and transverse momentum () in the final state. It is obvious from Eq. 1 that non-abelian gluon bremsstrahlung is preferentially emitted along two directions specified by the beam “” axis and the transverse momentum transfer vector . The uniform rapidity-even, , distribution associated with moderate scattering is a unique feature of non-abelian bremsstrahlung in the kinematic range of interest associated with beam jets and is due to the triple gluon vertex. The uniform rapidity-even distribution is an especially important characteristic of non-abelian radiation. The combination of the two leads to a uniform rapidity “ridge” in the direction of the momentum transfer that fluctuates in both magnitude and direction from event-to-event but measurable in two or higher gluon correlation measurements. The rapidity-even bremsstrahlung ridge is of course kinematically limited to interval between the target and projectile rapidities. Independent but kinematically correlated multiple target and projectile beam jet bremsstrahlung sources can also naturally account for the triangle boost non-invariant rapidity density observed in as emphasized in Ref.Adil:2005qn ().

For scattering of color neutral dipoles considered in Gunion:1981qs () the Rutherford perturbative distribution of momentum transfers were modeled by color neutral form factors of the form . For GB radiation the singularity is also regulated by such a form factors. Therefore the color neutralization scale also regulates the singularity in Eq. 1 as well. That and dependence of that scale arises naturally in small models based the gluon saturation scale  Kharzeev:2004bw (); Lappi:2006fp (); Dusling:2012cg (). Our emphasis here however is to explore the general characteristics of from the perturbative QCD perspectives that allows us to derive analytically many of the observed remarkably simple scaling relations between azimuthal harmonic cumulants, , as a basic coherent state semi-classical wave interference effect without invoking hydrodynamic local equilibrium assumptions.

The screened single inclusive GB perturbative gluon distribution is

(2)
(3)

where is the azimuthal angle of and is the azimuthal angle of and abbreviations

(4)
(5)
(6)

were introduced a kinematic rapidity envelope factor corresponding to approximately uniform rapidity dependence of the non-abelian bremsstrahlung Gunion:1981qs () regulated with kinematic spectator power counting  Brodsky:1977bu (); Kharzeev:2004bw (). Note for gluon production from the scattering of two color neutral dipoles in the large limit. The rapidity envelopes can be used to build up multi beam jet boost non-invariant triangular as in the BGK Brodsky:1977de () model and also to model the intrinsic boost non-invariance of even in symmetric A+A collisions as with HIJING Adil:2005qn ().

Figure 2: (Color online) Single GB beam jet bremsstrahlung azimuthal Fourier moments, from Eq. (12) are shown versus for for solid(dashed).

Figure 3: (Color online) Single GB beam jet bremsstrahlung azimuthal Fourier moments, , averaged over with are shown versus for for solid(dashed).

Figure 4: (Color online) Ideal power scaling of averaged with (see Eq.11) breaks down at higher because in the limit of non-abelian bremsstrahlung limits (see Eq. (8)).

The single gluon azimuthal moments, in cumulant notation, from a single GB color antenna defined by the momentum transfer with azimuthal angle are defined by

(7)

where we defined , so that and . Note that there are two simple real poles . Since , only contributes to the unit contour integral, resulting in the final analytic expression above. Note that the azimuthal averaged single gluon inclusive () bremsstrahlung distribution with is then

(8)

This has a collinear divergence at in the limit in addition to the usual abelian beam axis divergence. The first is regulated by the color neutral dipole form factor in the GB model.

The azimuthal Fourier moments are however finite in Eq. (7) even in the case of vanishing and depend analytically on and via

(9)
(10)
(11)
(12)

Note that in the limit, all reach unity at but vanish for . For finite , all moments maximizing at with . Figure 2 illustrates the magnitude of GB moments as a function of for and two different .

Note the remarkable power law scaling with (for fixed ) of the azimuthal moments of gluon bremsstrahlung from a single GB color antenna:

(13)

that is similar to the scaling observed by ALICE, CMS and ATLAS Aamodt:2011by (); Chatrchyan:2012wg (); Aad:2012gla () at LHC at least for the higher moments dominated by purely geometric fluctuations. This scaling is of course not expected to hold perfectly for ensemble averaged ratios of of di-hadron inclusive rates. One of our aims below is to test the survival of the above ideal scaling in Eq. (13) to ensembles averages in two gluon inclusive processes.

However, note that by rotation invariance all harmonics vanish for single inclusive GB antennas when averaged over the momentum transfer azimuthal angle . We show below in section V that the finite rms fluctuating harmonics of two particle inclusive survive with similar magnitude and dependence as in Figs. 1,2.

In Fig. 4 we see that the simple fixed power law scaling of Eqs. (11,13) holds for but gradually breaks down at higher when ensemble averaged over in .

Iii All Orders in Opacity VGB Generalization of Gunion-Bertsch radiation

A recursive reaction operator method was originally developed in GLV Gyulassy:2000er (); Gyulassy:2003mc () to compute final-state multiple collision-induced gluon bremsstrahlung and elastic collisional energy loss Gyulassy:2002yv () to all orders in opacity for applications to jet quenching. Extensions of the method to final state heavy quarks jet energy loss was given in Djordjevic:2003zk (); Wicks:2005gt ().

Vitev further extended the reaction operator method to compute non-abelian energy loss in cold nuclear matter in Ref. Vitev:2007ve (). In addition to Final-State (FS) bremsstrahlung , Vitev solved the cold matter Initial-State (IS) bremsstrahlung problem to all orders in opacity and also the generalization of the first order in opacity Gunion-Bertsch Gunion:1981qs () non-abelian bremsstrahlung problem to all orders in opacity for asymptotic () boundary condition. We refer here to the Vitev all-order in opacity generalized GB radiation solution as VGB.

In Vitev:2007ve () the VGB solution was regarded to be of mainly academic interest, since the focus there was on induced initial state IS and final state FS gluon bremsstrahlung associated with hard processes in  Qiu:2004da (); Neufeld:2010dz (); Kang:2012kc (). In this paper, we focus entirely on the application of the VGB solution to low to moderate transverse momentum  few GeV gluon radiation from multiple beam jets in the same spirit as in GB Gunion:1981qs (), where the aim was to understand the general qualitative characteristics of inelastic high energy single inclusive processes from low order perturbative QCD perspective.

Our aim here is to calculate azimuthal asymmetry moments, , arising from basic perturbative QCD bremsstrahlung effects in high energy p+A interactions. The physical picture approximates p+A scattering as the scattering of an incoming color dipole at an impact parameter, , of high (positive) rapidity with nuclear target participant nucleons with high (negative) in the CM. The target participant dipoles at a fixed transverse coordinate are separated longitudinal separations fm in the cold nucleus target rest frame. However they act coherently when emitting gluons near mid rapidity due to Lorentz contraction in the CM and long formation time of gluons in the lab frame.

However, the target participants are distributed in the transverse direction by transverse separations  fm which can be resolved for GeV. This leads incoherent groups of target nucleons that radiate mid-rapidity gluons with ,  GeV gluons coherently. We propose in section IV below a simple percolation model to estimate the partially coherent target recoil bremsstrahlung . However, we concentrate in this section on the coherent projectile bremsstrahlung contribution.

The complete all orders in opacity, , VGB solution derived by Vitev in Vitev:2007ve () is

(14)

where the transverse vector “antenna” amplitudes are defined in terms of differences between“cascade” vector amplitudes as

(15)
(16)

Indices here keep track of combinations of non-vanishing momentum transfers from direct versus virtual diagrams contributing at a given opacity order of the opacity expansion. The partial summed momentum transfers are being the singular directions of non-abelian bremsstrahlung that also control the inverse formation times

(17)

Here is the energy of the gluon in a frame where the energy of the proton projectile is assumed to be large .

There are two simple limits depending on the kinematic range of interest. In the fully coherent limit, where , we can approximate all the cosines by unity. This is the limit we are interested in for our present applications to mid-rapidity multi-particle production not too close to projectile and target fragmentation regions, i.e .

The target scattering centers are ordered in this VGB problem as with for . is the local inverse mean free path of a gluon the nuclear target at position z impact parameter in the target rest frame. The denote normalized distributions of transverse momentum transfers at scattering center .

In the coherent scattering limit of relevance to near mid-rapidity radiation and neglecting possible dependence of the screening scale of the normalized distribution , we can write more explicitly at impact parameter

(18)

In order extract the the physical interpretation of the above complete but unwieldy expression, we derive in the Appendix A the linked cluster theorem version of Eq. 18 to be

(19)

where is the probability density that after elastic scatterings the cumulative total momentum transfer is ,

(20)

that is independent of the azimuthal direction of by rotation invariance. This distribution also arose naturally in the reaction operator derivation of the link cluster theorem for multiple elastic scattering in Ref. Gyulassy:2002yv ().

Eq. (19) is clearly the intuitive factorization limit where at each order only the total accumulated momentum transfer, , controls the azimuthal and momentum transfer dependence of the bremsstrahlung distribution.

Figure 5: (Color online) Schematic diagram corresponding to coherent bremsstrahlung from the projectile dipole from Eqs. (19,20). At opacity order the azimuthal distribution is enhanced for transverse momenta near the total accumulated momentum transfer where groups of recoiling target dipoles.

Figure 6: (Color online) Schematic diagram corresponding to partial coherent gluon bremsstrahlung from Eqs. (23). At opacity order the azimuthal distribution is enhanced in transverse momenta near the recoil momentum transfers where labels incoherent target groups of color dipoles fragmenting toward the negative rapidity region.

By rotation invariance can only depend on the k and Q azimuthal angles through their difference. After integrating over , the azimuthal angle of , then of course cannot depend on the azimuthal angle of . Therefore, it is obvious that at the single inclusive level all vanish for . To observe the intrinsic fluctuating azimuthal asymmetries event-by-event we turn to two particle correlations to extract non-vanishing second moments like . First we discuss the bremsstrahlung contribution from recoil target participants.

Iv bremsstrahlung  from Recoiling Target Participants

Incoherent groups of transversely overlapping recoiling target dipoles radiate gluon bremsstrahlung dominantly into the negative rapidity hemisphere, as illustrated in Fig. 6. In a given event when a projectile nucleon penetrates through a target nucleus at impact parameter , the projectile nucleon moving with positive rapidity is approximated as in Ref. Gunion:1981qs () by a color dipole with a separation . The target nucleons moving toward negative rapidities, , are however distributed with transverse coordinates , according to a Glauber nuclear profile distribution over a large area fm scale. Each target nucleon dipole is assume to have a separation . Projectile target dipole-dipole interactions with low transverse momentum transfer are suppressed by dipole form factors approximated by . Therefore, the projectile interacts dominantly with only nearby target dipoles in the transverse plane with . This leads to a fluctuating number of target participants with probability that follows also from the GLV opacity expansion Gyulassy:2000er (); Gyulassy:2002yv (); Vitev:2007ve ().

For a given target participant number, , the target dipoles naturally cluster near the projectile impact parameter as illustrated in Figs. (5,6). In a specific event, there are in general overlapping clusters that radiate coherently toward the negative rapidity hemisphere as illustrated in Fig.(6). The distribution of the number of recoiling coherent groups depends on , and the momentum exchanges with the projectile that build up to the total exchange to the projectile

(21)

where is a particular subset of the indices that the emitted gluon with transverse wavenumber (and generally ) cannot resolve, and is the contribution from group to the total momentum transfer to the projectile.

A simple percolation model for identifying clusters of coherently recoiling target groups of dipoles is to require that all members in a cluster have separation in the transverse plane in modulus less than the produced gluon transverse momentum resolution scale, i.e.

(22)

where is of order unity. If and as well as , then is added to if its . The M clusters are percolation groups in the above sense. Of course many other variants of transverse clustering algorithms exist. For our purpose of illustrating analytically dynamical sources in p+A compared to peripheral it suffices to study the dependence of on the number of independent recoil antennas with fixed by Glauber participant geometry. In future applications via Monte Carlo generators such as HIJING Wang:1991hta () the sensitivity of results to more realistic multi beam jet geometric fluctuations can be studied. Note that independent target participant beam jet clusters are cylindrical cuts into the target frame near the impact parameter with diameters . We expect typically independent recoil clusters even for the most central collisions, as illustrated in Figs. (5,6). This picture is similar to the CGC model picture except that no classical longitudinal fields are assumed in our entirely perturbative QCD dynamical bremsstrahlung approach here.

In a given event, recoil bremsstrahlung contribution to the single inclusive gluon distribution from coherently acting but transversely resolvable target antenna clusters is given by

(23)

where specifies different rapidity profile functions for each cluster required to produce the characteristic BGK Brodsky:1977de () boost non-invariant triangular enhancement of the rapidity density, , growing toward the value near the target rapidity and dropping toward unity near the projectile rapidity .

In the special doubly coherent projectile and target limit with , reduces to

(24)

with . Note that in the high energy small gluon saturation dynamics correlates with rapidity instead of the simple factorization assumed in Eq. (24). In our simple perturbative dipole picture this correlation can be implemented parametrically by taking Kharzeev:2004bw (); Lappi:2006fp (); Dusling:2013oia ().

The fully coherent projectile bremsstrahlung contribution is

(25)

For scattering with , the sum reduces in the CM to

(26)

which is symmetric with respect to changing the sign of the total momentum transfer, , as well as to reflecting .

In the more general partially coherent target case with independent clusters of dipole antennas, the total single inclusive radiation distribution in mode is

(27)

where we defined to be able to include the projectile contribution into the summation over target clusters. The numerator factor is defined using Eqs. (5,6) to be

(28)

For a fixed set of of independent recoil momenta, the single gluon inclusive azimuthal Fourier moments are given by linear combinations of from Eqs.  (7)-(12). However, since all the terms in the sum contribute with one of factors, averaging over rotations again causes all ensemble averaged to vanish for . In order to extract information about the relative fluctuating , we therefore turn to two gluon correlations in the next section.

V Multi gluon cumulant azimuthal harmonics, , from Color Scintillation Antenna (CSA) arrays

Multiple bremsstrahlung gluons are radiated over long ranges (“ridges”) in from multiple kinematically and transverse space correlated beam jets that form “Color Scintillation Antenna” (CSA) arrays that fluctuate from event to event. Depending on the transverse space geometry, and the transverse momentum transfers , , and their distributions, the CSA bremsstrahlung leads to fluctuating patterns of azimuthal correlations among the radiated gluons. Gluon bremsstrahlung from a single beam jet color dipole antenna builds up a “near side” correlations. Kinematic recoil momentum correlated participant target and projectile antennas, however, also naturally radiate with in complex fluctuating azimuthal harmonic bremsstrahlung patterns. At much higher transverse momenta , collinear factorized back-to-back hard jet production dominates over multiple beam jets bremsstrahlung and leads to very strong away side correlations that must be subtracted in order to reveal the moderate correlations that we compute here. We also assume that we can neglect a possibly large in magnitude transverse isotropic non-perturbative bulk background through appropriate experimental mixed event subtraction schemes.

Assuming that antenna clusters out of the target participants radiate independently - i.e., assuming that each cluster in the CSA array produces approximately a semi-classical coherent state of gluon radiation with random phase with respect to other clusters (see analogous partially coherent pion interferomentry formalism in Ref.Gyulassy:1979yi ()) - the even number inclusive gluons distribution factorizes as

(29)

where is defined in Eq. (28) and again the summation range includes the projectile contribution with . We emphasize that the total gluon inclusive has in addition to an isotropic and a highly away side correlated components that we assume can be subtracted away. Implicitly we also assume here the greatly simplified “local parton hadron” duality hadronization prescription as in CGC models. Of course, in CGC saturation models the details, especially the and will differ, but it is useful to explore here the basic consequences of this simple analytic model to get a feeling of how much of the azimuthal fluctuation phenomenology may have its roots in low order Low-Nussinov/Gunion-Bertsch pQCD interference phenomena. Quenching of bremsstrahlung harmonics due especially to more realistic hadronization phenomenology Wang:1991hta (); Andersson:1986gw (); Sjostrand:1993yb () in the few GeV minijet scale will also need to be investigated in the future.

Even with uncorrelated gluon number coherent state product ansatz for the multi gluon inclusive distribution above, the even number gluons with to become correlated through the CSA geometric and kinematic recoil correlations.

Consider, for example, the case (see Appendix B) of two recoiling target dipole antennas that emit preferentially near and near , at two different recoil azimuthal angles and , while the projectile dipole emits preferentially near at a third azimuthal angle. Such a three color antenna system then naturally leads to two particle triangularity due to dynamical correlations between and . As we also show below in section V, special cases of symmetric antenna arrays illustrate “perfect” bremsstrahlung leading to a pure two particle harmonic.

Consider in detail the prototype VGB antenna case again but for gluon cumulant relative harmonic moments. For a fixed impulse,

(30)

Note that by construction even gluon number are rotation invariant about the beam axis and thus independent of the random orientation, , of the reaction plane defined by the transverse momentum transfer . Of course odd gluon number cumulants vanish after averaging over the reaction plane.

Here are the poles inside the unit circle that contribute to the nth harmonics. For odd number of gluons all harmonics vanish but for even numbers all harmonics both even and odd are generated already by one color GB bremsstrahlung antenna. For , two recoiling GB antennas, and all odd moments vanish by symmetry. An odd number of antennas are needed to generate odd harmonics through even number of gluon correlators.

In the “mean recoil” approximation , we see that a single GB antenna satisfies the generalized power scaling law in case that subsets of the gluons have identical momenta. Suppose there are distinct momenta with such of the gluons have momenta equal to a particular value such that . In this case

(31)

This approximate mean recoil factorization and remarkable power scaling of coherent state semi-classical bremsstrahlung wave harmonics leads to an apparent “perfect fluid collective flow” interpretation.

Higher order cumulant harmonic correlations were proposed vnflow (); Bzdak:2013rya (); ATLASvn2k (); vncumurefs to help remove “non-flow” sources of correlations such as momentum conservation, back to back dijet, and Bose statistics effects and isolate true collective bulk fluid flow azimuthal asymmetries. The -particle cumulant suppresses “non-flow” contribution by eliminating the correlations which act between fewer than particles (see. e.g., fig.9 of ATLASvn2k ()). The first few cumulants for (notation from from Ref. Bzdak:2013rya (); ATLASvn2k ()) are

(32)

The observed ATLASvn2k () near equality of for in at LHC has been interpreted as strong evidence for perfect fluid flow. The similarity of “elliptic flow” in p+Pb and Pb+Pb observed by ATLASAad:2012gla () and also for “triangular flow”