Higher-order anisotropies in the Blast-Wave Model - disentangling flow and density field anisotropies
EEPJ manuscript No. (will be inserted by the editor)
Higher-order anisotropies in the Blast-Wave Model –disentangling flow and density field anisotropies
Jakub Cimerman , , Boris Tom´aˇsik , , M´at´e Csan´ad , and S´andor L¨ok¨os Czech Technical University in Prague, FNSPE, Bˇrehov´a 7, 11519 Prague 1, Czech Republic Comenius University, FMPI, Mlynsk´a Dolina F1, 84248 Bratislava, Slovakia Univerzita Mateja Bela, FPV, Tajovsk´eho 40, 97401 Bansk´a Bystrica, Slovakia E¨otv¨os Lor´and University, P´azm´any P. s. 1/a, H-1117 Budapest, Hungary
Abstract.
We formulate a generalisation of the blast-wave model which is suitable for the description ofhigher order azimuthal anisotropies of the hadron production. The model includes anisotropy in the densityprofile as well as an anisotropy in the transverse expansion velocity field. We then study how these two kindsof anisotropies influence the single-particle distributions and the correlation radii of two-particle correlationfunctions. Particularly we focus on the third-order anisotropy and consideration is given averaging overdifferent orientations of the event plane.
PACS.
The hot matter excited in ultrarelativistic heavy-ion col-lisions at colliders like the LHC or RHIC exhibits a size-able anisotropy in particle production perpendicularly tothe beam direction [1,2,3,4,5]. The azimuthal anisotropyof hadron momentum distributions, measured in terms ofFourier coefficients, is caused by the anisotropy of the fire-ball at freeze-out in spatial density and expansion pattern.That, in turn, results from its evolution which starts fromanisotropic initial conditions [6,7,8,9,10] and may furtherreceive anisotropic excitations on the way [11,12,13,14].The evolution depends on the Equation of State and thetransport coefficients [15,16,17]. Thus by measuring thefinal state anisotropies one gets an access to the intrinsicproperties of the matter [18]. Note that the anisotropiesare unique in each event and a large fraction of them, espe-cially higher-order anisotropies, are averaged out if mea-sured in a sample consisting of a large number of events.Hadrons are emitted at the moment of freeze-out andthis is when their distributions are formed. Hence, the di-rect information that they are carrying is about the stateof the fireball at this very moment. The present paperdeals with this relation. It is then the task for hydrody-namic and/or transport simulations to conclude about thepreceding evolution of the fireball.There are two kinds of anisotropies of the fireball at themoment of freeze-out, that may cause an anisotropy of thehadron distribution [19,20,21,22]. Firstly, if the transverseexpansion velocity in some directions is higher than in theothers, the stronger blueshift of the momentum spectrain those directions will cause momentum anisotropy. Sec-ondly, alone an expanding and spatially anisotropic fire- ball may also produce an anisotropy of the momentumdistribution. Unfortunately, the two mechanisms cannotbe distinguished by mere measurement of the momentumanisotropies.For the second-order anisotropies it has been shownthat the solution is offered by the azimuthal dependenceof the femtoscopic correlation radii, which are more sensi-tive to spatial anisotropy. Detailed studies with the helpof blast-wave [19] and Buda-Lund [20] models have beenperformed. After PHENIX has published the azimuthaldependence of the correlation radii [23] with respect tothe third-order event plane, the problem has been recon-sidered at that order in [21]. It has been demonstratedin framework of a toy model that at RHIC the spatialanisotropy is the driving feature which determines thephase of the oscillation of the correlation radii. This is inagreement with the second-order results [19,20]. Today,higher experimental statistics allows for more detailed in-vestigations and the data on third-order azimuthal depen-dence of correlation radii call for more detailed theoreti-cal studies [25,26,27]. The third-order anisotropy togetherwith azimuthal dependence of the correlation radii havebeen investigated by some of us in detail in frameworkof the Buda-Lund model recently [22]. The dependenceof the oscillation amplitudes on parameters of the modelwhich gauge the anisotropies in space and flow has beencalculated in great detail up to 6th order.The present paper is analogical to [22]. We reconsiderthe problem in framework of the blast-wave model. This isperhaps the most commonly used model for the analysis ofsoft hadron data in high energy heavy-ion physics. There-fore, we first systematically extend it for anisotropies of a r X i v : . [ nu c l - t h ] J u l J. Cimerman, B. Tom´aˇsik, M. Csan´ad, and S. L¨ok¨os: Anisotropies in the Blast-Wave Model higher order, although we later use it only up to the thirdorder. By doing this we actually modify the formulationof the model as proposed in [32], since that one was ad-equate only up to second order. Note that our extensionalso slightly differs from the one proposed in [24], sincewant to allow for varying transverse size of the fireball.Then, with the generalised blast-wave model we in-vestigate in quite detail how the oscillation amplitudes ofthe correlation radii depend on parameters which mea-sure the flow and the shape anisotropy. On top of theschematic picture obtained in [21] we add the details byproviding parameter maps, i.e. contour plots of the de-pendence of oscillation amplitudes on both flow and spaceanisotropy, similarly as was done for the Buda-Lund modelin [22]. Such maps should allow, at least in principle, toinfer both values for flow and space anisotropy from themeasured data. We also go beyond the toy-model studyof [21] by including both the second and the third orderanisotropies into the calculation and integrating over theorder not actually being investigated, analogically to realexperiment.In the next Section we introduce and explain the ex-tension of the blast-wave model used in this study. Then,Section 3 is devoted to calculations of the anisotropiesof single-particle distributions. Oscillations of correlationradii are investigated in Section 4. We demonstrate our re-sults with the help of qualitative data analysis in Section5. All results are summarised in the concluding Section.Some technical details are explained in the Appendices.
Particle production is described with the help of the emis-sion function S ( x, p ), which is the Wigner function, i.e.the phase-space density of hadrons that are being emittedfrom the fireball. In the blast-wave model [28,29,30,31,32] it is parametrised as S ( x, p ) d x = g (2 π ) m t cosh( Y − η s ) r dr dθ τ dη dτ √ π∆τ exp (cid:18) − ( τ − τ ) ∆τ (cid:19) Θ ( r − R ( θ )) exp (cid:18) − p µ u µ T (cid:19) . (1)Here, g is the spin degeneracy factor, T is the local tem-perature, u µ is the expansion velocity field, and Θ ( x ) isthe Heaviside step function. We parametrise the momen-tum of a particle with the help of transverse momentum p t , transverse mass m t , rapidity Y and the azimuthal an-gle φ as p µ = ( m t cosh Y, p t cos φ, p t sin φ, m t sinh Y ) . (2)In this paper we shall denote the rapidity with capitalletter in order to distinguish it from the spatial coordinate.As spatial coordinates we use the radial coordinate r and the azimuthal angle θ , as well as the space-time ra-pidity η s = ln (( t + z ) / ( t − z )) and longitudinal proper time τ = √ t − z . Then x µ = ( τ cosh η s , r cos θ, r sin θ, τ sinh η s ) . (3)Furthermore, R ( θ ) is the transverse size of the fire-ball, depending on the azimuthal angle θ . The spatialanisotropy of the model is specified by the particular pre-scription for R ( θ ). The transverse size is then parametrisedas R ( θ ) = R (cid:32) − ∞ (cid:88) n =2 a n cos( n ( θ − θ n )) (cid:33) , (4)where the amplitudes a n and the phases θ n are modelparameters. Note that θ n ’s denote the orientations of theso-called n -th order event planes. Note that in the serieswe have skipped the first-order term which leads to mereshift of the shape. Also note that the amplitudes for theoscillation are parametrised in an unusual way, with thehelp of ( − a n ). In this way, the resulting v n is to first orderproportional to a n , as will be seen later.Note that in [19] a different parametrisation for theelliptic shape of the fireball was used, with radii R x and R y along the two axes of the ellipse R x = aR (cid:48) , R y = R (cid:48) a , (5)with R (cid:48) and a being parameters of the model. The advan-tage of that parametrisation is that when a is being tuned,the volume stays constant and proportional to R . Nev-ertheless, such a prescription cannot be naturally gener-alised to higher orders. Therefore, we will now use Eq. (4).Some comments on the relation of the two parametrisa-tions can be found in Appendix A.It is convenient for further calculation to define a di-mensionless transverse coordinate¯ r = rR ( θ ) . (6)Particle production in our model occurs for ¯ r in the range[0 , u µ . Veloc-ity includes longitudinal as well as transverse component u µ = (cosh η s cosh ρ, sinh ρ cos θ b , sinh ρ sin θ b , sinh η s cosh ρ ) (7)where θ b = θ b ( r, θ ) (8)is the angle of the transverse vector of the velocity andwill be specified below. Furthermore ρ = ρ (¯ r, θ b ) (9)is the rapidity connected with the transverse velocity, sothat the transverse velocity at midrapidity is v t = tanh ρ .The canonical (azimuthally symmetric) blast-wave mo-del is recovered if R ( θ ) and ρ (¯ r, θ ) do not depend on theangle and θ b = θ . Here we construct the extension to ar-bitrary order of anisotropy. . Cimerman, B. Tom´aˇsik, M. Csan´ad, and S. L¨ok¨os: Anisotropies in the Blast-Wave Model 3 In a fireball without azimuthal symmetry we mustspecify the direction of the transverse expansion velocity.In [19], two models were investigated which differed in thechoice of that direction. Note, however, that only second-order anisotropy was studied there. It turned out thatfemtoscopic data [25] agreed with the choice in which thetransverse velocity was always perpendicular to the sur-face of the fireball. We adopt this choice also here. Trans-verse velocity will be locally perpendicular to surfaces withconstant ¯ r . Note that this is the natural direction of trans-verse pressure gradient and thus the acceleration. Hence,we actually identify the direction of the velocity with thatof acceleration. Such a choice is expected to be valid if thefireball decouples fast.The azimuthal angle of the velocity θ b is then obtainedfrom tan (cid:16) θ b − π (cid:17) = dydx = dydθdxdθ , (10)where the derivative is taken along a surface with constant¯ r . The solution is straightforward and we summarise ittogether with the final result for θ b in Appendix B.Finally, let us define the magnitude of the transversevelocity, which is parametrised with the help of transverserapidity ρ (¯ r, θ b ) = ¯ rρ (cid:32) ∞ (cid:88) n =2 ρ n cos ( n ( θ b − θ n )) (cid:33) . (11)Overall transverse flow is tuned with the help of ρ and theanisotropies have amplitudes ρ n . Note that we choose thesame phase factors θ n as we did for the spatial anisotropy.They are related to the event planes measured experimen-tally. Note also the introduction of the factor 2 before ρ n ,unlike in Eq. (4).In this paper we will restrict ourselves to anisotropiesup to third order; higher orders will be omitted.Note also that we did not include corrections to themomentum distribution due to viscosity [33], as done e.g. in[34,35]. We plan to investigate this important issue in thefuture. The single-particle spectrum is obtained by integratingthe emission function N ( p t , φ, Y ) = d Np t dp t dY dφ = (cid:90) S ( x, p ) d x . (12)The normalisation is such that the integral of N ( p t , φ, Y )over all momenta gives the number of particles. For theintegrations of transverse directions in eq. (12) it wouldbe convenient to use polar coordinates r and θ . However,it is even more convenient to use ¯ r , defined in Eq. (6),instead of r . This requires a new Jacobian r dr dθ = ¯ r R ( θ ) d ¯ r dθ . (13) Before moving on towards the anisotropy it is inter-esting to explore if and how introducing spatial and flowanisotropy into the model modifies the azimuthally inte-grated single-particle spectrum. We have checked that ifwe introduce only a spatial anisotropy (i.e. the a n coeffi-cients may be non-vanishing, but all ρ n ’s are set to 0), thenthe normalisation may be slightly modified but the slopeis unchanged. This is not the case for the flow anisotropy,however. As shown in Fig. 1a and Fig. 1b, flow anisotropyleads to slightly flatter spectra. We also show in Fig. 1chow the azimuthally integrated spectrum depends on thephase difference of the second and third order event planes(cf. Eq. (4)) ∆ = θ − θ . (14)We do not expect any correlation between the second-order and the third-order event planes and to our knowl-edge there is no such correlation seen in the data. Hence,all phase differences are realised equally likely. In a datasample averaged over a large number of events the meanvalue of all observed curves would be measured.Now we move on to the anisotropies of spectra whichwill be obtained as v n = (cid:82) N ( p t , φ, Y ) cos( n ( φ − θ n )) dφ (cid:82) N ( p t , φ, Y ) dφ . (15)The single-particle distributions are calculated via Eq. (12).The anisotropy coefficients v n can be then expressed as v n ( p t ) = C n ( p t ) C ( p t ) (16)where C n ( p t ) = (cid:90) d ¯ r (cid:90) π dθ ¯ r R ( θ ) cos( n ( θ b ( θ ) − θ n )) × I n (cid:18) p t sinh ρ (¯ r, θ ) T (cid:19) K (cid:18) p t cosh ρ (¯ r, θ ) T (cid:19) (17)where I n and K are modified Bessel functions and theintegration over the azimuthal angle of the momentum φ was already performed here.The result of the calculation, however, would dependon the value of the phase difference ∆ . This dependenceis hidden in R ( θ ) and ρ (¯ r, θ ). In an experimental analysis,one effectively takes an average over all its possible values.It has been shown in [22] that the averaging may have aneffect on the results. We thus have to add this averagingand introduce¯ C n ( p t ) = (cid:90) π d∆ (cid:90) d ¯ r (cid:90) π dθ ¯ r R ( θ ) × cos( n ( θ b ( θ ) − θ n )) × I n (cid:18) p t sinh ρ (¯ r, θ ) T (cid:19) K (cid:18) p t cosh ρ (¯ r, θ ) T (cid:19) . (18)Then, the event-averaged v n is obtained as v n ( p t ) = ¯ C n ( p t )¯ C ( p t ) . (19) J. Cimerman, B. Tom´aˇsik, M. Csan´ad, and S. L¨ok¨os: Anisotropies in the Blast-Wave Model R a t i o t o ρ = ρ = 0.3 ρ = 0.25 ρ = 0.2 ρ = 0.15 ρ = 0.1 ρ = 0.0511.522.533.5 (b) R a t i o t o ρ = ρ = 0.3 ρ = 0.25 ρ = 0.2 ρ = 0.15 ρ = 0.1 ρ = 0.0511.021.041.061.081.10 0 0.5 1 1.5 2(c) R a t i o t o ∆ = p t [GeV] ∆ = 5 π /30 ∆ = 4 π /30 ∆ = 3 π /30 ∆ = 2 π /30 ∆ = π /30 Fig. 1.
The ratios of azimuthally integrated single particlespectra from fireballs with anisotropies to a reference spectrumcalculated for the same set parameters except all anisotropycoefficients set to 0. Calculated for (directly produced) pionsand T = 120 MeV, ρ = 0 . R = 7 fm, τ = 10 fm/ c .(a) Ratios of spectra with second-order flow anisotropy and a = a = ρ = 0. (b) Ratios of spectra with third-order flowanisotropy and a = a = ρ = 0. (c) Ratios of spectra with a = a = ρ = ρ = 0 . ∆ = θ − θ . − a − a − − − − a ρ − − − − a ρ Fig. 2.
Dependence of v n ’s on spatial anisotropy a n and flowanisotropy ρ n of the same order. Results are shown for pions(upper row) and protons (lower row) at p t = 300 MeV. Modelparameters are T = 120 MeV, ρ = 0 . R = 7 fm, τ = 10fm/ c . The thick line identify v or v equal to 0, the thin linescorrespond to increment or decrease of v ( v ) by 0.005. We calculated the dependence of v and v on theanisotropy coefficients a n and ρ n . We have checked thatthe v n ’s of given order basically depend only on coeffi-cients of the same order, therefore we shall only investi-gate such same-order dependences. In Fig. 2 we show thecontour plots where dependence on both spatial and flowanisotropy can be seen. A complex structure is observed.In general, we can conclude, similarly to [19,20,22], thatalone by measuring the anisotropy of single-particle dis-tribution one is unable to determine uniquely both spatialand flow anisotropy. For heavy particles, like protons, v n ’sseem to be driven by the spatial anisotropy. Nevertheless,flow anisotropy kicks in as the value of a n grows. There iseven a maximum assumed by v and v as a function ofthe corresponding a n , although it may well be beyond thephenomenologically relevant parameter region. The rea-son is that for high enough values of a n the outer sur-face of the fireball becomes concave and a smaller regionmoves transversely in the direction of the event plane. Al-though a unique combination of source parameters mightpossibly be determined from the combination of measure-ments with different particle species, a clear answer shallbe provided by measuring the azimuthal dependence ofthe correlation radii. The femtoscopic technique which uses two-particle cor-relations is standard tool for measuring the space-timecharacteristics of the emitting source. Here we employ thestandard formalism where the correlation function is de- . Cimerman, B. Tom´aˇsik, M. Csan´ad, and S. L¨ok¨os: Anisotropies in the Blast-Wave Model 5 fined as C ( p , p ) = N ( p , p ) N ( p ) N ( p ) = d Ndp dp d Ndp d Ndp . (20)Instead of the momenta of the two particles, the correla-tion function is usually parametrised in terms of the av-erage momentum K and the momentum difference qK = 12 ( p + p ) (21a) q = p − p . (21b)Due to the on-shell constraint K · q = 0, the time compo-nent q can be expressed as q = K K q = βq . (22)Hence, only three spatial components of q shall be takenas independent. Note that we have introduced the pairvelocity β . The analysis is performed in the standard out-side-longitudinal reference frame where the outward di-rection is identified with the direction of the transversecomponent of K and the longitudinal axis is parallel tothe beam. Correlation function is then measured for K from some interval and its inverse widths in q carry in-formation about space-time structure of the source. Atgiven K -range one does not measure the size of the wholefireball but rather its homogeneity lengths. Those are thesizes of homogeneity regions. The homogeneity region isa part of the whole fireball which produces hadrons withmomentum K from a given range. Due to expansion it isusually smaller than the whole fireball. We shall particu-larly look at how the homogeneity lengths vary with theazimuthal angle φ of the K vector.We will assume in what follows that the dependence ofthe correlation function on momentum difference q can bereasonably well parametrised by a Gaussian prescription C ( q, K ) − (cid:0) − R o q o − R s q s − R l q l − R os q o q s − R ol q o q l − R sl q s q l (cid:1) , (23)where R o , R s , R l , R os , R ol , and R sl are the correlationradii which can depend on K . They will be directly cal-culated from the emission function, see below.It is important to realise that the coordinate framein which correlations are measured is specified by thehadrons used in the measurement. It is different from anycoordinate system which is attached to the fireball. Therotation between the two frames defines the explicit an-gular dependence of the correlation radii. In addition tothis, due to collective expansion of the fireball, hadronsflying in different directions come from different parts ofthe fireball and carry information about their homogeneitylengths. This introduces the implicit angular dependenceof the correlation radii [36].Generally, in the out-side-longitudinal system it can bederived that the correlation radii are given by the space- time variances as R o ( K ) = (cid:10) (˜ x o − β o ˜ t ) (cid:11) (24a) R s ( K ) = (cid:10) x s (cid:11) (24b) R l ( K ) = (cid:10) (˜ x l − β l ˜ t ) (cid:11) (24c) R os ( K ) = (cid:10) (˜ x o − β o ˜ t )˜ x s (cid:11) (24d) R ol ( K ) = (cid:10) (˜ x o − β o ˜ t )(˜ x l − β l ˜ t ) (cid:11) (24e) R sl ( K ) = (cid:10) ˜ x s (˜ x l − β l ˜ t ) (cid:11) . (24f)Note that the space-time variances depend on average mo-mentum K . Here we have introduced the averaging overthe source (cid:104) f ( x ) (cid:105) = (cid:82) S ( x, p ) f ( x ) d x (cid:82) S ( x, p ) d x (25)and we also introduced the shifted coordinates as˜ x µ = x µ − (cid:104) x µ (cid:105) . (26)Recall that the coordinates x o , x s , x l are connectedwith the direction of the emitted particles. The explicitangular dependence is obtained simply by expressing theout-side-longitudinal coordinates in terms of the coordi-nates x , y , which are fixed with the fireball:˜ x o = ˜ x cos φ + ˜ y sin φ (27a)˜ x s = − ˜ x sin φ + ˜ y cos φ . (27b)where φ is the azimuthal angle of the emitted hadron pairs.This leads to R s = 12 (cid:0) (cid:104) ˜ x (cid:105) + (cid:104) ˜ y (cid:105) (cid:1) + 12 (cid:0) (cid:104) ˜ x (cid:105) − (cid:104) ˜ y (cid:105) (cid:1) cos 2 φ −(cid:104) ˜ x ˜ y (cid:105) sin 2 φ (28a) R o = 12 (cid:0)(cid:10) ˜ x (cid:11) + (cid:10) ˜ y (cid:11)(cid:1) − (cid:0)(cid:10) ˜ y (cid:11) − (cid:10) ˜ x (cid:11)(cid:1) cos 2 φ + (cid:104) ˜ x ˜ y (cid:105) sin 2 φ + β o (cid:10) ˜ t (cid:11) − β o (cid:10) ˜ x ˜ t (cid:11) cos φ − β o (cid:10) ˜ y ˜ t (cid:11) sin φ (28b) R l = (cid:68)(cid:0) ˜ z − β l ˜ t (cid:1) (cid:69) (28c) R os = (cid:104) ˜ x ˜ y (cid:105) cos 2 φ + 12 (cid:0)(cid:10) ˜ y (cid:11) − (cid:10) ˜ x (cid:11)(cid:1) sin 2 φ + β o (cid:10) ˜ x ˜ t (cid:11) sin φ − β o (cid:10) ˜ y ˜ t (cid:11) cos φ (28d) R ol = (cid:10)(cid:0) ˜ z − β l ˜ t (cid:1) ˜ x (cid:11) cos φ + (cid:10)(cid:0) ˜ z − β l ˜ t (cid:1) ˜ y (cid:11) sin φ − β l (cid:10)(cid:0) ˜ z − β l ˜ t (cid:1) ˜ t (cid:11) (28e) R sl = (cid:10)(cid:0) ˜ z − β l ˜ t (cid:1) ˜ y (cid:11) cos φ − (cid:10)(cid:0) ˜ z − β l ˜ t (cid:1) ˜ x (cid:11) sin φ . (28f)In what follows we want to study the azimuthal depen-dence of the correlation radii. To this end, they are cus-tomarily expanded into Fourier series. The space-time co-variances also depend non-trivially on φ , although we havesuppressed writing this out explicitly. They can be writ-ten out and inserted in the right-hand sides of Eqs. (28).Then, by combining all sine and cosine terms on the r.h.s.of the obtained equations one can analytically identify allterms of Fourier expansion of the correlation radii. This J. Cimerman, B. Tom´aˇsik, M. Csan´ad, and S. L¨ok¨os: Anisotropies in the Blast-Wave Model has been done earlier for the second order [36,37]. Theresults of the third and higher orders are presented in Ap-pendix C.In practical calculation, however, one can proceed dif-ferently. The whole azimuthal dependence can be calcu-lated from Eqs. (28a)-(28f). Then one can extract anyFourier coefficient from the result. With increasing com-plexity of higher order terms this procedure appears com-putationally more efficient.Moreover, these are not yet the correlation radii whichcorrespond to the measured ones, even if the assumptionof Gaussian correlation function is valid. In order to mea-sure the correlation radii in real collisions, hadron pairsmust be collected over a large number of events. Theevents must be rotated so that the event planes are allaligned. Otherwise the azimuthal dependence would beaveraged out. For measuring the second-order oscillationsone rotates the events so as to align the second-order eventplanes. For the third-order oscillation one aligns the third-order event planes. Since the event planes of different or-ders may be assumed to be uncorrelated these alignmentseffectively introduce averaging over the direction of theother event planes. This must be included in calculations.It has been investigated in [22] that this averaging mayintroduce a few percent effect on the resulting correla-tion radii. Hence, when calculating the correlation radiifrom Eqs. (28), one additional integral over θ or θ mustbe calculated, depending on which order of oscillations weshall be interested in. We include this averaging over otherevent planes in our calculation.Therefore, the second order and the third order Fourieramplitudes that we are going to calculate do not belongto the same Fourier series. In the former case the emissionfunction is averaged over all possible values of θ , in thelatter averaging runs over θ . This will be also reflected innotation: the correlation radii will be expanded into series R i ( φ ) = R i,j, + ∞ (cid:88) n =1 R i,j,n cos( n ( φ − φ n )) (29)where i = o, s , and j = 2 ,
3, depending on which eventplane θ j has been put to 0. In general, terms of the sameorder may differ, if they come from averaging with differ-ent event-planes fixed. For example, we note that R o, , (cid:54) = R o, , (30a) R s, , (cid:54) = R s, , . (30b)Second order oscillations have been calculated in [19],but the averaging over third order event plane was notperformed there. In order to fill this gap, we have donethe calculation here and show the results in Fig. 3 Themost important parameter which sets the scale of thetransverse correlation radii is R . We can see that eventhe average radii R i, , depend on both anisotropy pa-rameters a and ρ . For the higher-order Fourier terms,we would like to factorise out their trivial scaling with R , thus for the analysis we divide all amplitudes by thezeroth-order term. As it was observed previously [19], the − − R a R − − R /R a − − R /R − − − − − − − R /R a ρ − − − − − − R /R ρ − − Fig. 3.
The dependence of transverse correlation radii andtheir oscillation amplitudes on second-order anisotropies inspace and transverse flow. The values are calculated for out-ward pair momentum K o = 300 MeV. Model parameters usedin the calculation are T = 120 MeV, ρ = 0 . R = 7 fm, τ = 10 fm/ c . Third-order anisotropy parameters were set to a = ρ = 0 . second order oscillation amplitude is mainly set by thespatial anisotropy parameter a . The dependence on ρ isweak. This confirms the early conjecture that the second-order spatial deformation can be measured with the helpof correlation radii [19]. Both spatial and flow anisotropycan then be obtained from combined measurement of v and the azimuthal dependence of correlation radii. Fig-ure 3 also shows that in this model higher-order terms inthe decomposition of the correlation radii are very small.The fourth-order terms are smaller than the second-orderterms by two orders of magnitude. Although it shows aninteresting dependence on a and ρ , it is most likely be-low any reasonable experimental sensitivity.We have also looked at third-order oscillation in case ofaveraging over all possible directions of the second-orderevent plane. The resulting dependence of the correlationradii on a and ρ is plotted in Fig. 4.Again, even the azimuthally averaged radii show somedependence on both anisotropy parameters. The third- . Cimerman, B. Tom´aˇsik, M. Csan´ad, and S. L¨ok¨os: Anisotropies in the Blast-Wave Model 7 − − R a R − − R /R a − − − − − − − R /R − − − − − − − − − − − R /R a ρ − − − − − R /R ρ − Fig. 4.
The dependence of transverse correlation radii andtheir oscillation amplitudes on third-order anisotropies in spaceand transverse flow. The values are calculated for outward pairmomentum K o = 300 MeV. Model parameters used in thecalculation are T = 120 MeV, ρ = 0 . R = 7 fm, τ = 10fm/ c . Second-order anisotropy parameters were set to a = ρ = 0 . order spatial anisotropy is best reflected in the third-orderscaled amplitude of the outward radii R o, , /R o, , . Forthe sideward radius the third-order oscillation depends onboth a and ρ . Note, however, that the third-order os-cillation is typically smaller by an order of magnitude ifit is compared to the second-order oscillation in Fig. 3.Even more suppressed is the next higher order, which isthe sixth in this case. In absolute numbers the scaled am-plitudes are on the level of a few per mille or even less. Wedo not expect that such a weak signal could be reasonablymeasured in experiments. The present model has been designed with the aim to bet-ter characterise measured data on spectra and anisotropies.However, such an analysis requires to take into accountmany more issues and is technically much more involved. -0.030.00910.03-0.03 0.0069 0.03 a ρ Fig. 5.
Combined contour lines of constant v (red from topleft to bottom right) and constant R s, , /R s, , (green frombottom left to top right). Thick lines show the values of datafor p t = 863 MeV ( v ) [40] and K t = 877 MeV (correlationradii) [26]. The increment between neighbouring lines is 0.01. In order to reach physically relevant results, resonancesmust be included in the analysis [38]. This highly increasesthe complexity of calculations. Then, all data, i.e. identi-fied spectra, anisotropy coefficients, and correlations, shouldbe fitted simultaneously. The new Bayesian technique [18]seems well suitable to this aim. Such a thorough analysis,however, goes far beyond our scope here. Nevertheless,we want to illustrate the qualitative features presented inprevious sections with the help of comparison to data. Wehasten to stress that this comparison should be under-stood merely on qualitative level.The STAR Collaboration has measured data from Au+Aucollisions at √ s NN = 200 GeV. They have analysed second-order oscillations of correlation radii as functions of az-imuthal angle with the blast-wave model extended to thatorder in [39]. In that analysis, the temperature T andtransverse flow gradient ρ were inferred from a simultane-ous fit to pion, kaon, and proton p t spectra and v . Anal-ysis of the azimuthally sensitive correlation radii yieldedthe sizes of the fireball and the second-order anisotropyparameters. From this analysis of 10-20% centrality STARdata, the following model parameter values have been ex-tracted by the STAR Collaboration [39]: T = 98 MeV, ρ = 0 . ρ = 0 . τ = 7 . c , ∆τ = 2 .
59 fm/ c . Thesecond-order spatial anisotropy can be translated into ourmodel as R = 11 . a = 0 . v and third-order scaledamplitude R s, , /R s, , on a and ρ . It is shown in Fig. 5.The curves decreasing to the right are lines with constant v , the others correspond to constant R s, , /R s, , . Thicklines represent the data values by PHENIX [40,26]. Outof available data we had to choose bins in p t for v and J. Cimerman, B. Tom´aˇsik, M. Csan´ad, and S. L¨ok¨os: Anisotropies in the Blast-Wave Model v p t [GeV]PHENIXBlast − wave Fig. 6.
The p t dependence of v for identified pions measuredby PHENIX collaboration [40] compared with the theoreticalcurve from the blast-wave model with parameters determinedin Fig. 5. correlation radii, which overlap. Unfortunately, these twosets of data are measured in slightly different momentumranges. In order to use overlapping p t bins, we have taken v point for pions at p t = 863 MeV [40] and correlationradii for K t = 877 MeV [26]. The values extracted fromsuch a simple comparison with data are: a = 0 . ρ = 0 . v (Fig. 6) and the az-imuthal angle dependence of the correlation radii (Fig. 7).For the latter we chose to plot the radii measured for K t = 530 MeV/ c as was also done by the PHENIX Col-laboration in [26]. The model fails completely in repro-ducing the absolute size of the correlation radii. The cor-responding parameter, however, has been fixed from theSTAR analysis of second-order oscillations of correlationradii. The mean values of the radii should be very similarfor second and the third order oscillations. This clearlydemonstrates the need of simultaneous fit to all availabledata in single analysis if one wants to go beyond the qual-itative level.We also show in Fig. 8 the third-order scaled ampli-tudes of correlation radii as functions of K t . The experi-mental error bars are huge and we explained that we havebased our analysis on the point at highest K t . It seemsthat the PHENIX data require amplitudes of oppositesigns, which the blast-wave model can accommodate.We want to close this section with a few commentson the applicability of the model to preliminary ALICEdata from Pb+Pb collisions at √ s NN = 2 .
76 TeV [41].The data seem to indicate that the third-order oscillationamplitudes of both R s and R o are negative. By inspectingFig. 4 we find that such situation only happens in a smallregion in parameter space with a around 0 and positive ρ . Thus from combined measurements of outward andsideward radii one could deduce that the fireball at the R o2 [f m ] PHENIXBlast − wave 0 5 10 15 20 0 π /2 π π /2 2 π R s [f m ] φ PHENIXBlast − wave Fig. 7.
The azimuthal dependence of correlation radii with re-spect to the third-order event plane for K t = 530 MeV/ c . Databy PHENIX collaboration [26] are compared with results fromthe blast-wave model with parameters determined in Fig. 5. LHC has rather symmetric shape and the anisotropy isset by the transverse collective velocity field.
We have generalised the blast-wave model so that it in-cludes third-order anisotropies in both space and expan-sion. Analogically to the second order, from the combina-tion of Fig. 2 and Fig. 4 we can infer, that it is indeedpossible to reconstruct both anisotropy coefficients of thismodel: a and ρ , from measurements of the azimuthalanisotropy of single-particle momentum distributions andHBT radii.We have also pointed out the need for averaging overthe difference of second and third-order reaction planeswhen focusing on a selected order of Fourier decomposi-tion of the hadron distribution or the correlation radii.This is effectively done in data analysis when all eventsare aligned according to the event plane of the selectedorder.The contour plots shown in Fig. 4 exemplify the state-ment made in [21] that at fixed flow anisotropy the am-plitude of correlation radii oscillation can be tuned withthe help of spatial anisotropy and even a flip in the phasecan be obtained. Such a flip of the phase corresponds to achange of the sign of the amplitude. Keeping constant flow . Cimerman, B. Tom´aˇsik, M. Csan´ad, and S. L¨ok¨os: Anisotropies in the Blast-Wave Model 9 − − − − − R o , / R o , PHENIXBlast − wave − R s , / R s , K t [GeV]PHENIXBlast − wave Fig. 8.
The K t dependence of third-order oscillation ampli-tude of the correlation radii. Data by PHENIX collaboration[26] are compared with results from the blast-wave model withparameters determined in Fig. 5. anisotropy and changing space anisotropy corresponds tomoving vertically in the panels of Fig. 4 and the phaseflip corresponds to crossing the thick line in that Figure.Our results are much more detailed since we show the fulldependence on the two parameters.We have also calculated the subleading terms for thethird-order anisotropies in the oscillations of correlationradii. These are the sixth-order oscillations. They wereshown to be an order of magnitude smaller than the lowerorder and hardly measurable at current experimental statis-tics. We have actually derived expressions for oscillationamplitudes at general order, but there is currently no needto go to higher orders also with model studies as sufficientstatistics would hardly be available.It is interesting to compare our results to those ob-tained in an analogical study which used the Buda-Lundmodel [22]. In that model, the way in which observablesdepend on the combination of the space and flow anisotropyis different to the one presented here. Partially, this isdue to our definition of the spatial profile with the mi-nus sign in Eq. (4). The corresponding sign in the Buda-Lund model was kept to be plus. On the qualitative level,the oscillations of the correlation radii within the Buda-Lund model seem to be much more sensitive to the flowanisotropy than it is the case in this study. -10-5051015 (a)a = 0.2 φ = 0 y [f m ] (b) ρ = 0.2 φ = 0 -15-10-50510-15 -10 -5 0 5 10 (c)a = 0.2 φ = π /2 y [f m ] x [fm] -10 -5 0 5 10 15 (d) ρ = 0.2 φ = π /2 x [fm] Fig. 9.
Transverse profiles of the emission function for theblast-wave model with source parameters as in Fig. 4, for par-ticles with p t = 300 MeV in the indicated direction φ . Leftcolumn: only density profile anisotropy; right column: onlyflow anisotropy. Other source parameters are T = 120 MeV, τ = 10 fm, R = 7 fm, ∆τ = 1 fm, ρ = 0 .
8. Shown are theeffective sources which are calculated by integrating over alldirections of the third-order anisotropy with a = ρ = 0 . In order to explore the difference of the two modelsin more details, in Figs. 9 and 10 we plot transverse pro-files of the emitting sources according to the blast-waveand Buda-Lund models, respectively. Plotted are not theemission functions directly. We have assumed that thereare the third-order anisotropies, as well, which have theirthird-order event planes completely independent from thesecond-order event plane directions. Then, we have inte-grated over all possible directions of the third-order eventplane and obtained an effective emission function withonly second-order anisotropy. In the figures, we have as-sumed pions with p t = 300 MeV and two different az-imuthal angles of particle emission: φ = 0 (upper rows)and φ = π/ φ = 0and φ = π/ -10-5051015 (a) ε = 0.2 φ = 0 y [f m ] (b) χ = 0.2 φ = 0 -15-10-50510-15 -10 -5 0 5 10 (c) ε = 0.2 φ = π /2 y [f m ] x [fm] -10 -5 0 5 10 15 (d) χ = 0.2 φ = π /2 x [fm] Fig. 10.
Analogical to Fig. 9, but calculated with the Buda-Lund model, with source parameters T = 120 MeV, τ =10 fm, a = 0 . b = − . R = 7, Z = 15, H = 8, H z = 16, (cid:15) = χ = 0 .
1. The model is explained in [22]. The values of thesecond-order anisotropy parameters are indicated in the figure.The averaging over all directions of the third-order anisotropyis performed with ε = χ = 0 . direction of φ . Thus we see the much stronger dependenceon shape anisotropy than on flow anisotropy.Having the two models which appear so differently,the choice of the more suitable model should be decidedby data. Note, however, that neither this schematic study,nor the one of ref. [22] included meson production fromthe decays of resonances and the modification of the corre-lation function due to viscosity. Their influence should beinvestigated in the future in order to arrive at a conclusiveanswer. This work has been performed in framework of COST ActionCA15213 “Theory of hot matter and relativistic heavy-ion col-lisions” (THOR). Partial support by VEGA 1/0469/15 (Slo-vakia) and by MˇSMT grant No. LG15001 (Czech Republic)is acknowledged. M. Cs. was supported by the J´anos BolyaiResearch Scholarship of the Hungarian Academy of Sciences.
A Relation between different parametrisationsof second-order spatial anisotropy
In this Appendix we derive the relation between our pa-rametrisation of the transverse shape and the one used in[19].First of all, it should be clearly stated that the twoparametrisations are different. Hence, the elliptic shapethat has been used in [19] would be fully reproduced with the help of parametrisation (4) only if higher-order termsare included. Of course, the importance of higher ordersdrops with n .We shall assume here that the second-order anisotropyis small, i.e., the parameter a from Eq. (5) is close to 1.The ellipse of [19] includes points with coordinates r x = R x cos θ = aR (cid:48) cos θ (31a) r y = R y sin θ = R (cid:48) a sin θ . (31b)Thus the radius as function of the azimuthal angle is R = (cid:113) r x + r y = (cid:115) a R (cid:48) cos θ + R (cid:48) a sin θ . (32)This can be rewritten as R = R (cid:48) (cid:114) a + 12 a (cid:18) a − a + 1 cos(2 θ ) (cid:19) . (33)Now we assume that a → a − / ( a + 1)is very small. Thus we can Taylor-expand the bracket upto first order and obtain R ( θ ) ≈ R (cid:48) (cid:114) a + 12 a (cid:18) a − a + 1 cos(2 θ ) (cid:19) . (34)The mean radius R of the present model is to be iden-tified with R = R (cid:48) (cid:114) a + 12 a . (35)The amplitude of the oscillations is to be identifiedwith ( − a ) a = − a − a + 1 . (36)Inverting this relation gives a = (cid:18) − a a (cid:19) . (37) B The direction of transverse velocity
Here we derive the direction of the transverse velocity,which is given by the angle θ b . Since it is supposed to beperpendicular to the surface of constant ¯ r , we write downits coordinates x = ¯ rR (cid:32) − ∞ (cid:88) n =2 a n cos( n ( θ − θ n )) (cid:33) cos θ (38a) y = ¯ rR (cid:32) − ∞ (cid:88) n =2 a n cos( n ( θ − θ n )) (cid:33) sin θ . (38b)In further calculations, however, we shall truncate the ex-pansion after the third-order term. . Cimerman, B. Tom´aˇsik, M. Csan´ad, and S. L¨ok¨os: Anisotropies in the Blast-Wave Model 11 The (truncated) expressions (38) can be inserted intothe derivatives in Eq. (10). This gives θ b = π AB . (39)where A = 4 a sin θ + 3 a sin 3( θ − θ )cos θ − − a cos 2( θ − θ ) + a cos 3( θ − θ )sin θ (40) B = a cos θ + 3 a sin 3( θ − θ )sin θ − − a cos 2( θ − θ ) + a cos 3( θ − θ )cos θ (41) C Fourier amplitudes of the correlation radii
Here we give an overview of the Fourier amplitudes ofthe azimuthal dependence of outward, sideward, and out-side cross-term correlation radii. Note that these formu-las are model-independent. Dependence on a particularmodel comes into evaluation of individual space-time (co-)variances. Note that analogical relations have been de-rived in [21], where Milne coordinates were used insteadof the Cartesian ones.We define the amplitudes R o = (cid:0) R o (cid:1) + ∞ (cid:88) n =1 (cid:104)(cid:0) R o (cid:1) sn sin( nφ ) + (cid:0) R o (cid:1) cn cos( nφ ) (cid:105) (42a) R s = (cid:0) R s (cid:1) + ∞ (cid:88) n =1 (cid:104)(cid:0) R s (cid:1) sn sin( nφ ) + (cid:0) R s (cid:1) cn cos( nφ ) (cid:105) (42b) R os = (cid:0) R os (cid:1) + ∞ (cid:88) n =1 (cid:104)(cid:0) R os (cid:1) sn sin( nφ ) + (cid:0) R os (cid:1) cn cos( nφ ) (cid:105) (42c)In a similar way we shall expand the space-time (co-)va-riances (cid:104) ˜ x µ ˜ x ν (cid:105) = (cid:104) ˜ x µ ˜ x ν (cid:105) + ∞ (cid:88) n =1 [ (cid:104) ˜ x µ ˜ x ν (cid:105) sn sin( nφ ) + (cid:104) ˜ x µ ˜ x ν (cid:105) cn cos( nφ )] (43)The series (43) are inserted into the model-independentexpressions (28) and the sine and cosine terms are reor-ganised with the help of addition theorems. Then one cancollect the terms order by order and put them in equalitywith the corresponding expansion of the correlation radii. We obtain for the outward radius( R o ) = 12 (cid:10) ˜ x (cid:11) + 12 (cid:10) ˜ y (cid:11) − (cid:10) ˜ y (cid:11) c + 14 (cid:10) ˜ x (cid:11) c + 12 (cid:104) ˜ x ˜ y (cid:105) s + β o (cid:10) ˜ t (cid:11) − β o (cid:10) ˜ t ˜ x (cid:11) c − β o (cid:10) ˜ t ˜ y (cid:11) s (44a)( R o ) c = 34 (cid:10) ˜ x (cid:11) c + 14 (cid:10) ˜ y (cid:11) c − (cid:10) ˜ y (cid:11) c + 14 (cid:10) ˜ x (cid:11) c + 12 ( (cid:104) ˜ x ˜ y (cid:105) s + (cid:104) ˜ x ˜ y (cid:105) s ) + β o (cid:10) ˜ t (cid:11) c − β o (cid:10) ˜ t ˜ x (cid:11) − β o (cid:10) ˜ t ˜ x (cid:11) c − β o (cid:10) ˜ t ˜ y (cid:11) s (44b)( R o ) s = 14 (cid:10) ˜ x (cid:11) s + 34 (cid:10) ˜ y (cid:11) s + 14 (cid:10) ˜ y (cid:11) s − (cid:10) ˜ x (cid:11) s + 12 ( (cid:104) ˜ x ˜ y (cid:105) c − (cid:104) ˜ x ˜ y (cid:105) c ) + β o (cid:10) ˜ t (cid:11) s − β o (cid:10) ˜ t ˜ x (cid:11) s − β o (cid:10) ˜ t ˜ y (cid:11) + β o (cid:10) ˜ t ˜ y (cid:11) c (44c)( R o ) c = 12 (cid:10) ˜ x (cid:11) c + 12 (cid:10) ˜ y (cid:11) c − (cid:10) ˜ y (cid:11) − (cid:10) ˜ y (cid:11) c + 12 (cid:10) ˜ x (cid:11) + 14 (cid:10) ˜ x (cid:11) c + 12 (cid:104) ˜ x ˜ y (cid:105) s + β o (cid:10) ˜ t (cid:11) c − β o (cid:10) ˜ t ˜ x (cid:11) c − β o (cid:10) ˜ t ˜ x (cid:11) c − β o (cid:10) ˜ t ˜ y (cid:11) s + β o (cid:10) ˜ t ˜ y (cid:11) s (44d)( R o ) s = 12 (cid:10) ˜ x (cid:11) s + 12 (cid:10) ˜ y (cid:11) s + 14 (cid:10) ˜ y (cid:11) s − (cid:10) ˜ x (cid:11) s + (cid:104) ˜ x ˜ y (cid:105) − (cid:104) ˜ x ˜ y (cid:105) c + β o (cid:10) ˜ t (cid:11) s − β o (cid:10) ˜ t ˜ x (cid:11) s − β o (cid:10) ˜ t ˜ x (cid:11) s − β o (cid:10) ˜ t ˜ y (cid:11) c + β o (cid:10) ˜ t ˜ y (cid:11) c (44e)( R o ) cn = 12 (cid:10) ˜ x (cid:11) cn + 12 (cid:10) ˜ y (cid:11) cn − (cid:10) ˜ y (cid:11) cn − − (cid:10) ˜ y (cid:11) cn +2 + 14 (cid:10) ˜ x (cid:11) cn − + 14 (cid:10) ˜ x (cid:11) cn +2 + 12 (cid:104) ˜ x ˜ y (cid:105) sn +2 − (cid:104) ˜ x ˜ y (cid:105) sn − + β o (cid:10) ˜ t (cid:11) cn − β o (cid:10) ˜ t ˜ x (cid:11) cn − − β o (cid:10) ˜ t ˜ x (cid:11) cn +1 − β o (cid:10) ˜ t ˜ y (cid:11) sn +1 + β o (cid:10) ˜ t ˜ y (cid:11) sn − (44f)( R o ) sn = 12 (cid:10) ˜ x (cid:11) sn + 12 (cid:10) ˜ y (cid:11) sn − (cid:10) ˜ y (cid:11) sn − + 14 (cid:10) ˜ y (cid:11) sn +2 + 14 (cid:10) ˜ x (cid:11) sn − − (cid:10) ˜ x (cid:11) sn +2 + 12 (cid:104) ˜ x ˜ y (cid:105) cn − − (cid:104) ˜ x ˜ y (cid:105) cn +2 + β o (cid:10) ˜ t (cid:11) sn − β o (cid:10) ˜ t ˜ x (cid:11) sn − − β o (cid:10) ˜ t ˜ x (cid:11) sn +1 − β o (cid:10) ˜ t ˜ y (cid:11) cn − + β o (cid:10) ˜ t ˜ y (cid:11) cn +1 . (44g)Note that from the third order onwards we have givengeneral expressions for any order.Analogically we derived the series for the sideward ra-dius.( R s ) = 12 (cid:10) ˜ x (cid:11) + 12 (cid:10) ˜ y (cid:11) + 14 (cid:10) ˜ y (cid:11) c − (cid:10) ˜ x (cid:11) c − (cid:104) ˜ x ˜ y (cid:105) s (45a)( R s ) c = 14 (cid:10) ˜ x (cid:11) c + 34 (cid:10) ˜ y (cid:11) c + 14 (cid:10) ˜ y (cid:11) c − (cid:10) ˜ x (cid:11) c −
12 ( (cid:104) ˜ x ˜ y (cid:105) s − (cid:104) ˜ x ˜ y (cid:105) s ) (45b)( R s ) s = 34 (cid:10) ˜ x (cid:11) s + 14 (cid:10) ˜ y (cid:11) s + 14 (cid:10) ˜ y (cid:11) s − (cid:10) ˜ x (cid:11) s −
12 ( (cid:104) ˜ x ˜ y (cid:105) c − (cid:104) ˜ x ˜ y (cid:105) c ) (45c)( R s ) c = 12 (cid:10) ˜ x (cid:11) c + 12 (cid:10) ˜ y (cid:11) c + 12 (cid:10) ˜ y (cid:11) + 14 (cid:10) ˜ y (cid:11) c − (cid:10) ˜ x (cid:11) − (cid:10) ˜ x (cid:11) c − (cid:104) ˜ x ˜ y (cid:105) s (45d)( R s ) s = 12 (cid:10) ˜ x (cid:11) s + 12 (cid:10) ˜ y (cid:11) s + 14 (cid:10) ˜ y (cid:11) s − (cid:10) ˜ x (cid:11) s − (cid:104) ˜ x ˜ y (cid:105) + 12 (cid:104) ˜ x ˜ y (cid:105) c (45e)( R s ) cn = 12 (cid:10) ˜ x (cid:11) cn + 12 (cid:10) ˜ y (cid:11) cn + 14 (cid:10) ˜ y (cid:11) cn − + 14 (cid:10) ˜ y (cid:11) cn +2 − (cid:10) ˜ x (cid:11) cn − − (cid:10) ˜ x (cid:11) cn +2 − (cid:104) ˜ x ˜ y (cid:105) sn +2 + 12 (cid:104) ˜ x ˜ y (cid:105) sn − (45f)( R s ) sn = 12 (cid:10) ˜ x (cid:11) sn + 12 (cid:10) ˜ y (cid:11) sn + 14 (cid:10) ˜ y (cid:11) sn − + 14 (cid:10) ˜ y (cid:11) sn +2 − (cid:10) ˜ x (cid:11) sn − − (cid:10) ˜ x (cid:11) sn +2 − (cid:104) ˜ x ˜ y (cid:105) cn − + 12 (cid:104) ˜ x ˜ y (cid:105) cn +2 . (45g)Finally, for the cross-term we obtained( R os ) = 12 (cid:104) ˜ x ˜ y (cid:105) c + 14 (cid:10) ˜ y (cid:11) s − (cid:10) ˜ x (cid:11) s + β o (cid:10) ˜ x ˜ t (cid:11) s − β o (cid:10) ˜ y ˜ t (cid:11) c (46a)( R os ) c = 12 (cid:104) ˜ x ˜ y (cid:105) c + 12 (cid:104) ˜ x ˜ y (cid:105) c + 14 (cid:10) ˜ y (cid:11) s + 14 (cid:10) ˜ y (cid:11) s − (cid:10) ˜ x (cid:11) s − (cid:10) ˜ x (cid:11) s + β o (cid:10) ˜ x ˜ t (cid:11) s − β o (cid:10) ˜ y ˜ t (cid:11) − β o (cid:10) ˜ y ˜ t (cid:11) c (46b)( R os ) s = − (cid:104) ˜ x ˜ y (cid:105) s − (cid:104) ˜ x ˜ y (cid:105) s + 14 (cid:10) ˜ y (cid:11) c − (cid:10) ˜ y (cid:11) c − (cid:10) ˜ x (cid:11) c + 14 (cid:10) ˜ x (cid:11) c + β o (cid:10) ˜ x ˜ t (cid:11) − β o (cid:10) ˜ x ˜ t (cid:11) c − β o (cid:10) ˜ y ˜ t (cid:11) s (46c)( R os ) c = (cid:104) ˜ x ˜ y (cid:105) + 12 (cid:104) ˜ x ˜ y (cid:105) c + 14 (cid:10) ˜ y (cid:11) s − (cid:10) ˜ x (cid:11) s − β o (cid:10) ˜ x ˜ t (cid:11) s + β o (cid:10) ˜ x ˜ t (cid:11) s − β o (cid:10) ˜ y ˜ t (cid:11) c − β o (cid:10) ˜ y ˜ t (cid:11) c (46d)( R os ) s = − (cid:104) ˜ x ˜ y (cid:105) s + 12 (cid:10) ˜ y (cid:11) − (cid:10) ˜ y (cid:11) c − (cid:10) ˜ x (cid:11) + 14 (cid:10) ˜ x (cid:11) c + β o (cid:10) ˜ x ˜ t (cid:11) c − β o (cid:10) ˜ x ˜ t (cid:11) c − β o (cid:10) ˜ y ˜ t (cid:11) s − β o (cid:10) ˜ y ˜ t (cid:11) s (46e) ( R os ) cn = 12 (cid:104) ˜ x ˜ y (cid:105) cn − + 12 (cid:104) ˜ x ˜ y (cid:105) cn +2 + 14 (cid:10) ˜ y (cid:11) sn +2 − (cid:10) ˜ y (cid:11) sn − − (cid:10) ˜ x (cid:11) sn +2 + 14 (cid:10) ˜ x (cid:11) sn − + β o (cid:10) ˜ x ˜ t (cid:11) sn +1 − β o (cid:10) ˜ x ˜ t (cid:11) sn − − β o (cid:10) ˜ y ˜ t (cid:11) cn +1 − β o (cid:10) ˜ y ˜ t (cid:11) cn − (46f)( R os ) sn = − (cid:104) ˜ x ˜ y (cid:105) sn +2 + 12 (cid:104) ˜ x ˜ y (cid:105) sn − − (cid:10) ˜ y (cid:11) cn +2 + 14 (cid:10) ˜ y (cid:11) cn − + 14 (cid:10) ˜ x (cid:11) cn +2 − (cid:10) ˜ x (cid:11) cn − − β o (cid:10) ˜ x ˜ t (cid:11) cn +1 + β o (cid:10) ˜ x ˜ t (cid:11) cn − − β o (cid:10) ˜ y ˜ t (cid:11) sn +1 − β o (cid:10) ˜ y ˜ t (cid:11) sn − . (46g) References
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