Viscosity, non-conformal equation of state and sound velocity in Landau hydrodynamics
VViscosity, non-conformal equation of state and sound velocity in Landauhydrodynamics
Deeptak Biswas, Kishan Deka, Amaresh Jaiswal, and Sutanu Roy Department of Physics, Center for Astroparticle Physics and Space Science, Bose Institute, Kolkata 700091, India School of Physical Sciences, National Institute of Science Education and Research, HBNI, Jatni-752050, Odisha, India School of Mathematical Sciences, National Institute of Science Education and Research, HBNI, Jatni-752050, Odisha, India (Dated: July 30, 2020)We find an analytical solution to relativistic viscous hydrodynamics for a 1+1 dimensional Landauflow profile. We consider relativistic Navier-Stokes form of the dissipative hydrodynamic equation,for a non-conformal system with a constant speed of sound, and employ the obtained solution tofit rapidity spectrum of observed pions in √ s NN = 200, 17.3, 12.3, 8.76, 7.62, 6.27, 4.29, 3.83,3.28 and 2.63 GeV collision energies. We find that at the freeze-out hypersurface with improvedLandau’s freeze-out prescription, the viscous corrections do not affect the rapidity spectra. Wedemonstrate that the solution of the non-conformal Landau flow lead to a better agreement withthe experimental data compared to the conformal ideal solution. We also extract speed of soundfrom fit to the rapidity spectra for various collision energies and find a monotonous decrease withdecreasing collision energies. Appealing to the fact that viscosity has negligible effect on rapidityspectra for Landau’s freeze-out scenario, we argue that our calculations provides a framework forextracting the average value of speed of sound in relativistic heavy-ion collisions. PACS numbers: 25.75.-q, 24.10.Nz, 47.75+fKeywords: Heavy ion collision, Relativistic hydrodynamics, Viscosity
Introduction : Heavy ion collisions at relativistic ener-gies offer the possibility to create strongly interacting hotand dense matter over extended region [1–3]. The space-time evolution of this hot and dense matter, created atRelativistic Heavy ion Collider (RHIC) and the LargeHadron Collider (LHC), has been successfully modeledusing relativistic dissipative hydrodynamic simulations[4–12]. However, the first hydrodynamic approach to de-scribe nucleus-nucleus collisions was proposed by Landauin 1953 where he studied longitudinal expansion along thecollision axis [13]. In his approach, Landau considered anon-dissipative expansion of the medium and obtained anapproximate analytical solution of ideal hydrodynamicequations (commonly refered to as the Landau hydrody-namics) which led to Gaussian like rapidity distributionsof produced particles [13]. Later a plateau like rapiditydistribution was proposed in Hwa-Bjorken hydrodynam-ics which considered a boost-invariant framework [14, 15].However, the boost-invariance symmetry was only ap-plicable in the mid-rapidity region of ultra-relativisticheavy ion collisions. On the other hand, the transverse-momentum ( p T ) integrated yield over the whole rapidityregion has an overall better agreement with the Gaussianstructure suggested by Landau [16–21].Several problems in high-energy heavy ion collisions,such as heavy quark propagation and the interaction ofjets or quarkonia with the hot and dense matter, requiresa realistic but simple description of the evolution of theproduced medium. Successes of Landau hydrodynam-ics in explaining the total charged multiplicities, rapiditydistribution and limiting fragmentation [16–21] indicatethat it can provide such a framework. Landau hydrody-namics is based on the assumption that the initial stages of relativistic heavy ion collisions experiences a fast lon-gitudinal expansion accompanied by a slower expansionin the transeverse plane. One can obtain analytical solu-tion for the one-dimensional longitudinal expansion [13]which has been analyzed in great detail in the literature[22–43]. After sufficiently large transverse expansion, i.e.,when the magnitude of the transverse displacement be-comes larger than the initial transverse dimension, thefluid element expands with a frozen rapidity. The finalrapidity distribution of particles is then given by that atthe freeze-out time which leads to good agreement withthe experimental data [16–21, 33–43].Despite the fact that the gross features of many mea-sured quantities are reproduced well, the Landau hydro-dynamics can, at best, be considered a good first approx-imation. One can expect corrections and improvementsbased on physical considerations and the requirement toexplain experimental data. The need to improve up onLandau’s original rapidity distribution should not comeas a surprise, as the original Landau distribution wasintended to be qualitative. On the other hand a morequantitative approach in order to explain experimentalresults and hence extract additional information aboutthe properties of the QCD matter formed in relativisticheavy ion collisions is desirable. One such important im-provement to Landau hydrodynanics would be to includedissipation in the evolution which will provide a platformto analytically estimate the transport coefficient of theQCD matter by analyzing the rapidity spectra. Whilethere has been attempts to improve the Landau modelby including viscous corrections analytically [44–46], theresultant solutions were not conclusive and/or did notshow the correct trend towards explaining the observed a r X i v : . [ h e p - ph ] J u l experimental data.In this article, we consider the relativistic Navier-Stokes equation for viscous evolution of the hot and densematter formed in high energy heavy-ion collisions. Wefind analytical solution of the viscous evolution equa-tion, for non-conformal system having constant speed ofsound, with 1+1 dimensional Landau flow profile. Wefind that for Landau’s prescription of freeze-out scenario,the viscous effects do not effect the rapidity spectrum ofthe produced particles. We employ the obtained solutionto fit rapidity spectrum of observed pions in √ s NN =200, 17.3, 12.3, 8.76, 7.62, 6.27, 4.29, 3.83, 3.28 and 2.63GeV collision energies. We show that the obtained so-lutions of Landau flow with non-conformal equation ofstate leads to a better agreement with the experimentaldata compared to the conformal Landau flow solution.We also find that the value of speed of sound, obtained byfitting the experimental results, show monotonic decreasewith decreasing collision energies. Appealing to the factthat viscosity has negligible effect on rapidity spectra forLandau’s freeze-out scenario, we advocate that our cal-culations provide a framework for extracting the aver-age value of sound velocity of QCD medium produced inheavy-ion collisions. Viscous Landau flow : Hydrodynamic equations followfrom the principle of conservation of energy and momen-tum of a continuous medium, which leads to vanishingof four-divergence of the energy-momentum tensor for arelativistic system. In absence of any conserved charges,the energy-momentum tensor of a relativistic fluid, withdissipative terms from Navier-Stokes theory, can be writ-ten as [47] T µν = (cid:15) u µ u ν − ( P − ζθ ) ∆ µν + 2 ησ µν , (1)where (cid:15) is the local energy density, P is the thermody-namic pressure, u µ is the fluid four-velocity and, η and ζ are the coefficients of shear and bulk viscosity, respec-tively. We also define a projection operator ∆ µν ≡ g µν − u µ u ν which is orthogonal to u µ , four-divergence of fluidvelocity θ ≡ ∂ µ u µ and a derivative operator ∇ µ ≡ ∆ µν ∂ ν which is also orthogonal to u µ . We follow mostly minusmetric convention, i.e., g µν = diag(1 , − , − , − σ µν ≡ ( ∇ µ u ν + ∇ ν u µ ) − ∆ µν ∇ α u α . In this work, weshall use the non-conformal equation of state, P = c s (cid:15) ,where the speed of sound c s will be assumed to be con-stant for simplicity.Following the seminal work of Landau [13], we considerthe longitudinal dynamics in the collision of two identicalnuclei moving along the z -direction. For longitudinal ex-pansion, the hydrodynamic equation, ∂ µ T µν = 0, leadsto [13, 21] ∂T ∂t + ∂T ∂z = 0 , ∂T ∂t + ∂T ∂z = 0 , (2) where, we have used the notation ( t, x, y, z ) ≡ ( x , x , x , x ) for co-ordinate labels. For one-dimensional expansion along z -axis, we introduce the lon-gitudinal fluid rapidity, y , in terms of which we can rep-resent the non-zero velocity fields u = cosh y and u =sinh y . In terms of the light-cone variables, t ± ≡ t ± z ,Eq. (2) can be written as ∂∂t + [ c + (cid:15) − ξ ∇ u ] e y + ∂∂t − [ c − (cid:15) + ξ ∇ u ] = 0 , (3) ∂∂t + [ c − (cid:15) + ξ ∇ u ] + ∂∂t − [ c + (cid:15) − ξ ∇ u ] e − y = 0 , (4)where c ± ≡ ± c s and ∇ u ≡ ∂u /∂t + ∂u /∂z = e y ∂y/∂t + − e − y ∂y/∂t − is defined to simplify notationsand ξ ≡ ζ + 4 η/
3. In this article, we solve the above setof differential equations for evolution of energy densityassuming the flow profile to be given by ideal Landauhydrodynamics.For a flow which is not boost-invariant, the fluid ra-pidity y is not the same as the space-time rapidity η s ≡ ln( t + /t − ). The notion of Landau flow is basedon the assumption that the difference between the fluidrapidity and space-time rapidity is small. Therefore onecan express the fluid rapidity in terms of the space-timerapidity as [13] e y = f e η s = f t + t − , (5)where f is a slowly varying (logarithmic) function of t + and t − , of order unity, such that the derivatives of f andquadratic and higher-order terms in log f could be ne-glected. Keeping in mind that √ f + 1 / √ f (cid:39)
2, we findthat ∇ u = 1 / √ t + t − for the expansion profile given inabove equation. We further introduce another change ofvariables, y ± ≡ ln ( t ± / ∆), where ∆ = 2 R/γ is the lon-gitudinally Lorentz contracted diameter of each collidingnuclei. In terms of these new variables, Landau obtained f = (cid:112) y + /y − for ideal hydrodynamic evolution, which isindeed a logarithmic function of t + and t − of order unity[13]. With this velocity profile, the solution for evolutionof energy density in case of ideal hydrodynamics, i.e., ξ = 0, is given by (cid:15) id = (cid:15) exp (cid:20) − c c s ( y + + y − ) + c + c − c s √ y + y − (cid:21) , (6)where (cid:15) is related to the initial energy density. It is im-portant to note that, in the conformal limit, we reproducethe original results of Landau [13, 21].In the following, we assume the flow profile to be givenby ideal Landau hydrodynamics and find a solution forthe evolution of energy density. Changing evolution vari-ables to y ± , we see that Eq. (3) + Eq. (4) leads to f ∂(cid:15)∂y + + ∂(cid:15)∂y − + 1 + f (cid:20) c + (cid:15) − ξ ∆ e − ( y + + y − ) / (cid:21) = 0 . (7)We can generate another linearly independent equationfrom Eq. (3) and (4) by considering the combinationEq. (3) − Eq. (4), f ∂(cid:15)∂y + − ∂(cid:15)∂y − + ( f − c + c s (cid:15) − c s ∆ (cid:18) f ∂ξ∂y + − ∂ξ∂y − (cid:19) × e − ( y + + y − ) / = 0 . (8)It is important to note that for collision of two identi-cal nuclei, the reflection symmetry about the transverseplane in the centre-of-mass frame must lead to evolu-tion equations invariant under y + ↔ y − interchange.Therefore the hydrodynamic equations describing theevolution of the matter formed in these collisions shouldbe even under parity transformation. Keeping in mindthat we assume Landau flow profile, i.e., Eq. (5) with f = (cid:112) y + /y − , it is easy to see that the left hand side ofEq. (7) has even parity whereas that of Eq. (8) has oddparity. Therefore, the solution of Eq. (7) should lead tothe evolution of energy density for viscous Landau flowin symmetric nucleus-nucleus collisions.To make progress, we assume the ratio ξ/s to be a con-stant where s is the entropy density. While this is a validassumption in conformal case, it is not be strictly true fora non-conformal system. Therefore, for the case of con-stant ξ/s , one can write ξ = α (cid:15) / (1+ c s ) = α (cid:15) /c + , where α is a constant. Substituting in Eq. (7) and rearranging,we get f ∂(cid:15)∂y + + ∂(cid:15)∂y − = 1 + f (cid:104) α ∆ (cid:15) c + e − ( y + + y − ) − c + (cid:15) (cid:105) . (9)Using method of characteristics, we get, dy + f = dy − d(cid:15) (1 + f ) (cid:104) α ∆ (cid:15) c + e − ( y + + y − ) − c + (cid:15) (cid:105) . (10)The above equations can be solved analytically to obtain (cid:15) = (cid:20) F (cid:18) y + f − y − (cid:19) e − (1+ f ) c s y − − c s αc + c − ∆ e − ( y + + y − ) (cid:21) c + c s , (11)where, F is an arbitrary function of the argument givenin parenthesis and can be determined by comparing theabove solution with the ideal solution given in Eq. (6),in the limit of vanishing viscosity, i.e., α = 0. We get F (cid:18) y + f − y − (cid:19) = (cid:15) c s /c + exp (cid:20) c − − c + f (cid:18) y + f − y − (cid:19)(cid:21) . (12)Substituting the above value of F in Eq. (11), we ob-tain the final solution for energy density evolution withLandau flow profile (cid:15) = (cid:20) g ( α ) (cid:15) c s /c + id − c s αc + c − ∆ e − ( y + + y − ) / (cid:21) c + c s , (13) where g ( α ) is an arbitrary function of α such that g (0) =1. It is easy to see that the above form of energy densityindeed satisfy Eq. (7).We note that while the above solution is formally simi-lar to that obtained in Ref. [45], there is still some leftoverfreedom in our solution given in Eq. (13) for the func-tional form of g ( α ). One way to fix this residual free-dom is by considering the longitudinal boost-invariantBjorken limit of Eq. (13) and comparing with the corre-sponding solution of viscous hydrodynamics in Bjorkencase [48, 49]. Doing this exercise, we obtain g ( α ) = 1 + α c s c + c − (cid:15) c + /c s ∆ , (14)where (cid:15) is the energy density corresponding to the ini-tial proper time τ = ∆. Equations (13) and (14) to-gether constitute the analytical solution of viscous Lan-dau hydrodynamics and represents the main results of thepresent work which will be used subsequently for calcu-lating the rapidity spectra of produced hadrons in heavy-ion collisions. Nevertheless, as demonstrated in the fol-lowing, we find that the effect of α (i.e., viscosity) onrapidity spectra of produced particles is negligible whenwe consider Landau’s prescription for freeze-out hyper-surface. Rapidity Distribution : Following Landau’s arguments[13], we assume that the one-dimensional solution is ap-plicable until the expansion of the fluid element in thetransverse direction is of the same order as the trans-verse dimension of the system. We find that if one con-siders the longitudinal and transeverse expansion inde-pendently, having a slower expansion with constant ac-celeration in the transverse direction [13, 21], then thetransverse expansion does not get any correction due toviscosity. Under this approximation the freeze-out timealso remains the same as ideal case [21]. While thereare many possible freeze-out conditions, the successes ofLandau hydrodynamics suggests that Landau’s freeze-out criteria can be a good first approximation and thecorrection is likely to be small and scale with the Landaufreeze-out proper time.Following Wong’s modified prescription [21] of Lan-dau’s freeze-out criteria [13], we consider the transverseexpansion using the non-conformal equation of state. Inthis case, one obtains a freeze out hypersurface where thefreeze out time is given by, t FO = 2 R (cid:115) c s c s cosh y. (15)With the help of above expression rapidity variables atthe freeze-out hypersurface takes the form y ± = y (cid:48) b ± y where y (cid:48) b ≡ ln [ c + / (4 c s )] + y b and y b ≡ ln( √ s NN /m p ) isthe beam rapidity with m p being mass of the proton [21].Keeping in mind that the beam rapidity is the largestrapidity achievable by a fluid element, we see that theterm proportional to α in Eq. (13) becomes negligible atfreeze-out and could be ignored in the first approxima-tion. Moreover, keeping in mind that, e − ( y + + y − ) / = ∆ τ FO = 1 γ (cid:115) c s c s , (16)where τ FO is the proper-time at freeze-out, one can alsosee that this term in Eq. (13) is negligible because of thelarge Lorentz factor, γ , of the colliding nuclei. From theabove equation, we also see that the freeze-out hypersur-face has constant proper-time.In order to understand the physical implications of thisfreeze-out scenario, we first note that viscosity indeedaffects particle productions. This is well known fromviscous hydrodynamic simulations of heavy-ion collisionswhere it is observed that inclusion of viscosity leads tochange in the transverse momentum spectra [50, 51].However, our claim is that although viscosity influencesparticle production via the overall normalization factor g ( α ), the shape of the rapidity distribution is not affectedsignificantly. In this scenario, the bulk of observed parti-cles come from central hotter region where the viscositydoes not play significant role. Viscosity plays importantrole at the edges of the fireball where the gradients arelarge but the temperature is small, leading to negligiblecontribution in the rapidity spectra.It is important to note that for ideal evolution, theratio of entropy density to number density, s/n , is a con-served quantity. This is not expected to hold when onehas dissipation in the system. On the other hand, wealso observe that the entropy density does not get anydirect correction from dissipative term in the relativis-tic Navier-Stokes equation, i.e., s ∼ (cid:15) /c + and hence s/n is approximately conserved for viscous evolution. More-over, in the present work, we saw that the viscous correc-tion to energy density evolution can be ignored at freeze-out, and therefore the final expression for rapidity distri-bution turns out to be proportional to entropy densitywhich is given by, dNdy ∼ exp (cid:18) c − c s (cid:113) y (cid:48) b − y (cid:19) . (17)We see that by setting c s = 1 / √ s NN = 200, 17.3 and 4.29 GeV (red solid curves) com-pared with the fit result using conformal solution of Lan-dau hydrodynamics [21] (blue dashed curves) and the ex-perimental results (black symbols). From the figure, wesee that a better fit is obtained using the non-conformal solutions for these collision energies. We have also fittedthe rapidity spectrum of pions for √ s NN = 12.3, 8.76,7.62, 6.27, 3.83, 3.28 and 2.63 GeV and found that thereis an overall better fit with solutions from non-conformalequation of state.In Fig. 2, we show a plot of squared speed of sound, ex-tracted by fitting the pion rapidity spectra using Eq. (17),over various collision energies (red solid line). The errorbars in the plot corresponds to standard error from least-square fit of the fit parameters. We see that at √ s NN =200 GeV, the fitted value of c s is slightly larger than1 / c s isseen to decrease, which is in agreement with lattice QCDpredictions for temperature dependence of c s [57, 58].However, with lower collision energies, we do not finda minimum in the √ s NN dependence of c s which is incontrast with earlier calculations of Ref. [55]. This maybe due to the fact that the rapidity distribution given inEq. (17), which is used to fit the data, and the expressionof y (cid:48) b is different in the present work and in Ref. [55], asexplained below.Based on our analysis of the rapidity spectra, we foundthat the width of the Gaussian profile is controlled by c s and hence the equation of state of the medium. This iseasy to see from Eq. (17). For rapidities small comparedto the beam rapidity, one can rewrite Eq. (17) to obtainthe well known Gaussian rapidity distribution dNdy ∼ exp (cid:20) − (cid:18) c − c s y (cid:48) b (cid:19) y (cid:21) . (18)Note that the variance of the Gaussian distribution ob-tained here from the analytical solution of non-conformalLandau hydrodynamics, σ = 2 c s y (cid:48) b − c s , (19)is different from those obtained previously by other au-thors [29, 55, 56, 59, 60] but agree in the conformallimit. Moreover, in these works, the authors have used y (cid:48) b = ln[ √ s NN / m p ] following Landau’s original descrip-tion [13] whereas we have employed an improved pre-scription y (cid:48) b ≡ ln [ c + / (4 c s )]+ y b following the argumentsgiven in Ref. [21]. However, within error bars, we did notobtain a clear signature for minima in √ s NN dependenceof c s , even if we consider Landau’s original prescriptionfor y (cid:48) b along with Eq. (18) for the fit.In Fig. 1, we have also shown the fit result for ra-pidity spectrum using the Gaussian distribution given inEq. (18) (green dashed-dotted curves). We observe thatthe fit is very close to that obtained by using Eq. (17).Further, in Fig. 2, we show a plot of squared speedof sound, extracted by fitting the pion rapidity spec-tra using Eq. (18) (green dashed-dotted line), over var-ious collision energies (black dashed line). We see that √ s NN =200 GeV √ s NN =17.3 GeV √ s NN =4.29 GeV d N / d y y DataConformalNonconformalNonconf. Gaussian
FIG. 1. (Color online) Rapidity spectrum of pions fitted usingEq. (17) (red solid curves) and Eq. (18) (green dashed-dottedcurves), for three representative collision energies: √ s NN =200, 17.3 and 4.29 GeV. Also shown are the fit result us-ing conformal solution of Landau hydrodynamics [21] (bluedashed curves) and the experimental results (black symbols).Experimental data are from Refs. [17, 52–54]. the extracted value of c s is lower for higher √ s NN butagree at lower collision energies. We found that usingthis Gaussian form, the fitted value of c s obtained for √ s NN = 200 GeV matches with the conformal value of1 /
3. We see that even in this case, a minimum is not ob-tained in √ s NN dependence of c s . While viscosity couldhave affected the width of the Gaussian distribution, wehave argued above that the effect is negligible. There-fore we claim that, for lower collision energies, rapidityspectra will provide a testing ground for determination ofthe correct value of c s and our solutions provide a frame-work for extracting this quantity. One should also keepin mind that the initial conditions may have significanteffect on the evolution and should be correctly accountedfor in such analysis.At this juncture, we would like to reiterate that wehave used a constant velocity of sound to derive the an-alytical expression for the evolution of energy density ofthe medium. We have employed this analytical expres-sion to fit the observed rapidity spectra at different col-lision energies and extracted the numerical value of thisconstant sound velocity. However, lattice calculationspredict temperature dependence of the sound velocity inthe QCD medium [57, 58]. Therefore one might view thisvalue of extracted c s as the approximate time-averagedvalue for that particular collision energy [61, 62]. This isin the same spirit as considering constant shear viscos-ity to entropy density ratio in hydrodynamic simulationswhere, in principle, one should consider temperature de- c S √ s NN (GeV) NonconformalNonconf. Gaussian
FIG. 2. (Color online) Squared speed of sound, extracted byfitting the pion rapidity spectra using the rapidity distribu-tion obtained using non-conformal solution given in Eq. (17)(red solid line) and also using the Gaussian distribution givenin Eq. (18) (green dashed-dotted line), over various collisionenergies. The error bars corresponds to standard error fromleast-square fit of the fit parameters. pendence. Our analytical expressions presented here willprovide the platform to extract the average value of c s ofthe QCD medium formed in heavy-ion collisions. Summary and outlook : We have considered the rela-tivistic Navier-Stokes equation for viscous evolution ofthe hot and dense matter formed in high energy heavy-ion collisions. We found analytical solution of the viscousevolution equation, for non-conformal system having con-stant speed of sound, with 1+1 dimensional Landau flowprofile. We found that for Wong’s modified prescriptionof Landau’s freeze-out scenario, the viscous effects do noteffect the rapidity spectrum of the produced particles.We employed the obtained solution to fit rapidity spec-trum of observed pions in √ s NN = 200, 17.3, 12.3, 8.76,7.62, 6.27, 4.29, 3.83, 3.28 and 2.63 GeV collision ener-gies. We showed that the obtained solutions of Landauflow with non-conformal equation of state leads to a bet-ter agreement with the experimental data compared tothe conformal Landau flow solution. We also showed thatthe value of speed of sound, obtained by fitting the exper-imental results, show a monotonic decrease with decreas-ing collision energy as opposed to previous predictions ofa minima.The importance of the analysis performed here is itsimplications in extracting the value of c s of QCD mediumformed in heavy-ion collisions by analyzing the rapidityspectrum of produced particles. More importantly, it isnow well understood that hydrodynamic modeling of rel-ativistic nuclear collisions at energies of order 10 GeVrequires treatment of non-equilibrium effects + breakingof boost invariance + non-zero baryon density. In thisarticle, we have attempted to address the first two, i.e.,treatment of non-equilibrium effects + breaking of boostinvariance, within an analytical framework. Althoughwe have performed the present calculations at vanishingbaryon density, nevertheless, we venture to claim thatthe rapidity spectra will provide a testing ground for de-termination of the correct equation of state. We wouldlike to emphasize that a simple analytical expression isalways useful to understand the qualitative behavior ofthe physical system and helps in building intuition forthe dynamics. At the very least, one can always treatan analytical solution of hydrodynamic equations as abenchmark to calibrate numerical codes.Looking forward, it is of course important to obtain thesolution of relativistic hydrodynamics and perform theanalysis of rapidity spectra in presence of finite baryonchemical potential. Unlike Landau’s original prescrip-tion of freeze-out hypersurface, which is also employedin the present work with Wong’s modification, one needsto consider a finite temperature freeze-out in order toaccurately estimate the viscous corrections to rapidityspectra. We leave these problems for future work. Acknowledgments : We thank Samapan Bhadury,Sandeep Chatterjee and Najmul Haque for useful discus-sions. D.B. acknowledges support from UGC and thanksNISER for kind hospitality. A.J. is supported in partby the DST-INSPIRE faculty award under Grant No.DST/INSPIRE/04/2017/000038. S.R. is supported inpart by the SERB Early Career Research Award underGrant No. ECR/2017/001354. 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