The curvature of the chiral pseudocritical line from LQCD: analytic continuation and Taylor expansion compared
Claudio Bonati Massimo D'Elia Francesco Negro Francesco Sanfilippo Kevin Zambello







XXVIIth International Conference on Ultrarelativistic Nucleus-Nucleus Collisions
 (Quark Matter 2018)












infnpisa,difipi]Claudio Bonati
infnpisa,difipi]Massimo D'Elia
infnpisa]Francesco Negro
infnroma]Francesco Sanfilippo
parma]Kevin Zambello


[infnpisa]INFN Sezione di Pisa, Largo B. Pontecorvo 3, I-56127 Pisa, Italy
[difipi]Università di Pisa, Largo B. Pontecorvo 3, I-56127 Pisa, Italy
[infnroma]INFN Sezione di Roma3, Via della Vasca Navale 84, I-00146 Roma, Italy
[parma]Università di Parma and INFN, Gruppo Collegato di Parma, Parco Area delle Scienze 7/A, I-43124 Parma, Italy


We present a determination of the curvature κ of the chiral
pseudocritical line from N_f=2+1 lattice QCD at the physical point
obtained by adopting the Taylor expansion approach.
Numerical simulations performed at three lattice spacings
lead to a continuum extrapolated curvature
κ = 0.0145(25), a value that is in excellent agreement with
continuum limit estimates obtained via analytic continuation
within the same discretization scheme,
κ = 0.0135(20).
The agreement between the two calculations 
is a solid consistency check for both methods.




QCD phase diagram Lattice QCD 
















§ INTRODUCTION


Despite its great theoretical and phenomenological relevance, the temperature - baryon chemical potential (T-μ_B)
QCD phase diagram is far from being fully understood.
Even first principle and non-perturbative approaches like Lattice QCD cannot directly access the μ_B>0 region
because of the sign problem.
Anyhow, within the Lattice QCD framework, two methods have been quite extensively adopted to explore
the small μ_B part of the phase diagram: analytic continuation (AC) and Taylor expansion (TE).
We focus on the quadratic (in μ_B) bending
of the pseudocritical line which separates the low-T confined and chirally broken phase from the high-T QGP phase.
The pseudocritical line can be parametrized as

    T_c(μ_B)/T_c=1-κ  (μ_B/T_c)^2  +  O(μ_B^4)  ,

where κ is the curvature of the line.
In the literature, results obtained with the two methods,
even though the same discretization of QCD is adopted,
seem to indicate the presence of a tension: TE tends
to yield smaller values, about κ∼ 0.006 
from Ref. <cit.>,
as opposed to what is found via AC, i.e. about κ∼ 0.014
from Ref. <cit.>.
Such a tension is of more than two standard deviations.
The agreement between the methods would be necessary to be able to
state that all the possible systematics are under control.
We compare AC and TE adopting the same discretization to directly test such agreement (see Ref. <cit.> for more details).




§ NUMERICAL SETUP AND RESULTS


We adopted the same discretization setup that we used in our previous studies <cit.>,
in which we discretized the N_f=2+1 QCD partition function Z
using the tree level Symanzik improved gauge action and the staggered stout smearing improved quark action. The gauge coupling and the bare quark masses are tuned to stay at the physical pion mass.

We based the determination of the crossover temperature on the study of the light quark condensate
⟨ψ̅ψ⟩_l=(T/V)·∂log Z/∂ m_l=
⟨u̅u⟩+⟨d̅d⟩,
where V is the spatial volume and m_l is the bare light quark mass.
Two different prescriptions have been adopted to handle the additive and multiplicative renormalization this observable is affected by

    ⟨ψ̅ψ⟩_r1 (T)≡[
    ⟨ψ̅ψ⟩_l - 2m_l/ m_s⟨s̅s⟩](T)/[⟨ψ̅ψ⟩_l- 2m_l/ m_s⟨s̅s⟩](T=0)   and   ⟨ψ̅ψ⟩_r2 (T)=m_l/m_π^4(⟨ψ̅ψ⟩_l (T) -⟨ψ̅ψ⟩_l (T=0)) ,

introduced respectively in Ref. <cit.> and Ref. <cit.> (the μ_B dependence of finite T observables is understood). This allowed us to check for possible systematics related to the specific renormalization prescription.


 §.§ Analytic continuation approach

We briefly summarize the results we obtained in Ref. <cit.> within the analytic continuation framework.
We performed numerical simulations at non zero imaginary baryon chemical potential (μ_B,I = i μ_B), which is a sign problem free theory. 
We adopted a setup with degenerate light quark chemical potentials  μ_l,I/(π T)=μ_B,I/(3π T)
and zero strange quark chemical potential μ_s,I=0.
At fixed μ_l,I/(π T) chosen in the set {0,0.2,0.24,0.275}, we computed ⟨ψψ⟩_r and the renormalized chiral susceptibility χ_ψψ^r
at several temperatures around T_c.

The crossover temperature has been estimated by identifying the inflection point of the condensate (see left panel of Fig.<ref>) and the peak of its susceptibility for all the explored values of μ_l,I/(π T).
As shown in the center panel of Fig. <ref>, we extracted the curvature κ
by fitting the data of T_c(μ_l,I/(π T)) with a linear function in (μ_l,I/(π T))^2.
The same procedure has been repeated on lattices with N_t=6, 8, 10, 12 and then the
continuum limit for κ taken assuming O(1/N_t^2) finite lattice spacing effects (right panel of Fig.<ref>).
Our final estimate, which takes into account several possible systematics (among which a procedure to approach the continuum limit in a different way from that described here) is κ=0.0135(20).






 §.§ Taylor expansion approach

At μ_B=0, we define the crossover temperature T_c as the inflection point of the renormalized quark condensate.
To investigate the small μ_B region, we consider the Taylor expansion of ⟨ψ̅ψ⟩_r to order μ_B^2

    ⟨ψ̅ψ⟩_r (T, μ_B) = ⟨ψ̅ψ⟩_r (T, 0) + μ_B^2 ∂⟨ψ̅ψ⟩_r/∂ ( μ_B^2 ) (T,0) + O(μ_B^4).

It is quite natural to extend the definition of T_c 
by looking for an inflection point at fixed μ_B≠ 0:
a point where
∂^2 ⟨ψ̅ψ⟩^r(T,μ_B)/ ∂ T^2 = 0.
Since the formula for κ that can be derived from this equation proved to be highly numerically demanding,
we considered also an alternative prescription for κ (introduced in Ref. <cit.>),
where the pseudo-critical temperature at μ_B≠ 0 is defined as the temperature
where the renormalized condensate remains at the same value as at T_c for μ_B  = 0: ⟨ψ̅ψ⟩^r (T, μ_B^2)|_T=T_c(μ_B^2)≡⟨ψ̅ψ⟩^r (T_c, 0).
The formulas for κ, that can be obtained by imposing these two conditions, read respectively

    κ_inflection = T_c∂^2/∂ T^2 ( ∂⟨ψ̅ψ⟩^r(T,μ_B)/∂ (μ_B^2)|_μ_B=0 ) |_T=T_c/∂^3/∂ T^3⟨ψ̅ψ⟩^r(T,0) |_T=T_c   and   κ_fixed ⟨ψψ⟩_r = T_c ∂⟨ψ̅ψ⟩^r/∂ (μ_B^2)|_μ_B = 0, T=T_c/∂⟨ψ̅ψ⟩^r/∂ T|_μ_B = 0, T = T_c

We performed simulations on 24^3× 6, 32^3× 8 and 40^3× 10 lattices
at several temperatures and we estimated the T-derivatives appearing in Eq. (<ref>) by
fitting data with suitable functions: atan, tanh and a cubic polynomial for the condensate and a lorentzian function, a quadratic polynomial and a cubic spline for its μ_B^2 derivative.
An example is reported in the left panel of Fig. <ref> for the μ_B^2 derivative of the condensate.
Even on our coarsest lattice, statistical uncertainty for κ_inflection is very large, about 30%; anyhow the estimate we get
is compatible with κ_fixed ⟨ψψ⟩_r.
With the present statistics data for κ_fixed ⟨ψψ⟩_r are, on the other hand, 
precise enough to perform the continuum limit extrapolation (see center panel of Fig. <ref>), leading to the continuum
estimate κ=0.0145(25).






§ CONCLUSIONS


As anticipated,
the results we find with the two approaches agree:
κ=0.0135(20) for analytic continuation and
κ=0.0145(25) for Taylor expansion.
Also recent results by the HotQCD collaboration
(as presented by P. Steinbrecher at the Quark Matter 2018 conference)
indicate that the tension is getting solved. We report in the right panel
of Fig. <ref> an updated summary plot of the determinations of κ
in the literature from Ref. <cit.>
and from the mentioned talk by P. Steinbrecher for the HotQCD collaboration at QM2018.





§ ACKNOWLEDGEMENT

FN acknowledges financial support from the INFN HPC_HTC project.




























 00
Endrodi2011
  G. Endrodi, Z. Fodor, S. D. Katz and K. K. Szabo,
  
  JHEP 1104, 001 (2011)
  [arXiv:1102.1356].


crow
  C. Bonati, M. D'Elia, M. Mariti, M. Mesiti, F. Negro and F. Sanfilippo,
  
  Phys. Rev. D 90, 114025 (2014)
  [arXiv:1410.5758].

corvo2
  C. Bonati, M. D'Elia, M. Mariti, M. Mesiti, F. Negro and F. Sanfilippo,
  
  Phys. Rev. D 92, no. 5, 054503 (2015)

  [arXiv:1507.03571].

Bonati:2018nut 
  C. Bonati, M. D'Elia, F. Negro, F. Sanfilippo and K. Zambello,
  
  arXiv:1805.02960 [hep-lat].
  
  

Cheng:2007jq
  M. Cheng, N. H. Christ, S. Datta, J. van der Heide, C. Jung, F. Karsch, O. Kaczmarek and E. Laermann et al.,
  
  Phys. Rev. D 77, 014511 (2008)
  [arXiv:0710.0354].


Kaczmarek2011
  O. Kaczmarek, F. Karsch, E. Laermann, C. Miao, S. Mukherjee, P. Petreczky,
  C. Schmidt, W. Soeldner and W. Unger,
  
  
  Phys. Rev. D 83, 014504 (2011)
  [arXiv:1011.3130].

ccp
  P. Cea, L. Cosmai and A. Papa,
  
  Phys. Rev. D 89, 074512 (2014)
  [arXiv:1403.0821].



ntc
R. Bellwied, S. Borsanyi, Z. Fodor, J. Günther, S. D. Katz, C. Ratti and K. K. Szabo,
  
  Phys. Lett. B 751, 559 (2015)

  [arXiv:1507.07510].

ccp2
  P. Cea, L. Cosmai and A. Papa,
  
  Phys. Rev. D 93, no. 1, 014507 (2016)

  [arXiv:1508.07599].

hegde
P. Hegde et al. [Bielefeld-BNL-CCNU Collaboration],
  
  PoS LATTICE 2015, 141 (2016)
  [arXiv:1511.03378].







 

