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Stress-Tensor Distribution in Quark-Antiquark SystemTakumi Iritani (RIKEN)
for FlowQCD Coll. Ryosuke Yanagihara, Masakiyo Kitazawa, Masayuki Asakawa (Osaka Univ.), Tetsuo Hatsuda (RIKEN)
YKIS2018b Symposium “Recent Developments in Quark-Hadron Science”, June 11-15, 2018, YITP, Kyoto Univ. Reference arXiv:1803.05656[hep-lat]
The First Observation of the Energy-Momentum Tensor
around static Quark-Antiquark
Pulling Pushing
mid-plane
z
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y
mid-plane
1. Introduction: Quark-Antiquark System 2. Energy-Momentum Tensor (EMT) and Gradient Flow 3. EMT around static Quark-Antiquark
1. Introduction: Quark-Antiquark System 2. Energy-Momentum Tensor (EMT) and Gradient Flow 3. EMT around static Quark-Antiquark
Static Quark-Antiquark System
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Wilson loop: time evolution of Quark-Antiquark system
Wilson Loop
Quark Antiquark
Coulomb + Linear pot.
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“Tube”-structure between Quark and Antiquark• Origin of a linear rising potential — “tube” structure • a tube-like structure is established by lattice QCD
• Dual-superconductor scenario? (Nambu, ’t Hooft, Mandelstam
Fig. Lattice QCD calc. by Leinweber-1.6
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in the 1970s)
How to Observe the Tube from Lattice QCD?
• Spatial correlation of “action density” & Wilson loop
Wilson loop
Action densityQuark Anti-QuarkSpace
Probing into QQbar System with Various Operators
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TI-Cossu-Hashimoto ‘13Fig. Shibata-Kondo-Kato-Shinohara ‘12
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color flux: Original Yang-Mills (L/ε =8 , z/ε =4 )
EzEyExBxByBz
longitudinal chromoelectric fields are dominant components
See also Cea et al.
How to Extract Quantitative Information?
• Tube structures are observed from lattice QCD, but • Action density or Chromo-Electric/Magnetic Fields
renormalization ? continuum limit ?what about the relation between interquark force ?
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1. Introduction: Quark-Antiquark System 2. Energy-Momentum Tensor (EMT) and Gradient Flow 3. EMT around static Quark-Antiquark
Energy-Momentum Tensor (EMT)
• : generator of Poincaré group
energy momentum
stress
pressure
Energy-Momentum Tensor (EMT)
• : generator of Poincaré group • conservation law • trace anomaly
• energy density • stress tensor
bulk thermodynamics
fluctuations
transport coeffs.
At finite temperature
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force per area
Problems of EMT in Lattice Gauge Theory
• Lorentz sym. is broken: definition is non-trivial
• dim. 4 operator — UV fluctuation is difficult to control
several strategies: ex. matching bulk thermodynamic quantities, shifted boundary conditions, …
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Gradient Flow
Lüscher & Weisz ‘11Diffusion Eq. to “Extra-dim.”
renormalized ops. of original theory
flowed ops. UV finite
4-dim. boundary
Extra dim.
(4+1)-dim.
)B�(t, x))t
= *g20�S[B]
�B�(t, x), B�(0, x) = A�(x)
small-t expansion
Renormalized EMT by Gradient FlowH. Suzuki ’13
with fermion H. Makino & H. Suzuki ‘14
flow time = renormalization scale<latexit sha1_base64="kS3a093Kos3vSCIHd5H80NtxmZk=">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</latexit>
dim. 4 ops.
coefficients<latexit sha1_base64="8UBNTrZ2VhOlscjDXBkshWD06Ak=">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</latexit>
<latexit sha1_base64="5Hlhzv5f9Ri/JGxR1krg3JrBckU=">AAACIHicbVDLTgIxFO3gC8cX6tJNIzHBDc4YE1kS3bjERB4JMyGdUqCh86C9QyST+QB/wx9wq3/gzrjUD/A7LDALAU/S5PScc3Pb40WCK7CsLyO3tr6xuZXfNnd29/YPCodHDRXGkrI6DUUoWx5RTPCA1YGDYK1IMuJ7gjW94e3Ub46ZVDwMHmASMdcn/YD3OCWgpU6h6HhEJv0UO2wU8zHOriX7wlEjCUkF0nOdssrWDHiV2Bkpogy1TuHH6YY09lkAVBCl2rYVgZsQCZwKlppOrFhE6JD0WVvTgPhMucnsMyk+00oX90KpTwB4pv6dSIiv1MT3dNInMFDL3lT8z2vH0Ku4CQ+iGFhA54t6scAQ4mkzuMsloyAmmhAquX4rpgMiCQXd38IWBfwxNU1TF2Mv17BKGpdl2yrb91fF6k1WUR6doFNUQja6RlV0h2qojih6Qi/oFb0Zz8a78WF8zqM5I5s5Rgswvn8BKlqi9Q==</latexit>
: running coupling in MS-bar scheme
<latexit sha1_base64="KM9bPdFfJtN6bdmWYUTfSjWdNsY=">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</latexit>
EMT using Gradient Flow
Lattice regularization
1. gradient flow
2. continuum limit
dim. 4 ops.
continuum limit
v
3. perturbative formula
UV finite
sufficiently smeared
Ex. Thermodynamics of SU(3) YM from EMT• Trace anomaly
• Entropy density
1. Calc. EMT from “flowed fields”
2. Continuum limit
3. Flow time zero limit
<latexit sha1_base64="jWj1U6ZF77jmdLJli49mTTQ78Gk=">AAACG3icbVDLSsNAFJ3UV42vqktBBotQQUoigi6LblwqWBWaGCbTqQ6dmYSZG2kJ2fkb/oBb/QN34taFP+B3OH0stPXAhcM593LvPXEquAHP+3JKM7Nz8wvlRXdpeWV1rbK+cWWSTFPWpIlI9E1MDBNcsSZwEOwm1YzIWLDruHs68K8fmDY8UZfQT1koyZ3iHU4JWCmqbF9GeSCzQGXFbSAJ3GuZCwL1ogb7uLcXVape3RsCTxN/TKpojPOo8h20E5pJpoAKYkzL91IIc6KBU8EKN8gMSwntkjvWslQRyUyYD/8o8K5V2riTaFsK8FD9PZETaUxfxrZzcKqZ9Abif14rg85xmHOVZsAUHS3qZAJDggeh4DbXjILoW0Ko5vZWTO+JJhRsdH+2GOC9wnVdG4w/GcM0uTqo+17dvzisNk7GEZXRFtpBNeSjI9RAZ+gcNRFFj+gZvaBX58l5c96dj1FryRnPbKI/cD5/AGj0oP0=</latexit>
Ex. Double Limit Analysis• Smearing length: • Fitting window: smeared & perturbative region
1. Continuum limit 2. Flow zero limit
fine lattice is mandatory
Ex. Entropy Density and Trace Anomaly in SU(3)
• Consistent with integral method (w/o using EMT). • EMT from Gradient Flow works well!
Ref. [1] Boyd et al. ’96, Ref. [2] Borsanyi et al. ‘12
FlowQCD ‘16
1. Introduction: Quark-Antiquark System 2. Energy-Momentum Tensor (EMT) and Gradient Flow 3. EMT around static Quark-Antiquark
Ex. Maxwell Stress Tensor
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the length of the arrow<latexit sha1_base64="T9TRsZs7RiMlTIGUuWCt66loArg=">AAACFnicbVDLSsNAFJ34rPEVdSVuBovgqiZF0GXRjcsK9gFNDJPJpB06yYSZiVjS4G/4A271D9yJW7f+gN/h9LGwrQcGDufcw517gpRRqWz721haXlldWy9tmJtb2zu71t5+U/JMYNLAnHHRDpAkjCakoahipJ0KguKAkVbQvx75rQciJOXJnRqkxItRN6ERxUhpybcO3VTwVHE4dJlOhcjvD+9z56xa+FbZrthjwEXiTEkZTFH3rR835DiLSaIwQ1J2HDtVXo6EopiRwnQzSVKE+6hLOpomKCbSy8cnFPBEKyGMuNAvUXCs/k3kKJZyEAd6MkaqJ+e9kfif18lUdOnlNEkzRRI8WRRlDOqTR33AkAqCFRtogrCg+q8Q95BAWOnWZrZIRR8L0zR1Mc58DYukWa04dsW5PS/XrqYVlcAROAanwAEXoAZuQB00AAZP4AW8gjfj2Xg3PozPyeiSMc0cgBkYX7+L9p73</latexit>
Pulling Pushing
Potential
EMT around static Quark-Antiquark
the renormalized EMT distribution
<latexit sha1_base64="rxGUOJaqYB5eDqzl13unkyb983E=">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</latexit>
Operators:
double limit
set up
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x
y
: APE smearing for spatial link variables & multi-hit procedure for temporal link<latexit sha1_base64="5hXxxCtq9K0IIdbUE8qVRJlKlrk=">AAACAXicbVDLSsNAFL2prxpfVZduBotQQUoigi6LblxW6QvaUCbTSTt2MgkzE7GErvwBt/oH7sStX+IP+B1O2yxs64ELh3Pu5d57/JgzpR3n28qtrK6tb+Q37a3tnd29wv5BQ0WJJLROIh7Jlo8V5UzQumaa01YsKQ59Tpv+8GbiNx+pVCwSNT2KqRfivmABI1gbqdEs3Z/VTruFolN2pkDLxM1IETJUu4WfTi8iSUiFJhwr1XadWHsplpoRTsd2J1E0xmSI+7RtqMAhVV46vXaMTozSQ0EkTQmNpurfiRSHSo1C33SGWA/UojcR//PaiQ6uvJSJONFUkNmiIOFIR2jyOuoxSYnmI0MwkczcisgAS0y0CWhui9LsaWzbtgnGXYxhmTTOy65Tdu8uipXrLKI8HMExlMCFS6jALVShDgQe4AVe4c16tt6tD+tz1pqzsplDmIP19QsHQZX7</latexit>
<latexit sha1_base64="NprR39ZVCtU2sMNGmhQASdJ1HMw=">AAACF3icbVDLSsNAFJ3UV42vqjvdDBahgpREBF0W3bis0Bc0pUym03boZBJmbqQlBPwNf8Ct/oE7cevSH/A7nLRd2OqBC4dz7uXee/xIcA2O82XlVlbX1jfym/bW9s7uXmH/oKHDWFFWp6EIVcsnmgkuWR04CNaKFCOBL1jTH91mfvOBKc1DWYNJxDoBGUje55SAkbqFIy8gMKREJLW0m3hB7Mk4LcE5Hp91C0Wn7EyB/xJ3Topojmq38O31QhoHTAIVROu260TQSYgCTgVLbS/WLCJ0RAasbagkAdOdZPpDik+N0sP9UJmSgKfq74mEBFpPAt90ZhfrZS8T//PaMfSvOwmXUQxM0tmifiwwhDgLBPe4YhTExBBCFTe3YjokilAwsS1s0cDHqW3bJhh3OYa/pHFRdp2ye39ZrNzMI8qjY3SCSshFV6iC7lAV1RFFj+gZvaBX68l6s96tj1lrzprPHKIFWJ8/SbCfWQ==</latexit>
: flowed link variables
<latexit sha1_base64="ZcBgI6uQjukCodmRvO7tPT+kdOE=">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</latexit>
Setup: SU(3) Yang-Mills (quenched QCD)
EMT around Quark-Antiquark
<latexit sha1_base64="ZqKYKFWqFTdzAtfcsqdD/Zn6Wmo=">AAACInicbVDLSsNAFJ34rPEVdelmsBTqpiQi6EYounFZoS9oY5hMp+00k0mYmYgl5Av8DX/Arf6BO3EluPY7nLZZ2NYDA4dz7uHeOX7MqFS2/WWsrK6tb2wWtsztnd29fevgsCmjRGDSwBGLRNtHkjDKSUNRxUg7FgSFPiMtP7iZ+K0HIiSNeF2NY+KGaMBpn2KktORZpbqX0lHGvdF9Wg5OM3gFu0zHe8gLIPfoTPWsol2xp4DLxMlJEeSoedZPtxfhJCRcYYak7Dh2rNwUCUUxI5nZTSSJEQ7QgHQ05Sgk0k2n38lgSSs92I+EflzBqfo3kaJQynHo68kQqaFc9Cbif14nUf1LN6U8ThTheLaonzCoIjjpBvaoIFixsSYIC6pvhXiIBMJKNzi3RSr6mJmmqYtxFmtYJs2zimNXnLvzYvU6r6gAjsEJKAMHXIAquAU10AAYPIEX8ArejGfj3fgwPmejK0aeOQJzML5/AaI8oxs=</latexit>
Distribution of the principal axes
the length of the arrow , finite flow time w/o cont. limit<latexit sha1_base64="T9TRsZs7RiMlTIGUuWCt66loArg=">AAACFnicbVDLSsNAFJ34rPEVdSVuBovgqiZF0GXRjcsK9gFNDJPJpB06yYSZiVjS4G/4A271D9yJW7f+gN/h9LGwrQcGDufcw517gpRRqWz721haXlldWy9tmJtb2zu71t5+U/JMYNLAnHHRDpAkjCakoahipJ0KguKAkVbQvx75rQciJOXJnRqkxItRN6ERxUhpybcO3VTwVHE4dJlOhcjvD+9z56xa+FbZrthjwEXiTEkZTFH3rR835DiLSaIwQ1J2HDtVXo6EopiRwnQzSVKE+6hLOpomKCbSy8cnFPBEKyGMuNAvUXCs/k3kKJZyEAd6MkaqJ+e9kfif18lUdOnlNEkzRRI8WRRlDOqTR33AkAqCFRtogrCg+q8Q95BAWOnWZrZIRR8L0zR1Mc58DYukWa04dsW5PS/XrqYVlcAROAanwAEXoAZuQB00AAZP4AW8gjfj2Xg3PozPyeiSMc0cgBkYX7+L9p73</latexit>
cf. classical electrodynamics
orange circles: Quark, Antiquark
First observation of “tube” structure by the EMT!
Pulling Pushing
Mid-plane
<latexit sha1_base64="4GGerwDRKwpmbj5Tpo5dXPvSz1g=">AAACoHicbVHLTsMwEHTCO7wKHLlYVDxOJUGVgAMSggs3QKKAaKrKcbetheNE9gZRonwNX8UP8B04bQW0sJZXo51ZrT0bpVIY9P0Px52ZnZtfWFzylldW19YrG5v3Jsk0hwZPZKIfI2ZACgUNFCjhMdXA4kjCQ/R8WfIPL6CNSNQdDlJoxaynRFdwhrbUrryHMcM+ZzK/K9o55/uFd0bDCHpC5amltHgtvAmN1gXdG54w9Gz6zYXYB2SjXPxoplRvb8UPM8XV60XJhKA63/Pblapf84dB/4JgDKpkHDftymfYSXgWg0IumTHNwE+xlTONgkuw/8kMpIw/sx40LVQsBtPKh2YWdNdWOrSbaHsV0mH1d0fOYmMGcWSV5cPNNFcW/+OaGXZPWrlQaYag+GhQN5MUE1puhnaEBo5yYAHjWti3Ut5nmnG0+5uYYrDciudZY4JpG/6C+6Na4NeC23r1/GJs0SLZJjvkgATkmJyTK3JDGoQ7S86hc+KcujvulXvt3o6krjPu2SIT4T59AT0iylo=</latexit>
mid-plane
z
x
y
degeneration in Maxwell
we focus on the mid-plane in cylindrical coordinates
<latexit sha1_base64="KNnaweqEzVcbftu+pDuLDRVNXk0=">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</latexit>
Renormalized Stress-Tensor in Mid-plane (1) R = 0.46 fm
<latexit sha1_base64="614n2+eDqul8UzBvV2WTgdi4SWI=">AAACZ3icdVHLSgMxFE3H9/hoVRDBTbQIuikzUtCFC9GNSytWhU4d7qS3NZrJjElGrMP8o1t/QPAH3Gpau/B5IHA45x5uchKlgmvjec8lZ2x8YnJqesadnZtfKFcWl851kimGTZaIRF1GoFFwiU3DjcDLVCHEkcCL6PZo4F/co9I8kWemn2I7hp7kXc7AWCms3AQCZE8gDWIw1wxEflZcnYZ5vV5sqe1ADc0wbwQRqLxRFDTQPMY795/Y4+M/sX3qhZWqV/OGoL+JPyJVMsJJWHkJOgnLYpSGCdC65XupaeegDGcCCzfINKbAbqGHLUslxKjb+bCTgm5apUO7ibJHGjpUvyZyiLXux5GdHDxB//QG4l9eKzPdvXbOZZoZlOxzUTcT1CR0UDDtcIXMiL4lwBS3d6XsGhQwY7/h2xZt+EPhuq4txv9Zw29yvlPzvZrfqFcPDkcVTZM1skG2iE92yQE5JiekSRh5Im/kvURKr07ZWXFWP0ed0iizTL7BWf8A7Rm7cA==</latexit>
<latexit sha1_base64="gm+WaKP9nfKmI2vwBt4pGVazhyM=">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</latexit>
• non-trivial degeneracies • partially restoration of trace anomaly
<latexit sha1_base64="KNnaweqEzVcbftu+pDuLDRVNXk0=">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</latexit>
<latexit sha1_base64="WFFxwdHIDWcNbMiEzBSF7uuA12E=">AAACGHicbVDLSsNAFJ3UV42vqMtuBotQNyURQTdC0Y3LCn1BG8JkOmmHziRhZiKWkIW/4Q+41T9wJ27d+QN+h5M2C9t6mIHDOfdy7z1+zKhUtv1tlNbWNza3ytvmzu7e/oF1eNSRUSIwaeOIRaLnI0kYDUlbUcVILxYEcZ+Rrj+5zf3uAxGSRmFLTWPicjQKaUAxUlryrMqAIzXGiKWtzEsHPNEvq4kzeA1tz6radXsGuEqcglRBgaZn/QyGEU44CRVmSMq+Y8fKTZFQFDOSmYNEkhjhCRqRvqYh4kS66eyIDJ5qZQiDSOgfKjhT/3akiEs55b6uzFeWy14u/uf1ExVcuSkN40SREM8HBQmDKoJ5InBIBcGKTTVBWFC9K8RjJBBWOreFKVLRx8w0TR2MsxzDKumc1x277txfVBs3RURlUAEnoAYccAka4A40QRtg8ARewCt4M56Nd+PD+JyXloyi5xgswPj6BU7Kn0k=</latexit>
cf. classical electrodynamics
mid-plane
z
x
y
<latexit sha1_base64="ATnmO05ZOuTLkwh4DBdbtjbI3os=">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</latexit>
cf. 6 GeV/fm3 ~ 1036 Pa
Renormalized Stress-Tensor in Mid-plane (2) R-dep.
<latexit sha1_base64="614n2+eDqul8UzBvV2WTgdi4SWI=">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</latexit>
<latexit sha1_base64="gm+WaKP9nfKmI2vwBt4pGVazhyM=">AAACcXicdVFdT9swFHUyxkfGoIyniRerFRLTpCpBSOwJIfayRzpRqNR00Y1721p1nGDfTFRRfij8gP2B/QHcjwc+r2Tr6Jx7fO3jtFDSUhjee/6HtY/rG5tbwaftzzu7jb0v1zYvjcCuyFVueilYVFJjlyQp7BUGIUsV3qTTn3P95i8aK3N9RbMCBxmMtRxJAeSopEGxAj1WyOMMaCJAVVf1n99JZUx9ZL7FZiEmVSdOwVSduuaxlRneBu/YYpogwXJ/54AzHiaNVtgOF8Vfg2gFWmxVl0njXzzMRZmhJqHA2n4UFjSowJAUCusgLi0WIKYwxr6DGjK0g2qRTs0PHTPko9y4pYkv2KeOCjJrZ1nqOuePsS+1OfmW1i9p9GNQSV2UhFosB41KxSnn86j5UBoUpGYOgDDS3ZWLCRgQ5D7k2RRL8q4OgsAFE72M4TW4Pm5HYTvqnLTOL1YRbbID1mRHLGKn7Jz9YpesywR78Ji35QXef/+rz/3mstX3Vp599qz874/13b5E</latexit>
non-trivial degeneracies
partially restoration of trace anomaly
mid-plane
z
x
y
<latexit sha1_base64="ATnmO05ZOuTLkwh4DBdbtjbI3os=">AAACNHicbVDLSgMxFM34rOOr6tJNsAi6KTMi1IWLohuXrfQFnTrcSdM2NJMZkoxYhvkUf8MfcKsfILgTXfoNpo+FtV4SOJxzLvfeE8ScKe04b9bS8srq2npuw97c2t7Zze/tN1SUSELrJOKRbAWgKGeC1jXTnLZiSSEMOG0Gw+ux3rynUrFI1PQopp0Q+oL1GAFtKD9f8jiIPqfYC0EPCPC0lt3d+qkXJuZlJ/LUkxODn1a9AGRazTJ8iR0/X3CKzqTwInBnoIBmVfHzX143IklIhSYclGq7Tqw7KUjNCKeZ7SWKxkCG0KdtAwWEVHXSyYEZPjZMF/ciab7QeML+7kghVGoUBsY5PkP91cbkf1o70b2LTspEnGgqyHRQL+FYR3icFu4ySYnmIwOASGZ2xWQAEog2mc5NUZo9ZLZtm2DcvzEsgsZZ0XWKbvW8UL6aRZRDh+gInSAXlVAZ3aAKqiOCHtEzekGv1pP1bn1Yn1PrkjXrOUBzZX3/AJG6q1Y=</latexit>
Potential
Interquark Force from the Stress-Tensor
EMT
<latexit sha1_base64="D2ymK/8iTeTtoKsxcYSoketvcYw=">AAACIXicbZDLSsNAFIYn9VbjLerSzWAV6sKSiKAboSiIy1rsBdpSJpNJO3RyYeZELCEv4Gv4Am71DdyJO3Hvczi9LGz1h4Gf/5zDOfO5seAKbPvTyC0sLi2v5FfNtfWNzS1re6euokRSVqORiGTTJYoJHrIacBCsGUtGAlewhju4GtUb90wqHoV3MIxZJyC9kPucEtBR1zq47rYDAn0ZpHEEpQxf4GPc9iWhqVcvVo+y1KtmXatgl+yx8F/jTE0BTVXpWt9tL6JJwEKggijVcuwYOimRwKlgmdlOFIsJHZAea2kbkoCpTjr+TYYPdeJhP5L6hYDH6e+JlARKDQNXd44uV/O1UfhfrZWAf95JeRgnwEI6WeQnAkOER2iwxyWjIIbaECq5vhXTPtEoQAOc2aKAP2SmaWowzjyGv6Z+UnLsknN7WihfThHl0R7aR0XkoDNURjeogmqIokf0jF7Qq/FkvBnvxsekNWdMZ3bRjIyvH8GhoqY=</latexit>
<latexit sha1_base64="a4RbdMMMvAwpuyLwwnKpVkmDA1c=">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</latexit>
the “action-at-a-distance” interquark force is derived from “local” stress tensor
Potential
non-trivial consistency check
Summary
EMT around static Quark-Antiquark• “First direct measurement” of
the stress-tensor distribution around the static Quark-Antiquark. • the “action-at-a-distance” force is derived
from the local properties from the stress-tensor • Deeper & Correct understandings of
the Quark-Antiquark system from local properties.
“Gradient Flow” is useful approach to the EMT Outlooks: Quark-Antiquark system at finite temperature, Stringy excitations of Quark-Antiquark system, Three Quark system, Full QCD, …
Backup
Setups
beta a [fm] Ns Nconf R1/a R2/a R3/a T/a
6.304 0.058 48 140 8 12 16 8
6.465 0.046 48 440 10 20 10
6.513 0.043 48 600 16 10
6.600 0.038 48 1500 12 18 24 12
6.819 0.029 64 1000 16 24 32 16
w0-scaling 0.46 fm 0.69 fm 0.92 fm
SU(3) Wilson gauge action
Wilson loop: APE smearing for the spatial links, & multi-hit procedure for the temporal link
Double Limit at the Center of Mass<latexit sha1_base64="VRoYFFF3gbTRiFgTqT+mJG4GnRk=">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</latexit>
(1) continuum limit (2) flow time zero limit
mid-plane
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x
y
Pressure distribution
z
x
y
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Double Limit of EoS (1) Continuum Limit
• different discretization— O(a2) systematic errors
• continuum limit— converge
Double Limit of EoS (2) Flow Time Zero Limit
• Smearing length: • Fitting window: smeared & perturbative region
linear fit Range-1: [0.01:0.015] Range-2: [0.005:0.015] Range-3: [0.01:0.02]
Integral Method J. Engels et al. ‘90
action:
integration:
exp. value of plaq.
scaling func.ex. Edwards-Heller-Klassen ‘98
Cross Section along z-axisfixed flow time w/o cont. limit
dot-dashed vertical line: location of the sources
The flux-tube at mid-plane (z=0) at R = R1 is affected by the peak structure around quark and anti-quark.
