Gradient  Flowと  エネルギー運動量テンソル  	Masakiyo  Kitazawa  (Osaka  U.)for  FlowQCD  Collabora?on

Asakawa,  Hatsuda,  Iritani,  Itou,  MK,  Suzuki  FlowQCD,  PRD90,011501R  (2014)		
「熱場の量子論とその応用」・理研・2014年9月4日	

momentum	
energy	Poincaresymmetry	stress	 pressure	:  nontrivial  observable        on  the  laUce	
Deﬁni?on  of  the  operator  is  nontrivial  ①  	because  of  the  explicit  breaking  of  Lorentz  symmetry	
ex:	Its  measurement  is  extremely  noisy② due  to  high  dimensionality  and  etc.	  	If  we  have	Thermodynamics	
direct  measurement  of  expecta?on  values	

Fluctua'ons	  and	  Correla'ons	
viscosity,  speciﬁc  heat,  ...  

If  we  have	Ø conﬁnement  stringØ EM  distribu?on  in  hadrons	

Hadron	  Structure	Ø vacuum  conﬁgura?onØ mixed  state  on  1st  transi?on	

Vacuum	  Structure	Gradient  Flow	YM  Gradient  Flow	t:  “ﬂow  ?me”
dim:[length2]	

Luescher,  2010	

YM  Gradient  Flow	

Luescher,  2010	

t:  “ﬂow  ?me”
dim:[length2]	p transform  gauge  ﬁeld  like  diﬀusion  equa?onp diﬀusion  lengthp This  is  NOT  the  standard  cooling/smearingp All  composite  operators  at  t>0  are  UV  ﬁnite  Luescher,Weisz,2011

Applica?ons  of  Gradient  Flow	① scale  seUng② running  coupling③ topology④ operator  construc?on⑤ autocorrela?on,  etc.	

Small  Flow  Time  Expansionof  Operators  and  EMT	Operator  Rela?on	

an  operator  at  t>0Luescher,  Weisz,  2011

remormalized  operatorsof  original  theory	Operator  Rela?on	

an  operator  at  t>0

Luescher,  Weisz,  2011

remormalized  operatorsof  original  theory	Construc?ng  EMT	p gauge-­‐invariant	  dimension	  4	  operators	

Suzuki,  2013

Construc?ng  EMT  2	

Suzuki  coeﬀs.	

Suzuki,  2013

Construc?ng  EMT  2	

Suzuki  coeﬀs.	

Remormalized	  EMT	Suzuki,  2013

Numerical  Analysis
on  the  LaUce	Gradient  Flow  Method	
laUce  regularizedgauge  theory	

gradient    ﬂow	measurement  on  the  laUce	
analy?c(perturba?ve)	
con?nuum  theory(with  dim.  reg.)

gradient    ﬂow	

con?nuum  theory(with  dim.  reg.)Caveats	

Gauge  ﬁeld  has  to  be  suﬃciently  smeared!	

laUce  regularizedgauge  theory	

gradient    ﬂow	measurement  on  the  laUce	
analy?c(perturba?ve)	
con?nuum  theory(with  dim.  reg.)

gradient    ﬂow	

con?nuum  theory(with  dim.  reg.)Caveats	

Gauge  ﬁeld  has  to  be  suﬃciently  smeared!	

laUce  regularizedgauge  theory	
Perturba?ve  rela?on  gradient    ﬂow	has  to  be  applicable!	
analy?c(perturba?ve)	
con?nuum  theory(with  dim.  reg.)

gradient    ﬂow	con?nuum  theory(with  dim.  reg.)in  con?nuum	non-­‐perturba?veregion	O(t)  eﬀect	

classical	

in  con?nuum	non-­‐perturba?ve
region	O(t)  eﬀect	

classical	

on  the  laUce	p tà0  limit  with  keeping  t>>a2  	Numerical  Simula?on	Ø SU(3)  YM  theoryØ Wilson  gauge  ac?on

twice  ﬁner  laUce

Simula'on	  1	  Simula'on	  2	  (arXiv:1312.7492)	

(new,  preliminary)	

• laUce  size:  323xNt• Nt  =  6,  8,  10• β  =  5.89  –  6.56
• ~300  conﬁgura?ons	

• laUce  size:  643xNt• Nt  =  10,  12,  14,  16• β  =  6.4  –  7.4• ~2000  conﬁgura?ons	

using	 SX8  @  RCNP

SR16000  @  KEK	using  BlueGeneQ  @  KEKeﬃciency  ~40%	!	

Thermodynamics	
Thermodynamics	
direct  measurement  of  expecta?on  values	“Trace  Anomaly”  at  T=1.65Tc	

Emergent  plateau!  	
Nt=6,8,10~300  confs.	

the  range  of  t  where  the  EMT  formula  is    successfully  used!	

“Trace  Anomaly”  at  T=1.65Tc	

Emergent  plateau!  	
Nt=6,8,10~300  confs.	

the  range  of  t  where  the  EMT  formula  is    successfully  used!	

Entropy  Density  at  T=1.65Tc	
Emergent  plateau!  	
systema?cerror	

Nt=6,8,10~300  confs.	

Direct  measurement  of  e+p  on  a  given  T!	
NO  integral  /  NO  vacuum  subtrac?on	Con?nuum  Limit	
Boyd+1996	

323xNtNt  =  6,  8,  10
T/Tc=0.99,  1.24,  1.65	

0	Con?nuum  Limit	
Boyd+1996	

323xNtNt  =  6,  8,  10
T/Tc=0.99,  1.24,  1.65	

Comparison  with  previous  studies	0	Numerical  Simula?on	Ø SU(3)  YM  theoryØ Wilson  gauge  ac?ontwice  ﬁner  laUceSimula'on	  1	  Simula'on	  2	  (arXiv:1312.7492)	

(new,  preliminary)	

• laUce  size:  323xNt• Nt  =  6,  8,  10• β  =  5.89  –  6.56
• ~300  conﬁgura?ons	

• laUce  size:  643xNt• Nt  =  10,  12,  14,  16• β  =  6.4  –  7.4• ~2000  conﬁgura?ons	

using	 SX8  @  RCNP

SR16000  @  KEK	using  BlueGeneQ  @  KEKeﬃciency  ~40%	!	

Entropy  Density  on  Finer  LaUces		!
!
yranimiPrelFlowQCD,2013
T=1.65Tc	

T  =  2.31Tc
643xNtNt  =  10,  12,  14,  162000  confs.

p The  wider  plateau  on  the  ﬁner  laUcesp Plateau  may  have  a  nonzero  slope

Con?nuum  Extrapola?on	 • T=2.31Tc• 2000  confs• Nt  =  10  ~  16	aà0  limit  with  ﬁxed  t/a2	
1312.7492	

Preliminary!!	
Nt=16	 14	 12	10	0	Con?nuum  extrapola?onis  stable	

EMT  Correlators	
Fluctua'ons	  and	  Correla'ons	
viscosity,  speciﬁc  heat,  ...  EMT  Correlator	
p Kubo  Formula:  T12  correlator  ßàshear  viscosity	Ø Hydrodynamics  describes  long  range  behavior  of  Tµν	p Energy  ﬂuctua?on  ßàspeciﬁc  heat	

EMT  Correlator  :  Noisy…	With  naïve  EMT  operators	

Nakamura,  Sakai,  PRL,2005

Nt=8
improved  ac?on~106  conﬁgura?ons	

Nt=16
standard  ac?on
5x104  conﬁgura?ons	…  no  signal	Energy  Correla?on  Func?on	
T=2.31Tc
b=7.2,  Nt=16
2000  confs
p=0  correlator	

Preliminary!!	
smeared	Energy  Correla?on  Func?on	
T=2.31Tc
b=7.2,  Nt=16
2000  confs
p=0  correlator	

Preliminary!!	
smeared	p τ independent  const.      à  energy  conserva?on	Energy  Correla?on  Func?on	
T=2.31Tc
b=7.2,  Nt=16
2000  confs
p=0  correlator	

Preliminary!!	
smeared	p speciﬁc  heat	à  Novel  approach  to  measure  speciﬁc  heat!	
Gavai,  Gupta,  Mukherjee,  2005

diﬀeren?al  method  /  cont  lim.	

Summary	Summary	
EMT	  formula	  from	  gradient	  ﬂow	This  formula  can  successfully  deﬁne  and  calculate  the  EMT  on  the  laUce
This  operator  provides  us  with  novel  approaches  to  measure  various  observables  on  the  laUce!
This  method  is  direct,  intui?ve  and  less  noisy	

Many	  FThermodynamics	
direct  measurement  of  expecta?on  values	

Fluctua'ons	  and	  Correla'ons	

uture	  Studies!
!	

viscosity,  speciﬁc  heat,  ...  

now  we  have	Ø conﬁnement  stringØ EM  distribu?on  in  hadrons	

Hadron	  Structure	Ø vacuum  conﬁgura?onØ mixed  state  on  1st  transi?on	

Vacuum	  Structure	and  scale  seUng,  full  QCD  Makino,Suzuki,2014,  etc.