Transverse-velocity scaling of femtoscopy in proton proton collisions

Transcription

1 Journal of Physcs G: Nuclear and Partcle Physcs PAPER Transverse-velocty scalng of femtoscopy n proton proton collsons To cte ths artcle: T J Humanc 018 J. Phys. G: Nucl. Part. Phys Vew the artcle onlne for updates and enhancements. Related content - Extractng the hadronzaton tmescale n \sqrt{s}=7 TeV proton--proton collsons from pon and kaon femtoscopy T J Humanc - K 0_s K 0_s correlatons n 7 TeV pp collsons from the ALICE experment at the LHC T J Humanc and the ALICE Collaboraton - Femtoscopy from the ALICE experment A Ksel and ALICE Collaboraton Ths content was downloaded from IP address on 10/04/018 at 16:46

2 Journal of Physcs G: Nuclear and Partcle Physcs J. Phys. G: Nucl. Part. Phys. 45 (018) (9pp) Transverse-velocty p scalng of femtoscopy n ffffff s ¼ 7 TeV proton proton collsons T J Humanc Department of Physcs, Oho State Unversty, Columbus, OH, Unted States of Amerca E-mal: humanc.1@osu.edu Receved 13 January 018, revsed 16 March 018 Accepted for publcaton 19 March 018 Publshed 6 Aprl 018 Abstract Although transverse-mass scalng of femtoscopc rad s found to hold to a good approxmaton n heavy-on collson experments, t s seen to fal for hgh-energy proton proton collsons. It s shown that f nvarant radus parameters are plotted versus the transverse velocty nstead, scalng wth the transverse velocty s seen n s = 7 TeV proton proton experments. A smple sem-classcal model s shown to qualtatvely reproduce ths transverse velocty scalng. Keywords: boson femtoscopy, 7 TeV proton proton collsons, transversevelocty scalng, nvarant radus parameters (Some fgures may appear n colour only n the onlne journal) 1. Introducton A common feature seen n dentcal-partcle hgh-energy heavy-on collson femtoscopy experments s transverse-mass scalng,.e. m T scalng, of the extracted radus parameters [1, ], where mt = kt + m0, k T s the average momentum of the partcle par,.e. k = + ¾ T pa pb, and m 0 s the partcle rest mass. Transverse-mass scalng manfests tself when plottng radus parameters extracted from dentcal-partcle pars of varous masses from the same colldng system versus m T, and observng that the downward trend of the radus parameters wth ncreasng m T does not depend on the partcle mass. A good example of ths for the case of one-dmensonal femtoscopy where the nvarant radus parameter s extracted s seen n fgure 8 of [1]. Ths fgure plots nvarant radus parameters versus m T for dentcal pon, kaon and proton pars extracted n Pb Pb collsons at snn =.76 TeV from the ALICE experment. As seen n ths fgure, m T scalng for the nvarant radus parameters holds to a good approxmaton for all of the measured collson centraltes. Ths dependence /18/ $ IOP Publshng Ltd Prnted n the UK 1

3 Fgure 1. Expermental nvarant radus parameters versus transverse mass, m T, for varous charged-partcle multplcty ranges (gven by the numbers n the legend) extracted from one-dmensonal femtoscopc charged pon, charged kaon and neutral kaon analyses n s = 7 TeV proton proton collsons by the ALICE experment [4, 5]. Adapted from [5]. CC BY 3.0. s normally explaned as a sgnature of radal flow n these collsons, whch can be reproduced by varous models [, 3]. The stuaton s seen to be dfferent n hgh-energy proton proton collsons. Femtoscopc experments can show a smlar downward trend of the extracted radus parameters wth ncreasng m T for a sngle-mass partcle par n hgh-multplcty charged-partcle collsons, but the m T scalng seen n heavy-on experments when plottng the radus versus m T for several masses s not seen. Fgure 1 shows an example of ths lack of m T scalng. The fgure shows expermental nvarant radus parameters versus the average m T n that bn for varous charged-partcle multplcty (N ch ) ranges extracted from one-dmensonal femtoscopc charged pon, charged kaon and neutral kaon analyses n s = 7 TeV proton proton collsons by the ALICE experment [4, 5] (Ths s a reproducton of fgure 4 n [5].) As seen n the fgure, for the hgher multplcty ranges, N ch 1 and N ch >, the nvarant radus parameter, R nv, decreases for ncreasng m T ndvdually for the pon and kaon analyses, however there s no apparent scalng between the pons and kaons wth m T. A feature that s common to both pons and kaons s the overall ncrease n R nv wth hgher multplcty, whch s also seen n heavy-on collsons. Ths can be attrbuted to an ncreased volume of the collson producng larger multplcty and rad due to, for example, fnal-state hadronc rescatterng [6 8]. Another nterestng feature seen n ths fgure s the rsng m T dependence of R nv for the pons and charged kaons n the lowest multplcty range, N ch 1 11, also not showng m T scalng. So far there has been no satsfactory explanaton n the lterature for ths lack of m T scalng observed n femtoscopc nvarant radus measurements n hgh-energy proton proton collsons. One can at least pont out the dfferences between hgh-multplcty pp collsons

4 Fgure. Same data as n fgure 1 replotted versus transverse velocty, β T. and central heavy-on collsons that mght contrbute to ths dfference n observed scalng. In central heavy-on collsons, partcle producton proceeds va many soft nucleon nucleon collsons creatng thousands of partcles, mostly pons, whch then undergo fnal-state rescatterng that thermalzes the system and results n radal flow. If the scatterng cross sectons of dfferent partcle speces are averaged-out n ths dense medum, then all partcles should partcpate equally n the radal flow and the source rad are expected to approxmately follow a common m T scalng for all partcle speces []. Ths stuaton s n contrast to partcle producton n pp collsons whch proceed through a sngle hard pp collson that produces at most partcles. Even for these hghest-multplcty pp collsons, sgnfcant collectve effects of the same nature as heavy-on collsons would not be expected. Thus t s not surprsng that m T scalng s not expermentally observed n pp collsons. The goal of the present work s to (1) show that a dfferent scalng s seen for proton proton collsons and () develop a smple sem-classcal model to qualtatvely reproduce ths new scalng.. Transverse-velocty scalng Fgure shows the data n fgure 1 replotted versus the average transverse velocty, β T, of the m T bn, where β T = k T /m T, and k T s the average value n the m T bn. As seen, there now appears to be a scalng of R nv on β T for each charged multplcty range,.e. the dependence of R nv s only on β T and not on the partcle mass wthn the uncertantes of the measurements. At ths pont, the b T scalng s presented as emprcal,.e. there s no theoretcal gudance for why ths scalng should work. In the next secton a smple sem-classcal model s 3

5 developed to try to descrbe ths scalng. Results from the model are then compared wth the expermental results n fgure. 3. Descrpton of a smple sem-classcal model to descrbe the transversevelocty scalng In ths secton, a smple sem-classcal model s descrbed whch attempts to mock up the β T scalng seen n the measurements n fgure. In addton to showng β T scalng between the pons and kaons, the model also should account for the dfferent character of R nv on β T seen for the lowest multplcty range,.e. ncreasng R nv wth ncreasng β T for N ch 1 11, and the hgher ranges,.e. larger overall R nv that does not vary much wth β T for N ch 1 and that s decreasng for ncreasng β T for N ch >. It wll be assumed that the hgher multplcty ranges are dfferent from the lowest one due to (1) longer meson hadronzaton tmes makng R nv larger, and () the presence of a sgnfcant radal flow actng before hadronzaton whch ultmately results n the decreasng dependence of R nv on β T for the hghest multplcty range. It wll be assumed that the addtonal energy stored n the radal flow s converted to addtonal partcle producton upon hadronzaton, although ths s not explctly addressed n ths smple model. A Monte Carlo calculaton s used. After hadronzaton, pars of charged partcles,.e. charged pon or kaon pars, are gven a combned Bose Ensten symmetrzaton and Coulomb nteracton weght, and a one-dmensonal correlaton functon n terms of the nvarant momentum dfference s then formed. The radus parameter, R nv, s then extracted by fttng a Gaussan spacal model to the correlaton functon, smlar to what was done by the experments that extracted the R nv values from data shown n fgure. A more detaled descrpton of ths calculaton now follows. The space-tme pont of the th partcle of rest mass m 0 at hadronzaton n the proton proton collson frame (x, y, z, t ) wth energy momentum (p x, p y, p z, E ) s determned n the model by a Gaussan dstrbuton for the hadronzaton tme of wdth σ t, such that dn dt t µ exp -, ( 1) st x = t b f y = t b f z = t p rad, cos rad, sn, ( ) E z b rad, = b + b f 1 + bb f b pt =, ( 3) E where pt = px + p, f y s the azmuthal angle of the th partcle set randomly between 0 π, and β f s the average radal flow velocty. The quantty β rad, s the relatvstc sum of the radal flow velocty and transverse partcle velocty. The quanttes σ t and β f are free parameters to be adjusted to get the best agreement wth the measurements. The energy momentum of each partcle s determned from fts of exponental dstrbutons to ALICE expermental p T dstrbutons for charged pons and kaons from s = 7 TeV proton proton collsons [9], and from assumng a flat partcle rapdty dstrbuton n the range of 1 < y < 1, whch s close to the expermental rapdty range used. The assumpton of a flat rapdty dstrbuton n ths rapdty range s seen from expermental measurements to be a reasonably good approxmaton for both pons and kaons for pp collsons at s = 7 TeV [10, 11]. 4

6 Quantum statstcs and the Coulomb nteracton are mposed par-wse on charged boson pars a and b by weghtng them at ther hadronzaton phase-space ponts ( r a, ta, pa, Ea) and ( ¾ r, t, ¾ p, E ) b b b b wth where, and where W = G( h){ 1 + cos ( D r D p -DD t E)}, ( 4) ab D r = r - ¾ D = - ¾ a rb p pa pb D t = ta - tb D E = Ea - Eb ( 5) G ( h) s the Gamow factor, ph m0a G( h) = h = ( 6) exp( ph) - 1 qnv and q = D nv p - DE s the nvarant momentum dfference and α s the fne structure constant. The correlaton functon, C(q nv ), s formed by bnnng pars n terms of the nvarant momentum dfference as the rato of weghted pars, N(q nv ), to unweghted pars, D(q nv ), C ( q ) = nv Nq ( nv). ( 7) Dq ( ) nv Snce a fnal-state Coulomb nteracton, va the Gamow factor, s n the model between boson pars after hadronzaton to more closely mock up the expermental condtons for the femtoscopc analyss, a Gaussan functon wth the Gamow factor usng the Bowler Snyukov equaton [1, 13] s ftted to equaton (7) to extract the boson source parameters whch are compared wth experment, ft nv nv C ( q ) = a{ 1 - l + lg( h)[ 1 + exp (-q R )]}, ( 8) where R nv s the nvarant radus parameter whch, n prncple, s related to the sze of the boson source, λ s a parameter that reflects the strength of the quantum statstcs effect as well as the degree to whch the Gaussan functon fts to the correlaton functon, and a s an overall normalzaton parameter. Equaton (8) s the same functon as used by ALICE to extract R nv and λ from quantum statstcs n ther measurements for the charged-partcle pars, except that nstead of usng the Gamow factor to account for the Coulomb nteracton they use a factor calculated from Coulomb waves. However, for the small radus parameters that are extracted n these collsons,.e. 1 fm, the two methods of calculatng Coulomb yeld almost the same results [1]. nv 4. Results and dscusson Fgure 3 shows sample correlaton functons for charged pons and charged kaons from the model calculated from equaton (7). Fts of equaton (8) to the model ponts are also shown. As seen, the fts represent the model ponts reasonably well. Fgure 4 shows the same expermental results as fgure wth the model calculatons for N ch 1 11, N ch 1 and N ch > overlad as sold lnes for pons and dashed lnes for kaons. Table 1 gves the values of the free parameters σ t and β f used n the calculatons to adjust the model to best agree wth the measurements. The uncertantes n extractng σ t and β f are estmated to be ±0.1 fm/c and ±0.05, respectvely. As seen, the model calculatons represent the trends of the measurements reasonably well, as well as showng the β T scalng between the pons and kaons. It s also seen that R nv from the model sharply turns upward for bt 1 for all multplcty ranges. Ths asymptotc-lke behavor can be qualtatvely 5

7 Fgure 3. Sample correlaton functons from s = 7 TeV proton proton collsons for charged pons and charged kaons from the model usng equaton (7). Fts of equaton (8) to the model ponts are also shown. Fgure 4. Same as fgure wth model calculatons, shown as sold and dashed lnes for pons and kaons, respectvely, overlad over the expermental ponts. understood by expressng R nv approxmately n terms of the three-dmensonal Gaussan rad, nv long sde T out R» ( R + R + g R ) 3, ( 9) 6

8 Table 1. σ t and β f parameters used n fttng the model to the N ch 1 11, N ch 1 and N ch > data n fgure 4. The uncertantes n σ t and β f are estmated to be ±0.1 fm/c and ±0.05, respectvely. Parameter N ch range ππ K ch K ch s t (fm/c) > β f > where R long s the radus n the z-drecton, R out s the transverse radus n the radal drecton, R sde s the transverse radus perpendcular to the radal drecton, and gt = bt [, 3]. From equaton (9) t s seen that R nv s expected to ncrease asymptotcally for bt 1. The expermental results for N ch 1 11 also clearly show ths ncrease for bt 1, although for the hgher multplcty ranges ths behavor s not obvous due to the larger uncertantes n the measurements there. Lookng at table 1, both extracted parameters are seen to systematcally ncrease wth ncreasng multplcty for both the pons and kaons. For the N ch 1 11 model calculatons, t s assumed that no radal flow s present and thus b f was set to zero for both pons and kaons. The hadronzaton tme wdth, σ t, s seen to be slghtly larger for the kaons than for the pons, but both are seen to be relatvely small at 0.5 and 0.7 fm/c, respectvely. The β f values for the hgher multplcty ranges are also seen to be equal wthn the uncertantes for the pons and kaons, suggestng that the pons and kaons are experencng the same radal-flow feld for each multplcty range. For the N ch > model calculatons, very large values of β f for pons and kaons of 0.95 and 0.90, respectvely, are requred to produce decreasng R nv values wth ncreasng β T to agree wth the measurements wthn the expermental errors. The σ t values are larger than for N ch 1 11 and N ch 1 and are sgnfcantly dfferent between the pons and kaons at 0.95 fm/c and.5 fm/c, respectvely, to reproduce the β T scalng at the hghest multplcty. From the model pont of vew, the reason for the larger σ t for kaons s, due to ther larger mass, the velocty of a kaon n the z-drecton s sgnfcantly smaller than for the average velocty of pons, resultng n the sze of the source n the z-drecton beng smaller for kaons than pons for equal values of σ t. Ths can be seen n equaton (). Ths has a sgnfcant effect on R nv snce t represents a combnaton of szes n both the radal and the z-drecton, as seen n equaton (9). Ths model s clearly a toy model snce t s based on geometry, partcle knematcs and several ad hoc assumptons about the proton proton collson and hadronzaton of the mesons. One could try to connect a more physcal nterpretaton to t by assumng that the mesons are quas-partcles or d-quarks before they hadronze. In ths pcture, when quaspartcles are ntally formed from the proton proton collson, ther momentum dstrbutons are assumed to be smlar to the momentum dstrbutons measured n experments. In a low partcle multplcty envronment, hadronzaton takes place n a relatvely short tme,.e. as n a Lund strng model pcture used n PYTHIA [14] and lttle or no radal flow wll be present. In a hgh partcle multplcty envronment, the quas-partcle fnds tself n a quark-gluon envronment and t s assumed that a radal-flow feld has been set up. The radal flow velocty adds to the ntal velocty of the quas-partcle and the hadronzaton tme s ncreased due to 7

9 the extended quark-gluon envronment. When hadronzaton fnally occurs, t s assumed that the addtonal energy stored n the radal flow s converted to addtonal partcle producton. The hadronzaton tme for a kaon mght be expected to be longer than for a pon due to ts larger mass and requrement of a strange quark. In spte of ts lmtatons, the toy model seems to contan elements,.e. smple geometry and sem-classcal partcle trajectores, that allow β T scalng to occur wth the approprate choce of free parameters. In addton, the extracted values for σ t and β f may shed some lght on the hadronzaton process n these collsons. 5. Summary Although transverse-mass scalng of femtoscopc rad s found to hold to a good approxmaton n heavy-on collson experments, t s seen to fal for hgh-energy proton proton collsons. It s shown that f nvarant radus parameters are plotted versus the transverse velocty nstead, scalng wth the transverse velocty s seen n s = 7 TeV proton proton experments. A smple sem-classcal model s shown to qualtatvely reproduce ths transverse velocty scalng. The elements makng up the smple model,.e. geometry and semclasscal partcle trajectores, appear to be suffcent to allow the b T scalng to occur wth the approprate choces of the free parameters n the model. Clearly, t would be a good test for a more physcally motvated and detaled model calculaton to descrbe ths scalng seen n proton proton femtoscopy experments. Acknowledgments The author wshes to acknowledge fnancal support from the U.S. Natonal Scence Foundaton under grant PHY ORCID Ds T J Humanc References [1] Adam J et al (ALICE Collaboraton) 015 One-dmensonal pon, kaon, and proton femtoscopy n Pb Pb collsons at snn =.76 TeV Phys. Rev. C [] Lsa M A, Pratt S, Soltz R and Wedemann U 005 Femtoscopy n relatvstc heavy on collsons Annu. Rev. Nucl. Part. Sc [3] Ksel A, Galazyn M and Bozek P 014 Pon, kaon, and proton femtoscopy n Pb Pb collsons at s NN =.76 TeV modeled n 3+1D hydrodynamcs arxv: [4] Abelev B et al (ALICE Collaboraton) 01 Ks 0 - Ks 0 correlatons n pp collsons at s = 7 TeV from the LHC ALICE experment Phys. Lett. B [5] Abelev B et al (ALICE Collaboraton) 013 Charged kaon femtoscopc correlatons n pp collsons at s = 7 TeV Phys. Rev. D [6] Akkeln S and Snyukov Y 1995 The HBT nterferometry of expandng sources Phys. Lett. B [7] Truesdale D and Humanc T 01 Predctons from a smple hadron rescatterng model for pp collsons at the LHC J. Phys. G: Nucl. Part. Phys [8] Humanc T J 014 Extractng the hadronzaton tmescale n s = 7 TeV proton proton collsons from pon and kaon femtoscopy J. Phys. G: Nucl. Part. Phys

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