Initial energy density of $\sqrt{s}=$ 7 and 8 TeV p+p collisions at the LHC

Results from the RHIC and LHC experiments show, that in relativistic heavy ion collisions, a new state of matter, a strongly interacting perfect fluid is created. Accelerating, exact and explicit solutions of relativistic hydrodynamics allow for a simple and natural description of this medium. A finite rapidity distribution arises from these solutions, leading to an advanced estimate of the initial energy density of high energy collisions. These solutions can be utilized to describe various aspects of proton-proton collisions, as originally suggested by Landau. We show that an advanced estimate based on hydrodynamics yields an initial energy density in $\sqrt{s}=7$ and 8 TeV p+p collisions at LHC on the same order as the critical energy density from lattice QCD, and a corresponding initial temperature around the critical temperature from QCD and the Hagedorn temperature. The multiplicity dependence of the estimated initial energy density suggests that in high multiplicity pp collisions at the LHC, there is large enough initial energy density to create a non-hadronic perfect fluid.


Introduction
The interest in relativistic hydrodynamics grew in past years mainly due to the discovery of the nearly perfect fluidity of the experimentally created Quark-Gluon-Plasma (QGP) at the Relativistic Heavy Ion Collider (RHIC) [1]. Hydrodynamical models aim to describe the space-time picture of heavy-ion collisions and infer the relation between experimental observables and the initial conditions. Besides numerical simulations there is also an interest in models where exact solutions of the hydrodynamical equations are used. It is customary to describe the medium created in heavy ion collisions with hydrodynamic models, however, the proton-proton system is frequently considered as a system that might be too small for thermalization, or that might be not hot or dense enough to create a supercritical (non-hadronic) medium. Energy densities in √ s = 200 GeV p+p collisions are likely below this limit. It is however an interesting question, how high energy densities can be reached in p+p collisions at the LHC with √ s = 7 and higher collision energies. In this paper we apply the hydrodynamical solution of Refs. [2,3] to describe the pseudorapidity distribution in pp collisions at √ s = 7 and 8 TeV and use the results of these hydrodynamical fits to estimate the initial energy density in these reactions.

Hydrodynamics
The basic hydrodynamical equations are the continuity and energy-momentum-conservation equations: with n being a conserved charge, and T is the energy-momentum tensor. In case of a perfect fluid where is the energy density and p the pressure. The Equation of State (EoS) closes the set of equations: = κp while p = nT defines temperature T . An analytic hydrodynamical solution is a functional form of , p, T , u µ and n, which solves the above equations. The solution is explicit if these fields are explicitly given as a function of space-time coordinates x µ = (t, r) = (t, r x , r y , r z ), or in a general d-dimensional case, x µ = (t, r 1 , . . . , r d ).

Rapidity distributions
The differential rapidity distribution or rapidity density dN/dy (with N being then the total number of particles) was calculated in Refs. [2,3,4,5]: with α = 2λ−1 λ−1 , and N 0 is a normalization parameter. The rapidity distribution is approximately Gaussian, if λ > 1. At λ = 1, the distribution becomes flat, as in this is the Bjorken limit (corresponding to the Hwa-Bjorken solution), the solution becomes boost invariant. Also note that in order to describe experimental data, pseudorapidity distributions have to be calculated as well. See details in Refs. [2,3,4,5].

Energy density estimation
In this section we recapitulate how this model can be used for improving the famous energy density estimation made by Bjorken [7]. Let us focus on the thin transverse slab at mid-rapidity, just after thermalization (τ = τ 0 ), illustrated by Fig. 2 of Ref. [7]. The radius R of this slab is estimated by the radius of the colliding hadrons or nuclei, and the initial "fireball" volume is dV = (R 2 π)τ 0 dη 0 , where τ 0 dη 0 is the longitudinal size, as dη 0 is the pseudorapidity width at τ 0 . See Refs. [2,3,4,5] for details. The energy content is dE = E dN , where dN is the number of particles and E is their average energy near y = 0. So, as given in Bjorken's paper, the initial energy density is for a boost invariant solution, we obtain Bjorken's energy density estimation as Here τ 0 is the proper-time of thermalization, estimated by Bjorken as τ 0 ≈ 1fm. For accelerationless, boost-invariant Hwa-Bjorken flows η 0 = η f = y, however, for accelerating solutions one has to apply a correction to take into account the acceleration effects on the energy density estimation [4]. Thus for the hydrodynamical solutions where the acceleration parameter is λ > 0. the initial energy density is given by a corrected estimation corr as Here Bj is the Bjorken estimation, which is recovered if dN/dy is flat (i.e. λ = 1), but for λ > 1, both correction factors are bigger than 1. Hence the initial energy densities are under-estimated by the Bjorken formula, if the measured dn/dy distributions are not constant, but have a finite width. In Refs. [2,3,4,5] we performed fits to BRAHMS pseudo-rapidity distributions from Ref. [8], and these fits indicate that corr = 8.5 − 10 GeV/fm 3 in √ s NN = 200 GeV Au+Au collisions at RHIC.
The above corrections are exact results, that were derived in details for a special equation of state (EoS) of κ = 1 [4]. The correction factors in Eq. (6) take into account the work done by the pressure on the surface of a finite and accelerating, hot fireball. However, the relation of the pressure to the energy density is obviously EoS dependent, and as proposed in Refs. [2,3,5] the effects of a non-ideal equation of state can be estimated with the following formula: This conjecture satisfies several consistency requirements, for example, it goes back to the exact result of Eq. (6) in case of a super-hard EoS of c s = 1 and gives initial energy density values that were checked against numerical solutions [2]. From basic considerations [7], as well as from lattice QCD calculations [9], it follows that the critical energy density, needed to form a non-hadronic medium is around 1 GeV/fm 3 . From the lattice QCD calculations one gets crit = (6−8)×T 4 crit (in c = 1 units), and even with a conservative estimate of T crit = 170 MeV, one gets crit < 1 GeV/fm 3 . Thus initial energy densities above this value indicate the formation of a non-hadronic medium.

Initial energy densities in 7 and 8 TeV LHC p+p collisions
Let us estimate the quantities in Eq. (5). The average transverse momentum in √ s = 7 TeV p+p collisions is p t = 0.545 ± 0.005 stat ± 0.015 syst GeV/c [10], which corresponds to E = 0.562 GeV/c 2 at midrapidity (assuming most of these particles are pions).
It is a non-trivial question, how to estimate the initial transverse area R 2 π in p+p collisions, because the geometrical area relates to the total, elastic, inelastic and differential cross-sections in an involved and non-trivial manner. In case of the collisions of large heavy ions, the initial overlap area is evaluated based on nuclear density profiles, for example using the relation R ∝ A 1/3 . Basically these relations that determine the nuclear geometry were obtained from the analysis of the differential cross-sections of elastic electron-ion [11] and elastic proton-nucleus data [12]. Similarly, to get a reliable estimate of the initial transverse area in p+p collisions, we should rely on the analysis of elastic p+p scattering data.
Our analysis is based on Eqs. (117-119) and (124-126) of Ref. [13], that show that both for a grey disc and for a Gaussian scattering density profile, σ el = πR 2 A 2 and σ tot = 2πR 2 A, where A measures the "greyness" of the proton. Thus the geometrical area R 2 π actually can be estimated as We have cross-checked these estimates by evaluating the geometrical area from B, the slope of differential elastic scattering cross-section at zero momentum transfer, as we may also use the relation R 2 π = 4πB [13]. We found that within errors both methods yield the same estimate for the initial geometry of p+p collisions. Based on the results of Refs. [14,15], we have conservatively estimated R = 1.76 ± 0.02 fm. Furthermore, the formation time, τ 0 , may conservatively assumed to be 1 fm/c. The only remaining parameter is the pseudorapidity density at midrapidity. As measured by the LHC experiments, the charged particle pseudorapidity density at midrapidity (dn/dη| η=0 ) is found to be 6.01±0.01(stat) +0.20 −0.12 (syst) at ALICE [10], while 5.78±0.01 stat ± 0.23 syst at CMS [16], but in some multiplicity classes it may reach values of 25-30 (see Table I. of Ref. [17]). We will take the average of the first two values. The total multiplicity is then 3/2× the charged particle multiplicity. Substituting 1.5dn/dη| η=0 , based on Eq. (5) one gets: which is below the critical value. The advanced estimate is based on TOTEM pseudorapidity density dN/dη data, as these reach out to large enough η values so that the acceleration parameter can be determined. Fits to TOTEM data were performed via Eq. (4) , as shown in Fig. 1. The fit resulted in the acceleration parameter λ = 1.073 ± 0.001 stat ± 0.003 syst , where the systematic error is based on the point-to-point systematic error of the data points.
Assuming c 2 s = 0.1 (this is a quite realistic value, at least no harder EoS is expected at LHC, as similar EoS was found at RHIC as well [18,9,19]), one only needs a τ f value.
Thus if the freeze-out temperature (assumed to be around the Hagedorn-temperature or the critical temperature of lattice QCD) is T f = 140 MeV, then an initial temperature of T 0 = 170 MeV (needed in order to form a strongly interacting quark gluon plasma) corresponds to τ f being 5-6 times τ 0 , for c 2 s = 0.1 and λ = 1.1. Even if c 2 s and λ are higher, τ f /τ 0 seems to be a rather conservative value. With this, one gets the multiplicative correction factors of 1.146 and 1.101, thus corr (7 TeV) = 1.262 Bj (7 TeV) = 0.640 GeV/fm 3 which is still below the critical value. The c 2 s and τ f /τ i dependence of the correction factor is shown in Fig. 2.   Note that the average p+p multiplicity was used here, so this value represents an average energy density in p+p collisions -below the critical value of 1 GeV/fm 3 (c.f. Ref [20] where a possible cross-over starting at dn/dη| η=0 = 6 was conjectured). Based on Table I. of Ref. [17], much larger multiplicities have been reached however. The energy density results for these multiplicities is shown on Fig. 3. It is clear from this plot, that even for the original Bjorken estimate, supercritical enegy densities may have been reached in high multiplicity events, roughly from dn/dη| η=0 ≈ 12. The corrected estimate gives supercritical values for dn/dη| η=0 ≈ 9. We also calculated the initial temperature based on the ∝ T 4 relationship, assuming that 175 MeV corresponds to 1 GeV/fm 3 approximately. This is also shown in the left panel of Fig. 3, as well as the reachable pressure values. A temperature of 300-600 MeV may have been reached in 200 GeV central Au+Au collisions of RHIC [21]. Initial temperature values in 7 TeV p+p seem to be lower than that, 300 MeV can be reached in events with a multiplicity of > 50. However, 200 MeV may already be reached in events with a multiplicity of 16. Now let us estimate what happens at √ s = 8 TeV. As for the Bjorken-estimate, we need the change in charged particle multiplicity, average transverse energy and transverse size. CMS indicates dn/dη| η=0 = 6.20 ± 0.46 for a non-single diffractive enhanced data sample in Ref. [22], while dn/dη| η=0 = 6.13 ± 0.1 is given in Ref. [23]. The average of the two values is in good agreement with the approximate s dependence of dn/dη| η=0 of 0.715 · √ s 0.23 as estimated in Ref. [24]. Average transverse momentum s dependence is estimated as p t = 0.413 − 0.0171 ln s + 0.00143 ln 2 s in Ref. [10], which means a 1.53% increase in E . Transverse size can be estimated based on the σ tot measurement of Refs. [25], σ tot = 101.7 ± 2.9(syst) mb, which means a 3.8% increase in area compared to 7 TeV, and R = 1.799 ± 0.025 fm Based on Eq. (5), this altogether means a 2.4% increase in Bj , i.e.
This value is based on the average multiplicity in √ s = 8 TeV collisions. However, at a fixed multiplicity, there is almost no difference between the two collision energies: average transverse energy increases by 1.5%, but the transverse also increases by 2.4%. This means a roughly 1% decrease, which is much smaller than the systematic uncertainties in this estimate -to be discussed in the next section. We plot the multiplicity dependence of ini , T ini and p ini for the 8 TeV case in the right panel of Fig. 3. We may again observe, that supercritical values are reached for multiplicities higher than 10 in case of the corrected estimate; but even Bjorken's estimate yields supercriticality if dn/dη| η=0 > 13.
6 Uncertainty of the estimate Different sources of uncertainties are detailed in Table 1. The most important one comes from dn/dη at midrapidity. From Fig. 3 it is clear that for the Bjorken-estimate, energy density is above the critical value of 1 GeV/fm 3 if the multiplicity is larger than 12, while in case of the corrected initial energy density, dn/dη| η=0 > 9 is needed. Taking all sources 0.562 0.5% 3% dN/dη (7 TeV) 5.895 0.2% 3% Table 1: Sources of statistical and systematic errors for the 7 TeV estimate.
of uncertainties into account, the final result for the energy density corresponding to mean multiplicity density at 7 TeV is corr (7 TeV) = 0.64 ± 0.01(stat) +0.14 −0.10 (syst) GeV/fm 3 (13) and the main systematic error comes from the estimation of the ratio τ f /τ 0 . In the 8 TeV case, the estimate yields a somewhat larger number (0.644 versus 0.640 GeV/fm 3 ), but the uncertainties are somewhat higher due to extrapolations to 8 TeV.
An important source of systematic uncertainty is the use of the given hydrodynamic solution. This uncertainty may be estimated by using other hydrodynamic models that contain acceleration: the Landau model [26], the Bialas-Peschanski model [27], or numeric models of hydrodynamics, however, in the current paper we focus on the analytic results that can improve on Bjorken's famous initial energy density estimate. A more detailed, or a numerical hydrodynamical investigation is outside the scope of the present manuscript, however we can cross-check these results by fitting the simultaneously taken CMS and TOTEM data on dn/dη with the same model, to investigate the stability of our initial energy density estimate from the details of dn/dη at midrapidity. 7 Improved initial energy density using combined TOTEM+CMS dN/dη data Fit to TOTEM dN/dη data was shown in Section 5 and in particular in Fig. 1. This shows an advanced estimate and reaches out to a large η region. However, it may seem necessary to put more attention on central η region and perform fits on a combined central+forward η region. Thus in this section we present dN/dη distributions of charged particles from combined TOTEM+CMS datasets. In the left panel of Fig. 4 a fit to such combined data at 7 TeV [10,28] is shown. This fit to TOTEM+CMS data [10,28] yields an initial flow acceleration parameter λ = 1.076 for 7 TeV pp collisions. As seen from the terms on the right hand side in Eq. (7), the corrected initial energy density depends on the acceleration parameter λ, the driving force for the hydrodynamic expansion or the pressure gradients and volume element expansion. Thus the advanced hydrodynamic estimate is corr (7 TeV) = 1.273 × Bj (7 TeV) = 0.645 GeV/fm 3 .
Here we stress that compared to the corr estimate from fitting TOTEM only, the difference less than 1%, which is quite reasonable. The fit in the right panel of Fig. 4 shows the hydrodynamic dN/dη distribution and charged particle in 8 TeV pp collision measured by TOTEM and CMS [22] (including TOTEM data measured in collisions with a  [10] and TOTEM [28] at 7 TeV fitted with the result of our relativistic hydrodynamical solution described in this paper, with λ = 1.076. Right: Charged particle dN dη distributions from CMS and TOTEM 8 TeV data of Ref. [22,29] fitted with our results, with λ = 1.066. displaced interaction vertex [29]). The resulting acceleration parameter is λ = 1.066, this corresponds to a final result of corrected initial energy density form the advance estimate corr (8 TeV) = 1.235 × Bj (8 TeV) = 0.641 GeV/fm 3 (15) which differs from the previous estimate based on only TOTEM data by less than 0.5%.

Summary
We have shown, that based on an accelerating hydro solution and data of the TOTEM and CMS experiments at CERN LHC, the advanced estimate of the initial energy density yields a value that below the critical value of 1 GeV/fm 3 , but is not inconsistent with a supercritical state in high multiplicity 7 and 8 TeV proton-proton collisions. The energy density is proportional to the measured multiplicity, and so in high-multiplicity events, initial energy densities several times the critical energy density of 1 GeV/fm 3 have been reached. It means, that an important and necessary condition is satisfied for the formation of a non-hadronic medium in high multiplicity (dn/dη η=0 > 9) 7 and 8 TeV p+p collisions at CERN LHC, however, the exploration of additional signatures (radial and elliptic flow, volume or mean multiplicity dependence of the signatures of the nearly perfect fluid in p+p collisions, scaling of the HBT radii with transverse mass, and possible direct photon signal and low-mass dilepton enhancement) and their multiplicity dependence can be a subject of detailed experimental investigation even in p+p collisions at the LHC. It is also important to note, that even the measurement of differential elastic scattering cross-section has implications regarding this estimate. We recall here that the application of hydrodynamical expansion to data analysis in high energy p+p collisions is not an unprecedented or new idea, as Landau worked out hydrodynamics for p+p collisions [30], and Bjorken also notes this possibility in his paper [7] describing his energy density estimate.
It is also noteworthy that Hama and Padula assumed [31] the formation of an ideal fluid of massless quarks and gluons in p+p collisions at CERN ISR energies of √ s = 53-126 GeV. Alexopoulos et al. used Bjorken's estimate to determine the initial energy density of ∼ 1.1 ± 0.2 GeV/fm 3 at the Tevatron in √ s = 1.8 TeV p+p collisions in the E735 experiment [32], while Lévai and Müller argued [33], that the transverse momentum spectra of pions and baryons indicate the creation of a fluid-like quark-gluon plasma in the same experiment at the same Tevatron energies. However, these earlier works considered the quark-gluon plasma as an ideal gas of massless quarks and gluons, while the RHIC experiments pointed to a nearly perfect fluid of quarks where the speed of sound is measured to be c s ≈ 0.35 ± 0.05 that is significantly different from that of a massless ideal gas of quarks and gluons, characterized by a c s = 1/ √ 3 ≈ 0.57. Recently, Shuryak and Zahed also proposed [34] the application of hydrodynamics for high multiplicity p+p and p+A collisions at CERN LHC.
The main result of our study indicates, that the initial energy density is apparently large enough in high multiplicity p+p collisions at the √ s = 7 and 8 TeV LHC energies to create a strongly interacting quark-gluon plasma, so a transition with increasing multiplicity is expected, as far as hydrodynamical phenomena are considered.
Probably the most important implication of our study is the need for an e+p and e+A collider: as far as we know only in lepton induced proton and heavy ion reactions can one be certain that a hydrodynamically evolving medium is not created even at the TeV energy range. The results of lepton-hadron and lepton-nucleus interactions thus will define very clearly the particle physics background to possible collective effects. For example, recently azimuthal correlations were observed in high multiplicity p+p and p+A as well as in heavy ion reactions (the ridge effect [35,36]), whose origin is currently not entirely clear. If such a ridge effect appears also in e+p and e+A collisions, then most likely this effect is not of a hydrodynamical origin, while if it does not appear in e+p and e+A collisions in the same multiplicity range as in p+p and p+A reactions, than the ridge is more likely a hydrodynamical effect.
If indeed a strongly interacting non-hadronic medium is formed in high multiplicity p+p collisions, than purely the jet suppression in heavy ion collisions does not reveal the true nature of these systems: the proper measure would be energy loss per unit length (as proposed in Ref. [37]), which may be quite similar in these systems, even if the total suppression is different.
We are looking forward to measurements unveiling the nature of the matter created in proton-proton collisions. In experimental p+p data, one should look for the enhancement of the photon to pion ratio in high multiplicity events (as compared to low multiplicity ones) [38], for a hydrodynamic scaling of Bose-Einstein correlation radii or that of azimuthal asymmetry [39], or even the enhancement of low mass dileptons [40].