Magnetic Field Induced Polarization Difference between Hyperons and Anti-hyperons
MMagnetic Field Induced Polarization Difference between Hyperons and Anti-hyperons
Yu Guo, Shuzhe Shi,
2, 3
Shengqin Feng, ∗ and Jinfeng Liao † College of Science, China Three Gorges University, Yichang 443002, China. Department of Physics, McGill University, 3600 University Street, Montreal, QC, H3A 2T8, Canada. Physics Department and Center for Exploration of Energy and Matter,Indiana University, 2401 N Milo B. Sampson Lane, Bloomington, IN 47408, USA. (Dated: July 30, 2019)Recent STAR measurements suggest a difference in the global spin polarization between hyperonsand anti-hyperons, especially at relatively low collision beam energy. One possible cause of thisdifference is the potential presence of in-medium magnetic field. In this study, we investigate thephenomenological viability of this interpretation. Using the AMPT model framework, we quantifythe influence of different magnetic field evolution scenarios on the size of the polarization difference ina wide span of collision beam energies. We find that such difference is very sensitive to the lifetime ofthe magnetic field. For the same lifetime, the computed polarization difference only mildly dependson the detailed form of its evolution. Assuming magnetic polarization as the mechanism to enhanceanti-hyperon signal while suppress hyperon signal, we phenomenologically extract an upper limit onthe needed magnetic field lifetime in order to account for the experimental data. The so-obtainedlifetime values are in a quite plausible ballpark and follow approximately the scaling relation ofbeing inversely proportional to the beam energy. Possible implications on other magnetic fieldrelated effects are also discussed.
INTRODUCTION
Studies of strongly interacting fluid under the influenceof rotational motion have attracted significant interestsrecently, with much excitement particularly triggered bythe STAR Collaboration’s global polarization measure-ments in heavy ion collisions [1, 2]. On quite generalground, one expects the interplay between macroscopicfluid rotation and microscopic spin of individual particlescan lead to many novel effects. For example, individ-ual particle spins will be polarized on average along theglobal angular momentum. In the context of heavy ioncollisions, the colliding system in a non-central collisionscarries a large angular momentum along the directionperpendicular to the reaction plane and a global polariza-tion effect shall be expected from the produced hadronsin such collisions [3–7]. More precisely, the angular mo-mentum would turn into interesting vorticity patterns inthe QCD fluid and the vorticity structures further in-duce the particle spin polarization [8–22]. The presenceof nonzero vorticity can have nontrivial impact on theproperties of the underlying matter, such as the phasestructures and equation of state [23–33]. If the rotatingsystem consists of chiral fermions, the vorticity can alsoinduce anomalous transport phenomena known as Chi-ral Vortical Effects [34–38]. For recent reviews, see e.g.[39–41].The global polarization effect measurements by STARCollaboration in [1] show signals for both hyperons andanti-hyperons at the level of a few percent, with astrongly increasing trend toward lower collision energy.The data also clearly demonstrate a visible differencein the polarization between hyperons and anti-hyperons,with P ¯Λ > P Λ and with such difference also becoming stronger at lower energy. While the average polariza-tion signal could be quantitatively explained by hydrody-namic and transport modelings, the observed differencebetween hyperons and anti-hyperons remain a puzzle. Ef-forts were made to investigate various factors that maycontribute to such a splitting albeit without conclusiveanswer [14, 42–46]. At the moment, this is one of the im-portant unresolved challenges within the fluid-vorticityparadigm for the observed global polarization.One plausible proposal is to take into account an addi-tional polarization (apart from the vorticity-induced ef-fect) due to the existence of in-medium magnetic fieldwhich gives opposite polarization effect for hyperons andanti-hyperons [14, 43, 46]. Indeed, there is a very stronginitial magnetic field in an off-central heavy ion colli-sion [47–57] and if sufficiently long-lived could provide aconsiderable amount of magnetic polarization that distin-guishes particles from anti-particles. We note in passingthat strongly interacting matter under strong magneticfield has in itself been a very active topic of significantinterests with many developments (see recent reviews ine.g. [39–41, 58, 59]).The main objective of the present study is to ex-plore the phenomenological viability of such a magnetic-field-based interpretation for the observed difference inhyperon/anti-hyperon global polarization. Using theAMPT model framework and incorporating both rota-tional and magnetic polarization effects, we will quantifythe influence of different magnetic field evolution scenar-ios on the polarization difference in a wide span of colli-sion beam energies. We will use the polarization differ-ence to phenomenologically extract an upper limit on theneeded magnetic field lifetime in order to fully accountfor the experimental data. We will discuss the behaviorof so-obtained lifetime values and discuss possible impli- a r X i v : . [ nu c l - t h ] J u l cations on other magnetic field related effects. FORMALISM
In this part we provide a detailed description of theformalism we use to compute the Λ and ¯Λ polarization.For the overall bulk matter created in the collisions, weuse the transport model AMPT [60, 61] for a numberof reasons. First, it provides a reasonable descriptionof the bulk collective dynamics such as soft particles’yields, transverse momentum spectra and flow observ-ables. We use the same setup as in [61] which demon-strated very good agreement with experimental data forAu+Au collisions at RHIC. Furthermore it can be usedfor a wide span of collision beam energies. Another ad-vantage is that it allows explicit tracking of every partonor hadron’s motion during the evolution and of each fi-nal state hadron’s formation. This allows a relativelystraightforward procedure to extract the system’s vortic-ity structure as well as to incorporate the spin polariza-tion effect upon the hadron formation. The AMPT modelwas first extended to compute vorticity structures in [15]and later widely used for polarization studies [16, 19, 20].From AMPT simulations one obtains the four velocitydistribution u µ ( x ) as well as energy density distribution (cid:15) ( x ) in space-time x = ( t, (cid:126)x ) across the system, which canbe further used to evaluate various quantities of interest.The rotational polarization effect on particle spin ina relativistic fluid can be determined from the thermalvorticity (cid:36) µν defined as [8, 9]: (cid:36) µν = −
12 ( ∂ µ β ν − ∂ ν β µ ) (1)where β µ = u µ /T with T = 1 /β the local temper-ature. A related quantity is the kinetic vorticity de-fined by Ω µν = − ( ∂ µ u ν − ∂ ν u µ ). Obviously (cid:36) µν = β { Ω µν − [( β∂ µ T ) u ν − ( β∂ ν T ) u µ ] } . The thermal vortic-ity differs from the Ω µν /T by terms containing gradientsof temperature, ∼ ( β∂ µ T ) = [( ∂ µ T ) /T ]. While straight-forward to evaluate in hydrodynamic models, such termsare trickier to compute in transport models. As a proxy,we use the energy density (cid:15) to evaluate such termsvia ( ∂ µ T ) /T = ( ∂ µ (cid:15) ) / (4 (cid:15) ) with underlying assumption (cid:15) ∝ T . Such gradient terms make non-negligible contri-butions and should be taken into account.We now discuss the calculation of particle polarizatione.g. for the hyperons and anti-hyperons. In the case thatpolarization solely comes from vorticity, one has the fol-lowing ensemble-averaged spin 4-vector of the producedΛ and ¯Λ determined from the local thermal vorticity atits formation location, as [1, 8, 9, 14, 19]: S µ = − m (cid:15) µνρσ p ν (cid:36) ρσ (2)where p ν is the four-momentum and m the mass ofthe produced hyperons/anti-hyperons. Past calculations solely based on the vorticity-induced polarization can notdescribe the observed difference between signals of Λ and¯Λ. In fact, as we will show later, the polarization effectfrom just the vorticity of fluid rotation would be largerfor Λ than ¯Λ, quite the opposite to data, due to a subtleeffect related to particle formation timing.The existence of a magnetic field could indeed inducea difference in the spin polarization between Λ and ¯Λ dueto their opposite magnetic moments. Under the presenceof electromagnetic fields F µν , the spin 4-vector formulawill become different from that in Eq.(2) and should begiven instead by the following [14]:˜ S µ = − m (cid:15) µνρσ p ν [ (cid:36) ρσ ∓ eF ρσ ) µ Λ /T f ] (3)where µ Λ = . m N is the absolute value of thehyperon/anti-hyperon magnetic moment, with m N =938MeV being the nucleon mass. T f is the local tem-perature upon the particle’s formation. In the case withnonzero electromagnetic field, there will be a differencebetween Λ and ¯Λ spin polarization due to the secondterm in the above. Here we focus on the electromag-netic field component that is most relevant to the globalpolarization effect, namely B y = F = − F along theout-of-plane direction which is also the direction of globalangular momentum. It should be noted that the aboveEq.(3) assumes local equilibrium of polarization underelectromagnetic fields. In the rather dynamical environ-ment of heavy ion collisions, particle polarization may notnecessarily relax instantaneously to the expected valueand off-equilibrium corrections could be important. Thisis an interesting problem for future study. One impor-tant caveat for comparison with experimental data is theinfluence of secondary decays on the measured hyperonpolarization. Two important recent studies [44, 45] haveexcluded a major role of such decay contributions andtherefore justified the application of Eq.(3) for primaryhadrons in our study as a very good approximation.In (non-central) heavy ion collisions, there is a stronginitial magnetic field eB arising from the fast-movingspectator protons, which has been very well studied [47–49]. The key issue here is whether such a magneticfield would survive long enough to have nonzero impactaround the freeze-out time for hadron formation. Thereare proposals for certain mechanisms that could providedrelatively long-lived late time magnetic field, e.g. by wayof medium induction [50–55] or by rotating fluid withnonzero charge density [46]. Nevertheless currently themagnetic field time evolution in heavy ion collisions israther uncertain [62]. Alternatively, one may turn thisaround (as suggested in [43]) and consider the splittingbetween Λ/¯Λ polarization as a way to put an empiricalconstraint on the size of potentially existing late timemagnetic field. In the present study, we further exploitthis line of thought and address the following question:what kind of magnetic field time evolution B y ( τ ) wouldbe needed, if the observed polarization difference wouldbe entirely attributed to the in-medium magnetic field? t B ( fm ) ∫ F B ( t B , t ) d t FIG. 1: (color online) The time integrated value (cid:82) F ( t B , t ) dt (for t from 0 ∼ t B for differentmagnetic field time evolution: type-1 (red solid curve), type-2 (green dashed curve) and type-3 (blue dash-dotted curve).See text for details. In order to study this question, and given the uncer-tainty about the time dependence, we phenomenologi-cally investigate this problem by assuming B y ( t ; (cid:126)x ) = B ( (cid:126)x ) · F B ( t B , t ) and studying three different kinds ofparameterization for F B that have been adopted in theliterature for various studies of magnetic field effects:Type-1: , F B ( t B , t ) ≡ t − t ) /t B (see e.g. [63, 64]);Type-2: F B ( t B , t ) ≡ [ t − t ) /t B ] / (see e.g. [48]);Type-3: F B ( t B , t ) ≡ e −| t − t | /t B (see e.g. [43]).In all these parameterizations, the t B is the essen-tial magnetic field lifetime parameter that controls howrapidly the magnetic field would decrease with time.Note however due to their different functional forms,the same t B value gives slightly different magnetic fieldevolution. To give an idea of such difference, we showin Fig. 1 the time integrated value (cid:82) F ( t B , t ) dt (for t from 0 ∼ t B for compar-ing these three types of evolution. Defining t = 0 asthe time point of the very initial contact of the collisionprocess, the t ≡ R A / ( γ beam v beam ) is the time for fulloverlap of the two colliding nuclei, with R A being thenuclear radius, v beam and γ beam being the beam speedand the corresponding Lorentz factor. Note this is im-portant particularly for collisions at low beam energy.The initial magnetic field value B ( (cid:126)x ) is determined fromevent-by-event calculations with Monte-Carlo Glaubersimulations as in e.g. [49]. Note this field strongly de-pends on beam energy, following a trend B ∝ √ s NN .For example, at the center point (cid:126)x = 0, the initial strength eB ( (cid:126)x =0) m π (where m π is the pion mass) equals0 . , . , . , . , . , . , .
922 for beamenergy √ s NN = 11 . , . , . , , , ,
200 GeVrespectively. These values are determined from simulat-ing proton distributions in the initial conditions and areconsistent with other calculations. In this study we fo-cus on the (20 ∼ or more AMPT events for each givenbeam energy to ensure enough statistics. The hyperonand anti-hyperon polarization results are then computedfrom Eq.(3) for each type of magnetic field time evolu-tion with a chosen lifetime parameter. We present thedetailed results from such study in the next section. RESULTS
As a first step, let us examine how the key parameter,magnetic field lifetime t B , would influence the polariza-tion observable. To do this, we vary this parameter (foreach given type of time evolution) and examine how theobtained global polarization signals of Λ and ¯Λ wouldchange. In Fig. 2 we show such results for collisions atbeam energy √ s NN = 19 . , ,
39 GeV respectively.In Fig. 3, we also show and compare the different con-tributions to the Λ and ¯Λ polarization from kinetic vor-ticity term, from temperature gradient term and frommagnetic field term, suggesting a dominant role of ki-netic vorticity and a non-negligible temperature gradientcontribution. As one can see, with increasing magneticfield lifetime (which means stronger magnetic field at latetime in the collisions), the P ¯Λ steadily increases while the P Λ decreases at all collision energies. With long enough t B , eventually the P ¯Λ always becomes larger than P Λ .The occurrence of “crosspoint” (where P ¯Λ = P Λ ) requireslonger lifetime at lower beam energy. Another interestingobservation is that when t B → P = ( P ¯Λ − P Λ ) to the de-tails of the time evolution. To do that, we evaluate andcompare the ∆ P values computed from the three typesof time dependence, with the results shown in Fig. 4. t B ( fm / c ) P H ( % ) FIG. 2: (color online) The dependence on magnetic field life-time parameter t B of the global polarization signals P H forhyperons ( H → Λ, blue solid curves with filled symbols) andanti-hyperons ( H → ¯Λ, red dashed curves with open sym-bols) at beam energy √ s NN =19.6 (square), 27 (diamond),39 (circle) GeV respectively. - t B ( fm ) P H ( % ) s NN = Λ Red: Λ FIG. 3: (color online) Different contributions to the Λ and ¯Λpolarization from kinetic vorticity term (solid curves), fromtemperature gradient term (dashed curves) and from mag-netic field term (dash-dotted curves) respectively. See textfor details.
There, we plot ∆ P versus beam energy √ s NN for thetype-1,2,3 magnetic fields with two choices of lifetime t B (a shorter one of 1 fm/c and a longer one of 4 fm/c ).
10 10020 50 20001234 s NN ( GeV ) P Λ __ - P Λ ( % ) FIG. 4: (color online) The difference ∆ P = P ¯Λ − P Λ versuscollision beam energy, for the type-1 (square), type-2 (dia-mond) and type-3 (circle) time dependence. The red solidcurves are for t B = 1fm while the blue dashed curves arefor t B = 4fm. The black circles with error bars are STARexperimental data from [1, 2]. The comparison demonstrates that the magnetic field in-duced splitting ∆ P = ( P ¯Λ − P Λ ), while most sensitiveto the parameter t B , also mildly depends on the detailedform of the time evolution. It is also clear that for thesame t B value, the magnetic field effect is stronger athigher beam energy, simply due to its larger peak value B . We also show the STAR measured splitting on thesame plot, which indicates that a longer lifetime is re-quired for describing the ∆ P at lower beam energy.We now use the experimental data for ∆ P as a wayto constrain the magnetic field lifetime parameter. Ateach beam energy, we find the optimal value of ˜ t B thatwould give the amount of measured splitting. This al-lows us to extract from data the preferred lifetime asa function of beam energy in a scenario that the split-ting is caused by such magnetic field. The results foreach of the type-1, 2, 3 (as left, middle, right panels)are shown in Fig. 5. The error bars are converted fromthe corresponding experimental data error bars, whichat the moment are substantial but may be significantlyreduced in upcoming RHIC Beam Energy Scan II mea-surements [65]. Common to all three types, the neededlifetime ˜ t B decreases with beam energy √ s NN . For exam-ple, ˜ t B ∼ / c for √ s NN = 11 . t B ∼ . / cfor √ s NN = 200GeV. We note that these numbers maybe quite plausible. To quantify such dependence, we per-form a fitting analysis, with the dependence ˜ t B = A √ s NN .Such a scaling formula is based on Lorentz contracted
10 10020 50 20005101520 s NN ( GeV ) t ˜ B ( f m / c ) Type -
10 10020 50 2000510152025 s NN ( GeV ) t ˜ B ( f m / c ) Type -
10 10020 50 2000102030 s NN ( GeV ) t ˜ B ( f m / c ) Type - FIG. 5: (color online) The optimal value of magnetic field lifetime parameter ˜ t B extracted from polarization splitting ∆ P data for a range of collision beam energy √ s NN . The left, middle and right panels correspond to the type-1, 2 and 3 formsof magnetic field time evolution. The solid curves are from fitting analysis with a formula ˜ t B = A √ s NN . The error bars areconverted from the corresponding errors of experimental data in [1, 2].
10 10020 50 2000100200300 s NN ( GeV ) ℬ ( M e V ) FIG. 6: (color online) The time-integrated magnetic fieldstrength
B ≡ (cid:82) ( eB y ) dt at the center point (cid:126)x = 0 as a func-tion of beam energy, for the type-1 (red square), type-2 (greendiamond) and type-3 (blue circle) time dependence with op-timized parameter ˜ t B from polarization splitting. time for the passing-through period between two nuclei,˜ t B ∝ R A γ ∝ √ s NN . The fitting curves are shown inFig. 5 as solid curves, with the χ -optimized parame-ter A = 92 for type-1, A = 125 for type-2 and A = 128for type-3 (all bearing unit of GeV · fm / c). An averagesover these three types of time dependence in a (perhapsnaive) statistical way would suggest ˜ t B = A √ s NN with A = 115 ±
16 GeV · fm / c. Interestingly, this is consid-erably longer than the expected vacuum magnetic fieldlifetime without any medium effect, which could be es-timated by t vac (cid:39) R A γ (cid:39)
26 GeV · fm / c √ s NN . Such extendedmagnetic field lifetime, as indicated by polarization dif-ference, may imply a considerable role of the medium feedback on dynamical magnetic field evolution.A magnetic field, in addition to inducing splitting be-tween Λ/¯Λ polarization, can also lead to various otherinteresting phenomena [39, 40]. Many of these effects aredependent on the time-accumulative effect of the mag-netic field. With the above analysis of the magneticfield time evolution based on polarization splitting, wecompute a related quantity, the time-integral of mag-netic field strength B ≡ (cid:82) ( eB y ) dt at the center point (cid:126)x = 0. This is computed at each beam energy and foreach type of time evolution (along with optimized pa-rameter ˜ t B ), with the results shown in Fig. 6. We notethat this provides an estimate of the upper limit for ac-cumulative magnetic field strength based on polarizationsplitting data, which would be useful for constrainingother effects arising from the magnetic field. These re-sults suggest that the time-integrated in-medium mag-netic field could be considerable and much exceed thetime-integrated vacuum magnetic field as estimated ine.g. [66]. As shown by Anomalous-Viscous Fluid Dy-namics (AVFD) simulations [63, 64], an in-medium mag-netic field of this scale could make a substantial contribu-tion to the signal of Chiral Magnetic Effect (CME). It isalso interesting to note that the potential CME signal asextracted by STAR Collaboration [67] via the so-calledH-correlator from two-component decomposition analy-sis [68, 69] shows a very similar trend in its beam energydependence, first increasing and then decreasing with apeak around √ s NN = (40 ∼
60) GeV region.
SUMMARY
In summary, we have quantitatively investigated themagnetic field as a probable cause of the observed differ-ence in global polarization between hyperons and anti-hyperons. Using the AMPT model framework, we havequantified the influence of different magnetic field life-time and time dependence on the size of the splitting ina wide span of collision beam energies.Our main findings include: (1) At all beam energies, alonger the magnetic field lifetime leads to a larger polar-ization for anti-hyperons while a smaller polarization forhyperons; (2) the lifetime parameter sensitively controlsthe size of the splitting, which is also mildly dependenton the precise form of magnetic field evolution; (3) Theneeded magnetic field lifetime in order to fully account forthe observed splitting ∆ P is in a plausible ballpark andstrongly decreases from low to high beam energy, rangingfrom a few fm/c at the low end of RHIC BES energy toa fraction of one fm/c at top RHIC energy; (4) The so-extracted magnetic field lifetime follows approximatelythe scaling relation of being inversely proportional to thebeam energy.To conclude, the interpretation of observed polariza-tion difference in terms of lasting magnetic field couldbe a plausible one and the required magnetic field life-time appears not impossible. In the present study, we’venot addressed the question of precisely how the mag-netic field should evolve, which would require dynami-cal simulations that solve the magnetic field evolutionfrom Maxwell equations. This would be an importantand interesting next step to examine the viability of suchinterpretation, which we shall carry out in a future study.This work is partly supported by NSFC Grant No.11875178 and No. 11735007, by NSF Grant No. PHY-1913729, and by the U.S. Department of Energy, Officeof Science, Office of Nuclear Physics, within the frame-work of the Beam Energy Scan Theory (BEST) TopicalCollaboration. SS is grateful to the Natural Sciences andEngineering Research Council of Canada for support. JLthanks the Institute for Advanced Study of Indiana Uni-versity for partial support. ∗ Electronic address: [email protected] † Electronic address: [email protected][1] L. Adamczyk et al. 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