We discuss the sign problem based on our recent two papers [1,2].

The sign problem is one of the grand challenges in theoretical physics, 

and it is also a problem relevant to recent heavy-ion collision experiments.

In heavy-ion collisions at colliding eneriges of $\sqrt{s_{NN}}=5-20$ GeV,

recent data seem to suggest the existence of the first order QCD phase transition,

while the existence of massive neutron stars implies that the EOS of dense matter

needs to be stiff enough. It is desired to obtain the QCD phase boundary and EOS

in the first-principles method, Monte Carlo simulations of lattice QCD, but the sign

problem in finite density QCD prevents us to obtain precise results.

Recently developed two methods, the complex Langevin method (CLM) 

and the Lefschetz thimble method (LTM), are promising. In these methods,

real variables are extended to complex, integration path/regions are modified 

from the original path, and can solve or at least circumvent the sign problem.

Still, we have problems when singular points are located near the original integration path.

In the first part, we discuss the Lefschetz thimbles in the NJL model with vector interaction [1].

The NJL model has been utilized to discuss the QCD phase diagram, but it is not easy

to apply CLM and LTM. Even in the mean field treatment of the NJL model,

we have many singular points coming from the fermion determinant. Furthermore, 

when we introduce the vector interaction, another sign problem arises from the Wick rotation

of the temporal component of the auxiliary vector field (auxiliary sign problem). 

We have found that these two problems can be in principle solved in LTM.

It is possible to obtain thimbles by requiring that the momentum integration path is chosen 

to be on the same Riemann sheet along the flow trajectories. 

The latter problem can be handled by complexifying the vector field.

In practice, however, the flow trajectories stop after some time near the fixed points.

In the second part, we propose a new method, the path optimization method (POM),

to search for the integration path [2].

Since the flow equation in LTM blows up at some point in most of the actions,

it is favorable to obtain the new integration path without solving the flow equation.

One of the natural ideas is to apply the variational method. We first give the trial 

function which parametrize the integration path, and optimize the trial function 

by minimizing the weight cancellation (or by minimizing the cost function).

We apply POM to a gaussian model [3]. We have found that the optimized

path agrees with the thimbles around the fixed points, the local maxima 

of the Boltzmann weights, and the analytic results are well reproduced

even in the parameter region where the sign problem is very severe and 

CLM is found to fail. Thus POM seems to be another promising tool for 

the sign problem. In the presentation, we also show some more recent results of POM.

[1] Lefschetz thimbles in fermionic effective models with repulsive vector-field, 

Yuto Mori, Kouji Kashiwa, Akira Ohnishi, arXiv:1705.03646 [hep-lat].

[2] Toward solving the sign problem with path optimization method, 

Yuto Mori, Kouji Kashiwa, Akira Ohnishi, arXiv:1705.05605 [hep-lat].

[3]  New Insights into the Problem with a Singular Drift Term in the Complex Langevin Method,

J. Nishimura and S. Shimasaki, Phys. Rev. D92, 011501 (2015), arXiv:1504.08359.

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