Vol. 10 (2017)

Acta Physica Polonica B Proceedings Supplement

No 4MATTER–ANTIMATTER COEXISTENCE METHODFOR FINITE DENSITY QCD∗
Hideo Suganuma
Department of Physics, Graduate School of Science, Kyoto University
Kitashirakawaoiwake, Sakyo, Kyoto 606-8502, Japan(Received October 2, 2017)
We propose a “matter–antimatter coexistence method” for the finitedensity lattice QCD, aiming at a possible solution of the sign problem. In
this method, we consider matter and antimatter systems on two parallel
R4 -sheets in five-dimensional Euclidean space-time. For the matter systemM with a chemical potential µ ∈ C on a R4 -sheet, we also prepare theantimatter system M̄ with −µ∗ on the other R4 -sheet shifted in the fifth
direction. In the lattice QCD formalism, we introduce a correlation term
iag ÃνiagAν
in M̄ , such
between theν ≡e
P gauge variables Uν ≡ e † in M andP Ũ
1
2as Sλ ≡ x,ν 2λ{Nc − Re tr[Uν (x)Ũν (x)]} ' x 2 λa {Aaν (x) − Ãaν (x)}2with a real parameter λ. In the limit of λ → ∞, a strong constraintŨν (x) = Uν (x) is realized, and the total fermionic determinant is real andnon-negative. In the limit of λ → 0, this system goes to two separated
ordinary QCD systems with the chemical potential of µ and −µ∗ . On a
finite-volume lattice, if one takes a large enough value of λ, Ũν (x) ' Uν (x) isrealized and there occurs a phase cancellation approximately between two
fermionic determinants in M and M̄ , which is expected to suppress thesign problem and to make the lattice calculation possible. For the obtained
gauge configurations of the coexistence system, matter-side quantities areevaluated through their measurement only for the matter part M . By thecalculations with gradually decreasing λ and their extrapolation to λ = 0,
physical quantities in finite density QCD are expected to be estimated.
DOI:10.5506/APhysPolBSupp.10.989

1. Introduction
The lattice QCD Monte Carlo calculation has revealed many aspects of
the QCD vacuum and hadron properties in both zero and finite temperatures. At finite density, however, lattice QCD is not yet well-investigated,because of a serious problem called the “sign problem” [1, 2], which originates from the complex value including minus sign of the QCD action and∗Presented at “Excited QCD 2017”, Sintra, Lisbon, Portugal, May 7–13, 2017.

(989)

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H. Suganuma

the fermionic determinant at finite density, even in the Euclidean metric [3].
In fact, the Euclidean QCD action S[A, ψ, ψ̄; µ] at finite density with thechemical potential µ is generally complexZ/ + m + µγ4 )ψ ∈ C ,S A, ψ, ψ̄; µ = SG [A] + d4 x ψ̄(D(1)
with the gauge action SG [A] ∈ R and covariant derivative Dν ≡ ∂ ν + igAν .
Then, the action factor cannot be identified as a probability density in theQCD generating functional, unlike ordinary lattice QCD calculations.
In this paper, aiming at a possible solution of the sign problem, wepropose a new approach of a “matter–antimatter coexistence method” forlattice QCD at finite density with a general chemical potential µ ∈ C.
2. Matter–antimatter coexistence method
Our strategy is to use a cancelation of the phase factors of the fermionicdeterminants between a matter system with µ and an antimatter systemwith −µ∗ , and our method is based on the general property [3]∗
S A, ψ, ψ̄; µ = S A, ψ, ψ̄; −µ∗(2)for the Euclidean QCD action S[A, ψ, ψ̄; µ] in the presence of the chemical/ +m
potential µ ∈ C. Actually, the fermionic kernel DF corresponding to D†
generally satisfies DF = γ5 DF γ5 in lattice QCD, so that one finds∗= ψ̄ (DF − µ∗ γ4 ) ψ ,(3)Det (DF + µγ4 )∗ = Det (DF − µ∗ γ4 ) .

(4)ψ̄(DF + µγ4 )ψ

which leads to relation (2), and2.1. Definition and setup
In the “matter–antimatter coexistence method”, we consider matter andantimatter systems on two parallel R4 -sheets in five-dimensional Euclidean
space-time. For the matter system M with a chemical potential µ ∈ C on aR4 -sheet, we also prepare the antimatter system M̄ with −µ∗ on the otherR4 -sheet shifted in the fifth direction, as shown in Fig. 1.
We put an ordinary fermion field ψ(x) with the mass m and the gauge
variable Uν (x) ≡ eiagAν (x) at x ∈ R4 on the matter system M , and we put
the other fermion field Ψ (x) ≡ ψ(x+ 5̂) with the same mass m and the gauge
variable Ũν (x) ≡ eiagÃν (x) ≡ Uν (x + 5̂) on the antimatter system M̄ .

Matter–Antimatter Coexistence Method for Finite Density QCD

991

Fig. 1. The matter–antimatter coexistence system in five-dimensional Euclidean
space-time. We put the matter system M with µ, Uν (x) and ψ(x) on a R4 -sheet,and the antimatter system M̄ with −µ∗ , Ũν (x) = Uν (x + 5̂) and Ψ (x) = ψ(x + 5̂)on the other R4 -sheet shifted in the fifth direction.

In the lattice QCD formalism, we introduce a correlation term between
the gauge variables Uν (x) in M and Ũν (x) in M̄ at x ∈ R4 , such as
hioX n†Sλ ≡2λ Nc − Re tr Uν (x)Ũν (x)(5)
x,ν

with a real parameter λ (≥ 0), which connects two different situations:Ũν (x) = Uν (x) in λ → ∞ and two separated QCD systems in λ → 0. Near
the continuum limit, this additional term becomes
no2 Zno2X2aa4 1aa1λa'dx
λA(x)
−Ã
(x)
(6)
Sλ 'A(x)
−Ã
(x)
νν
ν
ν
22 physxwith λphys ≡ λa−2 .
In fact, the total lattice action in this method is written as
h i X h iX
S = SG [U ] +ψ̄ (DF [U ] + µγ4 ) ψ + SG Ũ +Ψ̄ DF Ũ − µ∗ γ4 Ψx+Xnhio
2λ Nc − Re tr Uν (x)Ũν† (x)

x(7)

x,ν

with the gauge action SG [U ] ∈ R and the fermionic kernel DF [U ] in lattice QCD. After integrating out the fermion fields ψ and Ψ , the generating
functional of this theory reads

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H. SuganumaZ
Z =−SG [U ]Z

DU e

Det (DF [U ] + µγ4 )−PionhNc −Re tr Uν (x)Ũν† (x)−Pnx,ν 2λh i
DŨ e−SG [Ũ ] Det DF Ũ − µ∗ γ4×eZZnh io=
DU DŨ e−(SG [U ]+SG [Ũ ]) Det (DF [U ] + µγ4 ) DF Ũ − µ∗ γ4×ex,ν 2λhioNc −Re tr Uν (x)Ũν† (x).

(8)

In the continuum limit, this generating functional is expressed asZZno/̃ + m − µ∗ γ4/ + m + µγ4 D
Zcont = DA DÃe−(SG [A]+SG [Ã]) Det D×e−

Rad4 x 12 λphys {Aaν (x)−Ãν (x)}2(9)

with the continuum gauge action SG [A] ∈ R and D̃ν ≡ ∂ ν + ig Ãν .
In the practical lattice calculation with the Monte Carlo method, thefermionic determinant in Z is factorized into its amplitude and phase factor asnhioZZP†−(SG [U ]+SG [Ũ ]) − x,ν 2λ Nc −Re tr Uν (x)Ũν (x)e
Z =DU DŨ en
h i
ohi
× Det (DF [U ] + µγ4 ) DF Ũ − µ∗ γ4Ophase U, Ũ , (10)and the phase factor of the total fermionic determinanthi∗
Ophase U, Ũ ≡ eiarg[Det{(DF [U ]+µγ4 )(DF [Ũ ]−µ γ4 )}] ∈ C(11)is treated as an “operator” instead of a probability factor, while all otherreal non-negative factors in Z are treated as the probability density.
2.2. Property and procedure
The additional term Sλ connects the following two different situations
as the two limits of the parameter λ.
1. In the limit of λ → ∞, a strong constraint Ũν (x) = Uν (x) is realized,
and the phase factors of two fermionic determinants Det(DF [U ] + µγ4 )and Det(DF [Ũ ] − µ∗ γ4 ) are completely cancelled, owing to Eq. (4).
Therefore, the total fermionic determinant is real and non-negativenhioDet (DF [U ] + µγ4 ) DF Ũ = U − µ∗ γ4≥ 0,(12)and the sign problem is absent [4]. Note, however, that this system
resembles QCD with an isospin chemical potential [5].

Matter–Antimatter Coexistence Method for Finite Density QCD

993

2. In the limit of λ → 0, this system goes to two separated ordinary
QCD systems with the chemical potential of µ and −µ∗ , although
the cancellation of the phase factors cannot be expected between thetwo fermionic determinants Det(DF [U ] + µγ4 ) and Det(DF [Ũ ] − µ∗ γ4 )for significantly different Uν (x) and Ũν (x), which are independently
generated in the Monte Carlo simulation.
On a four-dimensional finite-volume lattice, if an enough large value of λis taken, Ũν (x) ' Uν (x) is realized and there occurs the phase cancellation
approximately between the two fermionic determinants Det(DF [U ] + µγ4 )and Det(DF [Ũ ] − µ∗ γ4 ) in M and M̄ , so that one expects a modest behaviorof the phase factor Ophase [U, Ũ ] in Eq. (11), which leads to feasibility of thenumerical lattice calculation with suppression of the sign problem.
Once the lattice gauge configurations of the coexistence system are obtained with the most importance sampling in the Monte Carlo simulation,
matter-side quantities can be evaluated through their measurement only forthe matter part M with µ.
By performing the lattice calculations with gradually decreasing λ andtheir extrapolation to λ = 0, we expect to estimate the physical quantities
in finite density QCD with the chemical potential µ.
3. Summary, discussion and outlook
We have proposed a “matter–antimatter coexistence method” for thelattice calculation of finite density QCD. In this method, we have preparedmatter M with µ and antimatter M̄ with −µ∗ on two parallel R4 -sheets
in five-dimensional Euclidean space-time, and have introduced a correlationPPterm Sλ ≡ x,ν 2λ{Nc − Re tr[Uν (x)Ũν† (x)]} ' x 21 λa2 {Aaν (x) − Ãaν (x)}2
between the gauge variables Uν = eiagAν in M and Ũν = eiagÃν in M̄ . In thelimit of λ → ∞, owing to Ũν (x) = Uν (x), the total fermionic determinant isreal and non-negative, and the sign problem is absent. In the limit of λ → 0,
this system goes to two separated ordinary QCD systems with the chemical
potential of µ and −µ∗ .
For an enough large value of λ, Ũν (x) ' Uν (x) is realized and a phasecancellation approximately occurs between two fermionic determinants in
M and M̄ , which is expected to suppress the sign problem and to makethe lattice calculation possible. For the obtained gauge configurations of
the coexistence system, matter-side quantities can be evaluated by theirmeasurement only for the matter part M . By gradually reducing λ and theextrapolation to λ = 0, it is expected to obtain estimation of the physicalquantities in finite density QCD with µ.

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H. Suganuma

In this paper, we have demonstrated this method with taking Sλ in
Eq. (5) as the simplest correlation between Uν (x) in M and Ũν (x) in M̄ . In
this method, however, there is some variety on the choice of the correlation
between Uν (x) and Ũν (x). For instance, it may be interesting to consider
the other correlation term like!
io 3hXXn†S̄ξ ≡8ξ
Nc − Re tr Uν (x)Ũν (x)xZ'ν
4d x 81 a2 ξno2 3aaAν (x) − Ãν (x)(13)

with a dimensionless non-negative real parameter ξ. At the classical level,
this correlation is an irrelevant interaction and it gives vanishing contributions in the continuum limit a → 0, like the Wilson term − 12 arψ̄D2 ψ.
The next step is to perform the actual lattice QCD calculation at finite
density using this method. It would be useful to combine this method withthe other known ways such as the hopping parameter expansion [6], the complex Langevin method [1] and the reweighting technique [2]. For example,
if one utilizes the hopping parameter expansion, the quenched-level analysis
becomes possible in this method, since the additional term Sλ only includes
gauge variables.
Efficiency of this method would strongly depend on the system parameters, such as the space-time volume V , the quark mass m, the temperature T
and the chemical potential µ. In any case, this method is expected to enlarge
calculable area of the QCD phase diagram on (T, µ, m, V ).
The author would like to thank Prof. P. Bicudo for his useful commentand his hospitality during the Workshop. The author is supported in partby the Grant for Scientific Research [(C)15K05076] from the Ministry ofEducation, Science and Technology of Japan.

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[3] P. Hasenfratz, F. Karsch, Phys. Rep. 103, 219 (1984).
[4] H. Suganuma, arXiv:1705.07516 [hep-lat].
[5] D.T. Son, M.A. Stephanov, Phys. Rev. Lett. 86, 592 (2001).
[6] G. Aarts, E. Seiler, D. Sexty, I.-O. Stamatescu, Phys. Rev. D 90, 114505(2014).