高级检索

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

Chiral Crossover and Chiral Phase Transition Temperatures from Lattice QCD

Shengtai LI Hengtong DING

李胜泰, 丁亨通. 利用格点量子色动力学研究手征平滑过渡温度和手征相变温度[J]. 原子核物理评论. doi: 10.11804/NuclPhysRev.37.2019CNPC65
引用本文: 李胜泰, 丁亨通. 利用格点量子色动力学研究手征平滑过渡温度和手征相变温度[J]. 原子核物理评论. doi: 10.11804/NuclPhysRev.37.2019CNPC65
Shengtai LI, Hengtong DING. Chiral Crossover and Chiral Phase Transition Temperatures from Lattice QCD[J]. Nuclear Physics Review. doi: 10.11804/NuclPhysRev.37.2019CNPC65
Citation: Shengtai LI, Hengtong DING. Chiral Crossover and Chiral Phase Transition Temperatures from Lattice QCD[J]. Nuclear Physics Review. doi: 10.11804/NuclPhysRev.37.2019CNPC65

利用格点量子色动力学研究手征平滑过渡温度和手征相变温度

doi: 10.11804/NuclPhysRev.37.2019CNPC65
详细信息
  • 中图分类号: O571.53

Chiral Crossover and Chiral Phase Transition Temperatures from Lattice QCD

Funds: National Natural Science Foundation of China (11947237, 11775096)
More Information
    Author Bio:

    (1990–), male, born in Dandong, Liaoning Province, currently working as a postdoc in the Institute of Modern Physics, CAS on lattice QCD at nonzero temperature and density; E-mail: stli@impcas.ac.cn

    Corresponding author: E-mail: hengtong.ding@mail.ccnu.edu.cn.
图(6)
计量
  • 文章访问数:  10
  • HTML全文浏览量:  4
  • PDF下载量:  1
  • 被引次数: 0
出版历程
  • 收稿日期:  2020-03-01
  • 修回日期:  2020-08-09

Chiral Crossover and Chiral Phase Transition Temperatures from Lattice QCD

doi: 10.11804/NuclPhysRev.37.2019CNPC65
    基金项目:  National Natural Science Foundation of China (11947237, 11775096)
    作者简介:

    (1990–), male, born in Dandong, Liaoning Province, currently working as a postdoc in the Institute of Modern Physics, CAS on lattice QCD at nonzero temperature and density; E-mail: stli@impcas.ac.cn

    通讯作者: E-mail: hengtong.ding@mail.ccnu.edu.cn.
  • 中图分类号: O571.53

摘要: 回顾了最近关于手征平滑过渡温度和手征相变温度的研究结果。首先给出了在零重子化学势能下的手征平滑过渡温度为156.5(1.5) MeV,其次,给出了在非零重子化学势能下手征相转变曲线的二阶及四阶曲率分别为0.012(4)和0.000(4)。接着讨论了在格点QCD中第一次得到的量子色动力学的手征相变温度。在热力学极限、连续极限及手征极限下,我们得到手征相变温度为132$^{(+3)}_{(-6)}$ MeV。

English Abstract

李胜泰, 丁亨通. 利用格点量子色动力学研究手征平滑过渡温度和手征相变温度[J]. 原子核物理评论. doi: 10.11804/NuclPhysRev.37.2019CNPC65
引用本文: 李胜泰, 丁亨通. 利用格点量子色动力学研究手征平滑过渡温度和手征相变温度[J]. 原子核物理评论. doi: 10.11804/NuclPhysRev.37.2019CNPC65
Shengtai LI, Hengtong DING. Chiral Crossover and Chiral Phase Transition Temperatures from Lattice QCD[J]. Nuclear Physics Review. doi: 10.11804/NuclPhysRev.37.2019CNPC65
Citation: Shengtai LI, Hengtong DING. Chiral Crossover and Chiral Phase Transition Temperatures from Lattice QCD[J]. Nuclear Physics Review. doi: 10.11804/NuclPhysRev.37.2019CNPC65
    • One of ultimate goals of lattice QCD calculations is to map out the QCD phase diagram[1]. The current understanding of the QCD phase structure at zero baryon chemical potential is sketched in the so-called Columbia plot (see Fig. 1). The horizontal axis of Fig. 1 is the mass of degenerate up and down quarks, while the vertical axis is the mass of the strange quark. The physical point $ (m_{\rm{u,d}}^{\rm{phy}}, m_{\rm{s}}^{\rm{phy}}) $ corresponding to a physical pion mass of $ 140 $ MeV has been confirmed to possess a crossover type transition but not a true phase transition[2]. As the strange quark mass keeps its physical quark mass value $ m_{\rm{s}}^{\rm{phy}} $ and the light quark masses move towards zero, the chiral phase transition is supposed to become a true phase transition[3]. Based on recent studies, the chiral phase transition in the chiral limit of light quarks in $ N_{\rm{f}} $=2+1 QCD is expected to be a second order phase transition belonging to an $ O(4) $ universality class[4-5]. Since the chiral phase transition temperature $ T_{\rm{c}}^0 $ is a fundamental quantity of QCD and QCD-inspired model calculations predict that $ T_{\rm{c}}^{0} $ is about 25 MeV smaller than the pseudo-critical temperature $ T_{\rm{pc}} $ at the physical point[6-7], the determination of $ T_{\rm{c}}^{0} $ is thus important. The chiral phase transition temperature is also relevant to the search for the elusive QCD critical end point (CEP). As seen from Fig. 2, the 2nd order O(4) phase transition in the chiral limit of light quarks will terminate at a tri-critical point ($ \mu_{\rm{B}}^{\rm{tri}} $, $ T_{\rm{c}}^{\rm{tri}} $) at a sufficiently large baryon chemical potential. It has been found that in the chiral limit the curvature of the transition line is negative up to the 2nd order of $ \mu_{\rm{B}} $[8-9], which suggests $ T_{\rm{c}}^{0} > T_{\rm{c}}^{\rm{tri}} $. While on the other hand, the crossover transition line at the physical quark mass will end at a critical end point which possesses a 2nd order phase transition belonging to a Z(2) universality class. This critical end point is what people are looking for and is connected to the tri-critical point via a transition line belonging to the Z(2) universality class. The tri-critical temperature $ T_{\rm{c}}^{\rm{tri}} $ is expected larger than the critical end point temperature $ T_{\rm{c}}^{\rm{CEP}} $ from model studies, i.e. $ T_{\rm{c}}^{\rm{tri}}-T_{\rm{c}}^{\rm{CEP}}\left(m_{q}\right) \propto m_{q}^{2 / 5} $[10-12]. So the chiral phase transition $ T_{\rm{c}}^{0} $ can be regarded as an upper bound of the critical temperature $ T_{\rm{c}}^{\rm{CEP}} $ at the CEP. The determination of $ T_{\rm{c}}^{0} $ is thus helpful to constrain the location of the CEP.

      Figure 1.  (color online)QCD phase structure in the mass quark plane[1].

      Figure 2.  (color online)QCD phase structure in the 3-D plane of temperature ($T$), quark mass ($m_{\rm{u,d}}$) and baryon chemical potential ($\mu_{\rm{B}}$)[13].

      In the next two sections we will summarize our recent studies on the determination of chiral crossover transition temperature at small baryon chemical potentials, and the chiral phase transition temperature.

    • Since the transition at the physical point is not a true phase transition there is no unique `critical' temperature whose definition can be arbitrary. While the chiral phase transition region of 3-flavor QCD in the lower-left corner of the Columbia plot (see Fig. 1) is small and far away from the physical point[14-17], the 2nd order O(4) transition line in the chiral limit of light quarks should have more influence to the thermodynamics at the physical point. Based on the remnant scaling behavior at the physical point we define the chiral crossover temperature through the peak/inflection points of many chiral observables such that all of them converge to a single value in the chiral limit, i.e. the chiral phase transition temperature. The continuum extrapolation of the chiral crossover transition temperature is shown in Fig. 3. Note that the results do not reply on critical exponents of any universality class and the determination of chiral crossover temperature is done, e.g. by extracting the peak location of light quark susceptibilities and inflection point of light quark chiral condensates etc at each lattice cutoff. It is not expected that the transition temperature extracted via these various chiral observables will converge to a rather precise point or value of 156.5±1.5 MeV even at the physical values of light quarks. In Fig. 4 we show the dependence of chiral crossover transition temperature on the baryon chemical potential up to the order of $ (\mu_{\rm{B}}/T)^4 $. The transition line at nonzero $ \mu_{\rm{B}} $ is parameterized as

      Figure 3.  (color online)The continuum extrapolation of chiral crossover transition temperature at the physical pion mass[18].

      Figure 4.  (color online)Chiral crossover transition line in (2+1)-flavor QCD[18].

      $$\begin{split} T_{\rm{pc}}(\mu_{\rm{B}}) =& T_{\rm{pc}}(\mu_{\rm{B}} = 0) \left[1-\kappa_2^{\rm{B}}\left(\frac{\mu_{\rm{B}}}{T_{\rm{pc}}(0)}\right)^2 -\right.\\&\left. \kappa_4^{\rm{B}}\left(\frac{\mu_{\rm{B}}}{T_{\rm{pc}}(0)}\right)^4\right], \end{split} $$ (1)

      we found $ \kappa^{\rm{B}}_2 $=0.012(4) and $ \kappa^{\rm{B}}_4 $=0.000(4). These results are consistent with other studies on the lattice[19-22]. Also shown in Fig. 4 are the freeze-out temperatures $ T_{\rm{f}} $ obtained at RHIC and LHC energies. The freeze-out temperature at the LHC energy obtained from particle yields is in very good agreement with chiral crossover temperature at vanishing baryon chemical potential obtained from lattice QCD computations, while $ T_{\rm{f}} $ at RHIC energies are also in good agreement to QCD transition line at $ \mu_{\rm{B}} $ larger than about 80 MeV and the 200 GeV data point is about 1.5 sigma away from the transition line.

    • To determine the chiral phase transition line we start by introducing the subtracted chiral condensates and their susceptibilities which are free from UV divergences,

      $$ \begin{split} M =& \frac{m_{\rm{s}}\left(\langle\bar{\psi} \psi\rangle_{l}-\frac{2 m_{l}}{m_{\rm{s}}}\langle\bar{\psi} \psi\rangle_{\rm{s}}\right) }{ f_{K}^{4}},\\ \chi_{M} \equiv& \frac{\partial M}{\partial H} \equiv \frac{m_{\rm{s}}^{2} \chi_{l, {\rm { subtot }}} }{ f_{K}^{4}}, \end{split}$$ (2)

      where $ H $ represents the breaking field of chiral symmetry and is defined as the ratio of the light quark mass to the strange quark mass, i.e. $ H = m_l/m_{\rm{s}} $. We have normalized the chiral observables by multiplying proper powers of the kaon decay constant $ f_{K} = 156.1/\sqrt{2} $ MeV.

      As shown in Fig. 5, the pseudo-critical temperature $ T_{\rm{pc}}(H) $ (peak location) decreases with decreasing pion mass, and the peak height at each pion mass $ \chi_{M}(T_{\rm{pc}}, H) $ increases with decreasing pion mass which is consistent with the $ O(4) $ scaling relation.

      Figure 5.  (color online)Chiral susceptibilities obtained from (2+1)-flavor QCD with various quark masses as a function of temperature on $N_{\tau}=8$ lattices[23].

      To determine the chiral phase transition temperature $ T_{\rm{c}}^{0} $, we employ a novel estimators $ T_{60}(V,H) $ which was introduced in Refs. [23, 4 and 5] and denotes the temperature satisfying $ \chi_{M}(T_{60}, H,V) = 0.6\times \chi_{M}(T_{\rm{pc}},H,V) $ with $ T_{60}<T_{\rm{pc}} $, and $ T_{60}(V,H) $ will converge to $ T_{\rm{c}}^{0} $ in the chiral limit and thermodynamic limit according to the following $ O(4) $ scaling relation,

      $$ \begin{split} T_{X}(H, L) =& T_{\rm{c}}^{0}\left(1+\left(\frac{z_{X}\left(z_{L}\right)}{z_{0}}\right) H^{1 / \beta \delta}\right) +\\& c_{X} H^{1-1 / \delta+1 / \beta \delta}, \quad X = 60, \delta. \end{split} $$ (3)

      The advantage of using $ T_{60} $ as well as another quantity $ T_{\delta} $ as estimators for the chiral phase transition temperature is that these estimators are insensitive to the universality class. Based on the above $ O(4) $ scaling relation, we have performed thermodynamic limit, continuum limit and chiral limit extrapolation of chiral phase transition temperature[23]. As shown in Fig. 6, we perform the chiral limit using the data points $ T_{60} (V\to \infty, N_{\tau}\to \infty) $, $ T_{\rm{pc}} (V\to \infty, N_{\tau}\to \infty) $. We have considered two types of the continuum limit, leaving out or including our coarsest lattice $ N_{\tau} $=6. Here the black square (triangle) points give the thermodynamic extrapolated and continuum extrapolated results of chiral crossover temperature $ T_{\rm{pc}} (H) $ obtained from simulations with lattices with $ N_{\tau} = 6,8,12 $ ($ N_{\tau} = 8,12 $), where the data points at $ m_{l} = m_{\rm{s}}/27 $ (the physical point) stands for the transition temperature $ T_{\rm{pc}}(1/27) = 156.5(1.5) $ MeV. The decreasing behavior of the pseudo-critical temperature with decreasing light quark mass obeys the following $ O(4) $