Classical and quantum entropy of parton distributions

doi:10.1103/PhysRevD.97.094029

Title: Classical and quantum entropy of parton distributions

Abstract

We introduce the semiclassical Wehrl entropy for the nucleon as a measure of complexity of the multiparton configuration in phase space. This gives a new perspective on the nucleon tomography. We evaluate the entropy in the small-x region and compare with the quantum von Neumann entropy. We also argue that the growth of entropy at small x is eventually slowed down due to the Pomeron loop effect.

Authors:

Hagiwara, Yoshikazu ^[1]; Hatta, Yoshitaka ^[2]; Xiao, Bo-Wen ^[3]; Yuan, Feng ^[4]

Kyoto Univ. (Japan). Dept. of Physics
Kyoto Univ. (Japan). Yukawa Inst. for Theoretical Physics
Central China Normal Univ., Wuhan (China). Key Lab.of Quark and Lepton Physics (MOE) and Inst. of Particle Physics
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Nuclear Science Division

Publication Date:: 2018-05-29

Research Org.:: Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Nuclear Physics (NP) (SC-26)

OSTI Identifier:: 1439376

Alternate Identifier(s):: OSTI ID: 1461140

Grant/Contract Number:: AC02-05CH11231; 17J08072

Resource Type:: Published Article

Journal Name:: Physical Review D

Additional Journal Information:: Journal Volume: 97; Journal Issue: 9; Journal ID: ISSN 2470-0010

Publisher:: American Physical Society (APS)

Country of Publication:: United States

Language:: English

Subject:: 73 NUCLEAR PHYSICS AND RADIATION PHYSICS

Citation Formats


                    Hagiwara, Yoshikazu, Hatta, Yoshitaka, Xiao, Bo-Wen, and Yuan, Feng. Classical and quantum entropy of parton distributions.  United States: N. p., 2018. 
        Web.  doi:10.1103/PhysRevD.97.094029.

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                    Hagiwara, Yoshikazu, Hatta, Yoshitaka, Xiao, Bo-Wen, & Yuan, Feng. Classical and quantum entropy of parton distributions.  United States.  doi:10.1103/PhysRevD.97.094029.

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                    Hagiwara, Yoshikazu, Hatta, Yoshitaka, Xiao, Bo-Wen, and Yuan, Feng. Tue .  
        "Classical and quantum entropy of parton distributions".  United States.  doi:10.1103/PhysRevD.97.094029.

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@article{osti_1439376,

  title        = {Classical and quantum entropy of parton distributions},

  author       = {Hagiwara, Yoshikazu and Hatta, Yoshitaka and Xiao, Bo-Wen and Yuan, Feng},

  abstractNote = {We introduce the semiclassical Wehrl entropy for the nucleon as a measure of complexity of the multiparton configuration in phase space. This gives a new perspective on the nucleon tomography. We evaluate the entropy in the small-x region and compare with the quantum von Neumann entropy. We also argue that the growth of entropy at small x is eventually slowed down due to the Pomeron loop effect.},

  doi          = {10.1103/PhysRevD.97.094029},

  journal      = {Physical Review D},

  number       = 9,

  volume       = 97,

  place        = {United States},

  year         = {2018},

  month        = {5}

}

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DOI: 10.1103/PhysRevD.97.094029

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Works referenced in this record:

Quark Wigner distributions and orbital angular momentum in light-front dressed quark model
journal, July 2014

Mukherjee, Asmita; Nair, Sreeraj; Ojha, Vikash Kumar
Physical Review D, Vol. 90, Issue 1
DOI: 10.1103/PhysRevD.90.014024

Entanglement entropy and entropy production in the color glass condensate framework
journal, August 2015

Kovner, Alex; Lublinsky, Michael
Physical Review D, Vol. 92, Issue 3
DOI: 10.1103/PhysRevD.92.034016

Use of the Husimi distribution for nucleon tomography
journal, August 2015

Hagiwara, Yoshikazu; Hatta, Yoshitaka
Nuclear Physics A, Vol. 940
DOI: 10.1016/j.nuclphysa.2015.04.005

Elliptic gluon generalized transverse-momentum-dependent distribution inside a large nucleus
journal, December 2016

Zhou, Jian
Physical Review D, Vol. 94, Issue 11
DOI: 10.1103/PhysRevD.94.114017

Entropy production from chaoticity in Yang-Mills field theory with use of the Husimi function
journal, November 2016

Tsukiji, Hidekazu; Iida, Hideaki; Kunihiro, Teiji
Physical Review D, Vol. 94, Issue 9
DOI: 10.1103/PhysRevD.94.091502

Entropy for SU(3) c quark states
journal, May 2004

Miller, D. E.
The European Physical Journal C, Vol. 34, Issue 4
DOI: 10.1140/epjc/s2004-01746-0

On the relation between classical and quantum-mechanical entropy
journal, December 1979

Wehrl, Alfred
Reports on Mathematical Physics, Vol. 16, Issue 3
DOI: 10.1016/0034-4877(79)90070-3

Diffusive scaling and the high-energy limit of deep inelastic scattering in QCD at large
journal, July 2006

Hatta, Y.; Iancu, E.; Marquet, C.
Nuclear Physics A, Vol. 773, Issue 1-2
DOI: 10.1016/j.nuclphysa.2006.04.003

Wigner, Husimi, and generalized transverse momentum dependent distributions in the color glass condensate
journal, November 2016

Hagiwara, Yoshikazu; Hatta, Yoshitaka; Ueda, Takahiro
Physical Review D, Vol. 94, Issue 9
DOI: 10.1103/PhysRevD.94.094036

Semi-inclusive deep inelastic scattering at small-x
journal, November 2009

Marquet, Cyrille; Xiao, Bo-Wen; Yuan, Feng
Physics Letters B, Vol. 682, Issue 2
DOI: 10.1016/j.physletb.2009.10.099

Geometric Scaling as Traveling Waves
journal, December 2003

Munier, S.; Peschanski, R.
Physical Review Letters, Vol. 91, Issue 23
DOI: 10.1103/PhysRevLett.91.232001

Holographic pomeron and entropy
journal, July 2013

Stoffers, Alexander; Zahed, Ismail
Physical Review D, Vol. 88, Issue 2
DOI: 10.1103/PhysRevD.88.025038

Generalized parton correlation functions for a spin-1/2 hadron
journal, August 2009

Meißner, Stephan; Metz, Andreas; Schlegel, Marc
Journal of High Energy Physics, Vol. 2009, Issue 08
DOI: 10.1088/1126-6708/2009/08/056

Towards a Theory of Entropy Production in the Little and Big Bang
journal, March 2009

Kunihiro, T.; Muller, B.; Ohnishi, A.
Progress of Theoretical Physics, Vol. 121, Issue 3
DOI: 10.1143/PTP.121.555

Universal behavior of QCD amplitudes at high energy from general tools of statistical physics
journal, January 2005

Iancu, E.; Mueller, A. H.; Munier, S.
Physics Letters B, Vol. 606, Issue 3-4
DOI: 10.1016/j.physletb.2004.12.009

Pomeron loops in zero transverse dimensions
journal, May 2006

Shoshi, Arif I.; Xiao, Bo-Wen
Physical Review D, Vol. 73, Issue 9
DOI: 10.1103/PhysRevD.73.094014

Parton saturation at small x and in large nuclei
journal, October 1999

Mueller, A. H.
Nuclear Physics B, Vol. 558, Issue 1-2
DOI: 10.1016/S0550-3213(99)00394-6

Entropy production and isotropization in Yang–Mills theory using a quantum distribution function
journal, January 2018

Tsukiji, Hidekazu; Kunihiro, Teiji; Ohnishi, Akira
Progress of Theoretical and Experimental Physics, Vol. 2018, Issue 1
DOI: 10.1093/ptep/ptx186

Entropy, quantum decoherence and pointer states scalar “parton” fields
journal, February 1996

Elze, Hans-Thomas
Physics Letters B, Vol. 369, Issue 3-4
DOI: 10.1016/0370-2693(95)01538-8

Glittering Glasmas
journal, September 2009

Gelis, F.; Lappi, T.; McLerran, L.
Nuclear Physics A, Vol. 828, Issue 1-2
DOI: 10.1016/j.nuclphysa.2009.07.004

Nucleon parton distributions in a light-front quark model
journal, February 2017

Gutsche, Thomas; Lyubovitskij, Valery E.; Schmidt, Ivan
The European Physical Journal C, Vol. 77, Issue 2
DOI: 10.1140/epjc/s10052-017-4648-5

Generalized TMDs and the exclusive double Drell–Yan process
journal, August 2017

Bhattacharya, Shohini; Metz, Andreas; Zhou, Jian
Physics Letters B, Vol. 771
DOI: 10.1016/j.physletb.2017.05.081

Saturation effects in deep inelastic scattering at low $Q^{2}$ and its implications on diffraction
journal, November 1998

Golec-Biernat, K.; Wüsthoff, M.
Physical Review D, Vol. 59, Issue 1
DOI: 10.1103/PhysRevD.59.014017

On the Quantum Correction For Thermodynamic Equilibrium
journal, June 1932

Wigner, E.
Physical Review, Vol. 40, Issue 5
DOI: 10.1103/PhysRev.40.749

Quark Wigner distributions and orbital angular momentum
journal, July 2011

Lorcé, C.; Pasquini, B.
Physical Review D, Vol. 84, Issue 1
DOI: 10.1103/PhysRevD.84.014015

Accessing the gluon Wigner distribution in ultraperipheral $p A$ collisions
journal, August 2017

Hagiwara, Yoshikazu; Hatta, Yoshitaka; Pasechnik, Roman
Physical Review D, Vol. 96, Issue 3
DOI: 10.1103/PhysRevD.96.034009

Probing the Small- $x$ Gluon Tomography in Correlated Hard Diffractive Dijet Production in Deep Inelastic Scattering
journal, May 2016

Hatta, Yoshitaka; Xiao, Bo-Wen; Yuan, Feng
Physical Review Letters, Vol. 116, Issue 20
DOI: 10.1103/PhysRevLett.116.202301

Gluon saturation and entropy production in proton–proton collisions
journal, November 2011

Kutak, Krzysztof
Physics Letters B, Vol. 705, Issue 3
DOI: 10.1016/j.physletb.2011.09.113

Quark orbital angular momentum from Wigner distributions and light-cone wave functions
journal, June 2012

Lorcé, Cédric; Pasquini, Barbara; Xiong, Xiaonu
Physical Review D, Vol. 85, Issue 11
DOI: 10.1103/PhysRevD.85.114006

Quantum decoherence, entropy and thermalization in strong interactions at high energy. (I). Noisy and dissipative vacuum effects in toy models
journal, February 1995

Elze, Hans-Thomas
Nuclear Physics B, Vol. 436, Issue 1-2
DOI: 10.1016/0550-3213(94)00523-H

Dynamical entropy of dense QCD states
journal, February 2013

Peschanski, Robi
Physical Review D, Vol. 87, Issue 3
DOI: 10.1103/PhysRevD.87.034042

Statistical model for the nucleon structure functions
journal, July 1996

Bhalerao, R. S.
Physics Letters B, Vol. 380, Issue 1-2
DOI: 10.1016/0370-2693(96)00516-3

Deep inelastic scattering as a probe of entanglement
journal, June 2017

Kharzeev, Dmitri E.; Levin, Eugene M.
Physical Review D, Vol. 95, Issue 11
DOI: 10.1103/PhysRevD.95.114008

Gluon production in current-nucleus and nucleon-nucleus collisions in a quasi-classical approximation
journal, September 1998

Kovchegov, Yuri V.; Mueller, A. H.
Nuclear Physics B, Vol. 529, Issue 1-2
DOI: 10.1016/S0550-3213(98)00384-8

Multiplicity distributions in different pseudorapidity intervals at a CMS energy of 540 GeV
journal, October 1985

Alner, G. J.; Alpgård, K.; Anderer, P.
Physics Letters B, Vol. 160, Issue 1-3
DOI: 10.1016/0370-2693(85)91491-1

Unitarity and the BFKL pomeron
journal, March 1995

Mueller, A. H.
Nuclear Physics B, Vol. 437, Issue 1
DOI: 10.1016/0550-3213(94)00480-3

A statistical model of structure functions and quantum chromodynamics
journal, December 1989

Mac, E.; Ugaz, E.
Zeitschrift f�r Physik C Particles and Fields, Vol. 43, Issue 4
DOI: 10.1007/BF01550946

Husimi distribution for nucleon tomography
journal, January 2016

Hatta, Yoshitaka; Hagiwara, Yoshikazu
EPJ Web of Conferences, Vol. 112
DOI: 10.1051/epjconf/201611201010

Fock space distributions, structure functions, higher twists, and small x
journal, March 1999

McLerran, Larry; Venugopalan, Raju
Physical Review D, Vol. 59, Issue 9
DOI: 10.1103/PhysRevD.59.094002

A statistical approach for polarized parton distributions
journal, April 2002

Bourrely, C.; Soffer, J.; Buccella, F.
The European Physical Journal C, Vol. 23, Issue 3
DOI: 10.1007/s100520100855

Universality of unintegrated gluon distributions at small $x$
journal, May 2011

Dominguez, Fabio; Marquet, Cyrille; Xiao, Bo-Wen
Physical Review D, Vol. 83, Issue 10
DOI: 10.1103/PhysRevD.83.105005

Exploiting the equivalence of reggeon field theory in zero transverse dimensions and reaction–diffusion processes
journal, February 2007

Bondarenko, S.; Motyka, L.; Mueller, A. H.
The European Physical Journal C, Vol. 50, Issue 3
DOI: 10.1140/epjc/s10052-007-0218-6

Notes on the orbital angular momentum of quarks in the nucleon
journal, February 2012

Hatta, Yoshitaka
Physics Letters B, Vol. 708, Issue 1-2
DOI: 10.1016/j.physletb.2012.01.024

Quark imaging in the proton via quantum phase-space distributions
journal, April 2004

Belitsky, A. V.; Ji, Xiangdong; Yuan, Feng
Physical Review D, Vol. 69, Issue 7
DOI: 10.1103/PhysRevD.69.074014

Charged-particle multiplicity in proton–proton collisions
journal, July 2010

Grosse-Oetringhaus, Jan Fiete; Reygers, Klaus
Journal of Physics G: Nuclear and Particle Physics, Vol. 37, Issue 8
DOI: 10.1088/0954-3899/37/8/083001

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Similar Records

Title: Classical and quantum entropy of parton distributions

Abstract

Citation Formats

Quark Wigner distributions and orbital angular momentum in light-front dressed quark model journal, July 2014

Entanglement entropy and entropy production in the color glass condensate framework journal, August 2015

Use of the Husimi distribution for nucleon tomography journal, August 2015

Elliptic gluon generalized transverse-momentum-dependent distribution inside a large nucleus journal, December 2016

Entropy production from chaoticity in Yang-Mills field theory with use of the Husimi function journal, November 2016

Entropy for SU(3) c quark states journal, May 2004

On the relation between classical and quantum-mechanical entropy journal, December 1979

Diffusive scaling and the high-energy limit of deep inelastic scattering in QCD at large journal, July 2006

Wigner, Husimi, and generalized transverse momentum dependent distributions in the color glass condensate journal, November 2016

Semi-inclusive deep inelastic scattering at small-x journal, November 2009

Geometric Scaling as Traveling Waves journal, December 2003

Holographic pomeron and entropy journal, July 2013

Generalized parton correlation functions for a spin-1/2 hadron journal, August 2009

Towards a Theory of Entropy Production in the Little and Big Bang journal, March 2009

Universal behavior of QCD amplitudes at high energy from general tools of statistical physics journal, January 2005

Pomeron loops in zero transverse dimensions journal, May 2006

Parton saturation at small x and in large nuclei journal, October 1999

Entropy production and isotropization in Yang–Mills theory using a quantum distribution function journal, January 2018

Entropy, quantum decoherence and pointer states scalar “parton” fields journal, February 1996

Glittering Glasmas journal, September 2009

Nucleon parton distributions in a light-front quark model journal, February 2017

Generalized TMDs and the exclusive double Drell–Yan process journal, August 2017

Saturation effects in deep inelastic scattering at low Q 2 and its implications on diffraction journal, November 1998

On the Quantum Correction For Thermodynamic Equilibrium journal, June 1932

Quark Wigner distributions and orbital angular momentum journal, July 2011

Accessing the gluon Wigner distribution in ultraperipheral p A collisions journal, August 2017

Probing the Small- x Gluon Tomography in Correlated Hard Diffractive Dijet Production in Deep Inelastic Scattering journal, May 2016

Gluon saturation and entropy production in proton–proton collisions journal, November 2011

Quark orbital angular momentum from Wigner distributions and light-cone wave functions journal, June 2012

Quantum decoherence, entropy and thermalization in strong interactions at high energy. (I). Noisy and dissipative vacuum effects in toy models journal, February 1995

Dynamical entropy of dense QCD states journal, February 2013

Statistical model for the nucleon structure functions journal, July 1996

Deep inelastic scattering as a probe of entanglement journal, June 2017

Gluon production in current-nucleus and nucleon-nucleus collisions in a quasi-classical approximation journal, September 1998

Multiplicity distributions in different pseudorapidity intervals at a CMS energy of 540 GeV journal, October 1985

Unitarity and the BFKL pomeron journal, March 1995

A statistical model of structure functions and quantum chromodynamics journal, December 1989

Husimi distribution for nucleon tomography journal, January 2016

Fock space distributions, structure functions, higher twists, and small x journal, March 1999

A statistical approach for polarized parton distributions journal, April 2002

Universality of unintegrated gluon distributions at small x journal, May 2011

Exploiting the equivalence of reggeon field theory in zero transverse dimensions and reaction–diffusion processes journal, February 2007

Notes on the orbital angular momentum of quarks in the nucleon journal, February 2012

Quark imaging in the proton via quantum phase-space distributions journal, April 2004

Charged-particle multiplicity in proton–proton collisions journal, July 2010

Quark Wigner distributions and orbital angular momentum in light-front dressed quark model
journal, July 2014

Entanglement entropy and entropy production in the color glass condensate framework
journal, August 2015

Use of the Husimi distribution for nucleon tomography
journal, August 2015

Elliptic gluon generalized transverse-momentum-dependent distribution inside a large nucleus
journal, December 2016

Entropy production from chaoticity in Yang-Mills field theory with use of the Husimi function
journal, November 2016

Entropy for SU(3) c quark states
journal, May 2004

On the relation between classical and quantum-mechanical entropy
journal, December 1979

Diffusive scaling and the high-energy limit of deep inelastic scattering in QCD at large
journal, July 2006

Wigner, Husimi, and generalized transverse momentum dependent distributions in the color glass condensate
journal, November 2016

Semi-inclusive deep inelastic scattering at small-x
journal, November 2009

Geometric Scaling as Traveling Waves
journal, December 2003

Holographic pomeron and entropy
journal, July 2013

Generalized parton correlation functions for a spin-1/2 hadron
journal, August 2009

Towards a Theory of Entropy Production in the Little and Big Bang
journal, March 2009

Universal behavior of QCD amplitudes at high energy from general tools of statistical physics
journal, January 2005

Pomeron loops in zero transverse dimensions
journal, May 2006

Parton saturation at small x and in large nuclei
journal, October 1999

Entropy production and isotropization in Yang–Mills theory using a quantum distribution function
journal, January 2018

Entropy, quantum decoherence and pointer states scalar “parton” fields
journal, February 1996

Glittering Glasmas
journal, September 2009

Nucleon parton distributions in a light-front quark model
journal, February 2017

Generalized TMDs and the exclusive double Drell–Yan process
journal, August 2017

Saturation effects in deep inelastic scattering at low $Q^{2}$ and its implications on diffraction
journal, November 1998

On the Quantum Correction For Thermodynamic Equilibrium
journal, June 1932

Quark Wigner distributions and orbital angular momentum
journal, July 2011

Accessing the gluon Wigner distribution in ultraperipheral $p A$ collisions
journal, August 2017

Probing the Small- $x$ Gluon Tomography in Correlated Hard Diffractive Dijet Production in Deep Inelastic Scattering
journal, May 2016

Gluon saturation and entropy production in proton–proton collisions
journal, November 2011

Quark orbital angular momentum from Wigner distributions and light-cone wave functions
journal, June 2012

Quantum decoherence, entropy and thermalization in strong interactions at high energy. (I). Noisy and dissipative vacuum effects in toy models
journal, February 1995

Dynamical entropy of dense QCD states
journal, February 2013

Statistical model for the nucleon structure functions
journal, July 1996

Deep inelastic scattering as a probe of entanglement
journal, June 2017

Gluon production in current-nucleus and nucleon-nucleus collisions in a quasi-classical approximation
journal, September 1998

Multiplicity distributions in different pseudorapidity intervals at a CMS energy of 540 GeV
journal, October 1985

Unitarity and the BFKL pomeron
journal, March 1995

A statistical model of structure functions and quantum chromodynamics
journal, December 1989

Husimi distribution for nucleon tomography
journal, January 2016

Fock space distributions, structure functions, higher twists, and small x
journal, March 1999

A statistical approach for polarized parton distributions
journal, April 2002

Universality of unintegrated gluon distributions at small $x$
journal, May 2011

Exploiting the equivalence of reggeon field theory in zero transverse dimensions and reaction–diffusion processes
journal, February 2007

Notes on the orbital angular momentum of quarks in the nucleon
journal, February 2012

Quark imaging in the proton via quantum phase-space distributions
journal, April 2004

Charged-particle multiplicity in proton–proton collisions
journal, July 2010