Application of Tensor Network Scheme to Particle Physics CCS, Univ. - PowerPoint PPT Presentation

Application of Tensor Network Scheme to Particle Physics CCS, Univ. of Tsukuba Yoshinobu Kuramashi TNSAA 2018-2019@R-CCS, Kobe Japan, Dec. 3, 2018 1

Plan of Talk 1. Introduction to Particle Physics and Lattice Gauge Theory 2. An Overview of Current Status for application of TN Scheme to QFTs relevant to Particle Physics 3. Application to QFTs - 2D φ 4 Theory - One-Flavor Wilson Schwinger Model w/ and w/o θ term - 3D Finite Temperature Z 2 Gauge Theory 4. Summary

Introduction to Particle Physics Questions in history of mankind • What is the smallest component of matter? • What is the most fundamental interaction? � K.-I. Ishikawa@Hiroshima U. 3

Elementary Particles Known to Date +2/3e quark electric u c t � red,blue,green � charge −1/3e d s b µ t e −e electric lepton charge n e n µ n t 0 Higgs particle (LHC@CERN in 2012) 4

Fundamental Interactions Force Strength Gauge boson Theory Strong 1 Gluon QCD EM Photon QED 0.01 Weak 0.00001 Weak Boson Weinberg-Salam Gravity 10 −40 Graviton Superstring(?) Only four fundamental interactions is known so far 5

Unification of Fundamental Forces SuperString/ 10 −35 m � Beyond SM Grand/Unified/Theory/ SuperGravity/ 10 −32 m � (BSM) SuperSymmetry/ Standard/Model/ General/ Higgs particle was 10 −18 m � / Rela;vity/ QCD/ ElectroWeak/ the last piece of SM Spontaneous/ Symmetry/Breaking/ Standard model QED/ NonMAbelian/Gauge/ consists of Theory/ 10 −10 m � Quantum/Theory/ Special/Rela;vity/ gauge theories Maxwell/Eqs/ Classical/ ⇩ Mechanics/ Lattice QCD ElectroM/ Strong � Weak � /Gravity � 6 Magne;c �

Motivation of Lattice QCD Nonperturbative Phenomena in strong Interaction btw quraks • Confinement : quark can never be retrieved by itself • Asymptotic freedom � closer to each other, arbitrarily weaker • Hierarchy : 3 quarks ⇒ proton � neutron ⇒ nuclear • Finite temperature and finite density (finite T-μ): phase transition nucleus quark neutron proton finite T-μ 7

Difficulties in lattice gauge theory Partition function of lattice gauge theory after analytic integration of fermion field U : gauge field � D U det D ( { U } ) e − S g ( { U } ) Z = D : Dirac matrix � Z � Y e ¯ ψ D ½ U � ψ det D ½ U � ¼ d ψ n; α d ¯ ψ n; α n; α Expectation value of physical quantity Z � D U O ( { U, D − 1 } ) det D ( { U } ) e − S g ( { U } ) ⟨ O ⟩ = Monte Carlo should work for det D e −S >0 with importance sampling P = 1 Z det D ( { U } ) e − S g ( { U } ) In case that P is negative or a complex value, Importance sampling fails = statistical error becomes uncontrollable Sign problem / Complex action problem Another problem is computational cost for fermion system Direct treatment of Grassmann numbers is practically impossible

TN Scheme for Particle Physics Advantage • Free from sign problem and complex action problem in Monte Carlo method • Computational cost for L D system size ∝ D � log(L) • Direct treatment of Grassmann numbers • Direct measurement of Z itself Possible applications in particle physics Light quark dynamics in QED/QCD, Finite density QCD, Strong CP problem, Chiral gauge theories, Lattice SUSY etc. Disadvantage Computational cost increases for higher dimensions ⇒ better to start with lower (D≤3) dimensional models

Important Ingredients in Particle Physics • Quantum field theory • Gauge symmetry (U(1), SU(2), SU(3) etc.) • Fermion(quark, lepton), gauge boson(photon, gluon, weak boson), scalar particle(Higgs) • Spontaneous symmetry breaking It is important to investigate various lower dimensional models which contains the above ingredients

How to treat QFT w/ TN Scheme (1) Path-integral formalism � Scalar field(non-compact) How to discretize continuous d. o. f.? ⇒ Expansion w/ orthogonal functions. Shimizu Mod.Phys.Lett.A27(2012)1250035 Gauss-Hermite quadrature Kadoh et al. JHEP1803(2018)141, arXiv:1811.12376 � Gauge field(compact � Z N ,U(1),SU(2),SU(3) gauge groups) Character expansion � preserve gauge symmetry, good convergence Liu et al. PRD88(2013)056005 (w/o numerical demonstration) First successful calculation of 3D Z 2 gauge theory YK-Yoshimura arXiv:1808.08025[hep-lat] 11

How to treat QFT w/ TN Scheme (2) Path-integral formalism � Fermion field Nilpotency of Grassmann variables ⇒ finite numbers of terms in the expansion Coarse-graining procedure for Grassmann variables Shimizu-YK PRD90(2014)014508, Takeda-Yoshimura PTEP2015(2015)043B01 We have developed necessary tools to analyze QFT in path-integral formalism w/ tensor network scheme 12

Application of TN Scheme to Particle Physics (1) 2D models Ising model � Levin-Nave PRL99(2007)120601 X-Y model � Meurice et al. PRE89(2014)013308 CP(1)+θ � Kawauchi-Takeda PRD93(2016)114503 φ 4 theory(scalar field) � Shimizu Mod.Phys.Lett.A27(2012)1250035, Kadoh et al. arXiv:1811.12376 QED, QED+θ(fermion+U(1) gauge fields) � Shimizu-YK PRD90(2014)014508, PRD90(2014)074503, PRD97(2018)034502 Gross-Neveu model+μ(fermion) � Takeda-Yoshimura PTEP2015(2015)043B01 N=1 Wess-Zumino model(fermion+scalar fields) � Kadoh et al. JHEP1803(2018)141 13

Application of TN Scheme to Particle Physics (2) 3D models Ising � Xie et al. PRB86(2012)045139 Potts model � Wan et al. CPL31(2014)070503 Free Wilson fermion(fermion field) � Sakai-Takeda-Yoshimura PTEP2017(2017)063B07, Yoshimura et al. PRD97(2018)054511 Z 2 gauge theory(Z 2 gauge field) � YK-Yoshimura arXiv:1808.08025[hep-lat] 4D models Ising(φ 4 theory) � Akiyama et al. work in progress (parallel computation) 14

Selected Our Recent Work 1. 2D φ 4 theory Kadoh et al., arXiv:1811.12376 Scalar field, spontaneous breaking of Z 2 symmetry 2. One-Flavor Wilson Schwinger model w/ and w/o θ term Shimizu-YK, PRD90(2014)014508, PRD90(2014)074503, PRD97(2018)034502 Fermion+U(1) gauge fields, sign problem, complex action 3. 3D Finite Temperature Z 2 Gauge Theory YK-Yoshimura, arXiv:1808.08025[hep-lat] Gauge field, simplest gauge theory in 3D

Collaborators Y. Kuramashi, Y. Yoshimura U. Tsukuba S. Akiyama Y. Nakamura, (Y. Shimizu) R-CCS S. Takeda, R. Sakai Kanazawa U. D. Kadoh Chulalongkorn U./ Keio U. 16

TN Representation of 2D φ 4 theory (1) Kadoh et al. arXiv:1811.12376 Continuum action of 2D φ 4 theory 2 ( ∂ ρ φ ( x )) 2 + µ 2 � � 1 � 2 φ ( x ) 2 + λ 4 φ ( x ) 4 d 2 x 0 S cont . = Lattice action ⎧ ⎫ 2 ρ − φ n ) 2 + µ 2 1 n + λ ⎨ ⎬ � � 2 φ 2 0 4 φ 4 S = ( φ n +ˆ n 2 ⎩ ⎭ n ∈ Γ L ρ =1 Introduce a constant external field h to investigate spontaneous breaking of Z 2 symmetry � S h = S − h φ n , n ∈ Γ L Boltzmann weight is expressed as 2 e − S h = � � f ( φ n , φ n +ˆ ρ ) n ∈ Γ L ρ =1 2 ( φ 1 − φ 2 ) 2 − µ 2 � − 1 � − λ + h 0 φ 2 1 + φ 2 φ 4 1 + φ 4 � � � � f ( φ 1 , φ 2 ) = exp 4 ( φ 1 + φ 2 ) 2 2 8 16 ⇒ Need to discretize the continuous d. o. f.

TN Representation of 2D φ 4 theory (2) Kadoh et al. arXiv:1811.12376 Use of Gauss-Hermite quadrature � ∞ K d ye − y 2 g ( y ) ≈ � w α g ( y α ) −∞ α =1 Discretized version of partition function 2 � � � y 2 � � � � Z ( K ) = w α n exp f y α n , y α n +ˆ α n ρ { α } n ∈ Γ L ρ =1 SVD for f(φ,φ) K U α i σ i V † � f ( y α , y β ) = i β , i =1 Partition function with initial tensor � � Z ( K ) = T ( K ) x n t n x n − ˆ 1 t n − ˆ 2 n ∈ Γ L { x,t } � � � K T ( K ) ijkl = √ σ i σ j σ k σ l w α e y 2 α U α i U α j V † k α V † � l α . α =1

K dependence of <φ> Kadoh et al. arXiv:1811.12376 Expectation value of φ is calculated w/ insertion of an impurity tensor K T ( K ) ijkl = √ σ i σ j σ k σ l y α w α e y 2 α U α i U α j V † k α V † ˜ � l α , α =1 K dependence of <φ> near μ 0,c 2.2e-07 D =32 D =40 2e-07 D =48 1.8e-07 1.6e-07 1.4e-07 < φ > 1.2e-07 1e-07 λ=0.05, h=10 −12 , L=1024 8e-08 Symm. Phase near μ 0,c 6e-08 4e-08 2e-08 10 100 K K=256 is large enough 19

Susceptibility of <φ> Kadoh et al. arXiv:1811.12376 Critical point is determined from scaling property of susceptibility ⟨ φ ⟩ h,L − ⟨ φ ⟩ 0 ,L � µ 2 0 , c − µ 2 � − γ � � χ = A χ = lim h → 0 lim , 0 h L →∞ L dependence of <φ> h,L /h near μ 0,c h dependence of <φ> h,∞ /h near μ 0,c 1e+10 1e+10 h =10 -12 h =10 -10 1e+09 1e+09 h =10 -8 h =10 -6 1e+08 1e+08 1e+07 1e+07 1e+06 1e+06 < φ >/h < φ >/h 1e+05 1e+05 1e+04 1e+04 λ=0.05, D=32, K=256 λ=0.05, D=32, K=256 1e+03 1e+03 Symm. Phase near μ 0,c Symm. Phase near μ 0,c 1e+02 1e+02 1e+01 1e+01 1e+01 1e+02 1e+03 1e+04 1e+05 1e+06 1e+07 1e-12 1e-10 1e-08 1e-06 1e-04 1e-02 L h 20

Determination of Critical Coupling Kadoh et al. arXiv:1811.12376 Scaling property of susceptibility D dependence of λ/(μ c ) 2 6e-06 5e-06 4e-06 χ -1/1.75 3e-06 2e-06 λ=0.05, D=32, K=256 λ=0.05 1e-06 0 -0.1006180 -0.1006176 -0.1006172 -0.1006168 2 µ 0 Scaling property is well described by 2D Ising universality class (γ=1.75) Consider dimensionless quantity λ/(μ c ) 2 to take the continuum limit