Paper
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An elementary stringy estimate of transport coefficients of large temperature QCD
Authors
Francesco Bigazzi,Aldo L. Cotrone
Pedro Avellaneda,Gianpiero RUSSO,Tat Ching Fung,Yo-Ming Hsieh,Brett Sanders,Olga Shlykova,Марина Батюшкина,Andrey Zaplatnikov,Juha-Pekka Pitkänen,Na Deng,Francisco Javier Gómez Díez,Yurii Baraniuk,Vadim Melikyan,Michelle French,Nadiia Pylypenko,Антон Столітній,Vladimir Diev,Aleksandra Kotlova,Oksana Vazemiller,Oleg Pugachev,Баира Горяева,Anton Pereverzev,Юрий Мизерницкий,Анатолий Ваганов,Галина Комкова,Алла Басова,Marios Karouzos,Novikov Yurii,Antonín Zůbek,Maria Urban,António Topa,Peter Goodyear,Marianna Levrints (Lőrincz),Irina Stoianenko,Larisa Maltseva,Васиф Гаибов,Roman Revunov,Aleksandr Naboka,Levi Pérez,Павел Иванов,Nadezhda Glazko,Sergey Zelentsov,Oksana Lapaeva,Maria Volkovq,Elena Babaeva,John Hooker,Sergey Feskov,Nadezhda Solovieva,Oksana Bilonko,Felipe Silveira,Robert Sparkes,Daniel Batcheldor,Martin Goodchild,Timothy Rodigas,Srinivasan Jagannathan,Taewoong Lee,Mahendra Ramachandran,Michael Johnson,Luis Garcés-Erice,Anatoly Dymarsky,David Van Wie,Igor Timoshenkov,Tetsuji Kimura,Paul Ashford,Ivo Iori,Rudy GARGANO,César Adán Almendárez Sánchez,Kay Nieselt,Vera Bulygina,Chang Boon Patrick Lee,Vladimir Shats,Stefano Di Vita,Nathaniel Butlin,Irene Valenzuela,Valentina Giangreco Puletti,Yi Jun Wang,Liudmila Kalinina,Nikolai Grishin,Dmitry Prasolov,Olga Eremchenko,MEEI-MEI CHEN,Anzhelika Golovacheva,David Anderson,Nina Seylanova,Natalia Morozova,Evgenii Sviazov,Sanchayeeta Borthakur,William J Fischer,John Salzer,Yusuke Iida,Jacob Lynn,Oscar Trippella,Kirsten Larson,Xian-Hui Ge,Світлана Костю,Wei Chen,Mary Hägg,Romain NAJMAN,Andrew Royston,Colin Rennie,Gianluca Inverso,Constantinos Papageorgakis,Alexander Smirnov,Florence Mahuteau-Betzer,Lukas Witkowski,Gerhard Kresbach,Julia Earl,James Lombardi,Alberto Mariotti,Mateus ,Cristina Ferraris,Ryan Fagan,DOROTHEA TSEKOURA,Artem Biblin,Doug Beare,Richard Garavuso,Jaromir Tosiek,Vadim Romanov,Myroslava Kulynych,Saglara Mirzaeva,Olena Bielova,Evgeny Semchenko,Yuching Li,Vitalii Kotsur,Marco Silva,Aimee Fullerton,進發 程,Mohamed Sirajudeen ,Takeshi Sugawara,roman kashkovskiy,David Nelms,Mauricio Sánchez Barragán,Aynharan Sinnarajah,Luoping Zhang,Herlina Herlina,Meira Chaves Pereira,Takashi HOSOMA,Olga Konakova,Lysyuk Vladimir,Ірина Топалова,Humberto Gomez,Merchela Wassim,MARIA DE LA FUENTE,shuzhong sheng,Ada Arendt,Adrian Dawson,Cristiano de Sousa Zanetti,Pauline Fornarelli,Anna / Ганна Kovalenko / Коваленко,Eliningaya Kweka,Ragab Mousa,HSIEN CHUNG LEE,Maik Hammerschmidt,秀雲 王,Pierre Bénard,Ahmet Onur Girisgin,Francesco La Vigna,JULIO PABLO REYES FERNANDEZ,Alexander Shiler,Igor Nekhaev,Michael Beck,N/A ,Mohamed Ghazy,Kaixian Wu,Oleh Zhytnyk,Яна Погребная,Michael Mueller,Lila Aleksander,Michael Orloff,Isaías A. Moraes,Regner Trampedach,Zhaoxing Meng,Artur Shirinian,fernandojorgeoliveira ribeiro,Olga Kravtsova,Natascha Greisel,Marek Pluta,PYSMAK VIKTORIIA,Victoria Kryukova,Artem Vinogradov,Elena Ryabova,Matias Leoni-Olivera,Samir Ead,Галина Чеботарева,Katarzyna Wiejak,Aristomenis Donos,Volodymyr Sysoiev,Lakshmanan Chelladurai,Andrey Kabatskov,Kun Li,ibrahim han,Marta Orselli,Vittorio FIORE,Min He,zhaodong xu,Alexander Kandyba,xiaozhi liu,Lance Labun,chunshan lin,Hiran V Nath,Paula Schnurr,Christine Linehan,Bliss Cunningham,Petrina La Faire,Adilet Yesbayev,Amilcar Bezerra,Michael Zhang,Roman Doronkin,Dalis Latipov,Céline Zwikel,瑀如 戴,John Gamache,Ferhun Caner,Лусине Степанян,Luiz Sá,SUNG EUN CHO,Юлия Панимаш,Jun Zhong,Tommy Taylor,Маргарита Бойко,Paul Reasor,KUO-YU TSUI,Oscar Arratia,CHEN-DONG TSO,Oleg Nemtsev,Pietro Maria Liuzzo,Olga Rubleva,Bhavishya Mittal,Julio M. 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arXiv:1006.4634v2 [hep-ph] 6 Jul 2010
KUL-TF-10/06
An elementary stringy estimate of transport
coefficients of large temperature QCD
Francesco Bigazzi and Aldo L. Cotrone
Institute for theoretical physics, K.U. Leuven; Celestijnenlaan 200D, B-3001 Leuven,
Belgium.
francesco.bigazzi@fys.kuleuven.be, aldo.cotrone@fys.kuleuven.be
Abstract
Modeling QCD at large temperature with a simple holographic five dimensional theory encoding
minimal breaking of conformality, allows for the calculation of all the transport coefficients, up to
second order, in terms of a single parameter. In particular, the shear and bulk relaxation times are
provided. The result follows by deforming the AdS background with a scalar dual to a marginally
relevant operator, at leading order in the deformation parameter.
1
KUL-TF-10/06
An elementary stringy estimate of transport
coefficients of large temperature QCD
Francesco Bigazzi and Aldo L. Cotrone
Institute for theoretical physics, K.U. Leuven; Celestijnenlaan 200D, B-3001 Leuven,
Belgium.
francesco.bigazzi@fys.kuleuven.be, aldo.cotrone@fys.kuleuven.be
Abstract
Modeling QCD at large temperature with a simple holographic five dimensional theory encoding
minimal breaking of conformality, allows for the calculation of all the transport coefficients, up to
second order, in terms of a single parameter. In particular, the shear and bulk relaxation times are
provided. The result follows by deforming the AdS background with a scalar dual to a marginally
relevant operator, at leading order in the deformation parameter.
1
1 Introduction and results
The evolution of the Quark-Gluon Plasma (QGP) produced in heavy ion collisions is fairly
well described, after a short period of thermalization and before hadronization, by relativistic
hydrodynamics, consistently with the strongly coupled regime of such system [1]. Numerical
simulations of the hydrodynamic evolution of the QGP require as input the value of the
transport coefficients. While recent simulations indicate that the evolution of the QGP
should be quite insensitive to most of the second order coefficients (see for example [2, 3, 4]),
it is definitely sensitive to the value of the shear viscosity [5], and can be influenced in a
sizable way by the bulk viscosity and possibly the relaxation times [3, 4, 6]. Moreover,
a complete characterization of the Quark-Gluon Plasma of QCD up to second order still
requires the knowledge of the whole set of coefficients.
There are currently no first-principle reliable calculations of almost all the second order
coefficients for QCD at strong coupling: lattice results give some estimates of the viscosities
and the shear relaxation time [7], but they are affected by considerable uncertainties (see
for example [8]). In fact, actual simulations, lacking solid data for QCD, make often use,
as benchmark values of the transport coefficients, of the ones derived from the gravitational
dual of N = 4 SYM [9] (in some cases together with the bound on the bulk viscosity proposed
in [10] and a relation for the relaxation times from [11]). While the N = 4 SYM values for
the “shear” coefficients are expected to be in the right ballpark for QCD, they still concern
an exactly conformal theory, and in particular the bulk viscosity and many of the second
order coefficients are not determined.
In order to improve this situation, the first step is to break conformal invariance. Since
QCD is approximately conformal in the temperature window 1.5Tc . T . 4Tc, the confor-
mality breaking effects can be treated perturbatively. In this situation, probably the simplest
way of modeling QCD holographically is by a theory where conformality is slightly broken
by a marginally relevant operator.
The aim of this note is to point out that, in such a scenario, all the transport coefficients up
to second order for the uncharged plasma are given in terms of a single parameter (weighting
the conformality breaking) by making use of the results in [12, 11]: they are collected in Table
1. In particular, the behavior of the shear and bulk relaxation times is briefly discussed in
section 1.2.
There are surely more precise ways of modeling holographically QCD (none of which
is of course completely correct). Nevertheless, the model considered in this note has the
considerable advantage of full calculability, providing one of the few examples in which all
the second order transport coefficients are determined. Moreover, the results in Table 1
hold for any theory with gravity dual, where conformality is broken at leading order by a
marginally (ir)relevant operator (dual to a scalar with the simplest possible potential (8)),
including the cascading plasma [13] and the D3D7 plasmas [14].
2
The evolution of the Quark-Gluon Plasma (QGP) produced in heavy ion collisions is fairly
well described, after a short period of thermalization and before hadronization, by relativistic
hydrodynamics, consistently with the strongly coupled regime of such system [1]. Numerical
simulations of the hydrodynamic evolution of the QGP require as input the value of the
transport coefficients. While recent simulations indicate that the evolution of the QGP
should be quite insensitive to most of the second order coefficients (see for example [2, 3, 4]),
it is definitely sensitive to the value of the shear viscosity [5], and can be influenced in a
sizable way by the bulk viscosity and possibly the relaxation times [3, 4, 6]. Moreover,
a complete characterization of the Quark-Gluon Plasma of QCD up to second order still
requires the knowledge of the whole set of coefficients.
There are currently no first-principle reliable calculations of almost all the second order
coefficients for QCD at strong coupling: lattice results give some estimates of the viscosities
and the shear relaxation time [7], but they are affected by considerable uncertainties (see
for example [8]). In fact, actual simulations, lacking solid data for QCD, make often use,
as benchmark values of the transport coefficients, of the ones derived from the gravitational
dual of N = 4 SYM [9] (in some cases together with the bound on the bulk viscosity proposed
in [10] and a relation for the relaxation times from [11]). While the N = 4 SYM values for
the “shear” coefficients are expected to be in the right ballpark for QCD, they still concern
an exactly conformal theory, and in particular the bulk viscosity and many of the second
order coefficients are not determined.
In order to improve this situation, the first step is to break conformal invariance. Since
QCD is approximately conformal in the temperature window 1.5Tc . T . 4Tc, the confor-
mality breaking effects can be treated perturbatively. In this situation, probably the simplest
way of modeling QCD holographically is by a theory where conformality is slightly broken
by a marginally relevant operator.
The aim of this note is to point out that, in such a scenario, all the transport coefficients up
to second order for the uncharged plasma are given in terms of a single parameter (weighting
the conformality breaking) by making use of the results in [12, 11]: they are collected in Table
1. In particular, the behavior of the shear and bulk relaxation times is briefly discussed in
section 1.2.
There are surely more precise ways of modeling holographically QCD (none of which
is of course completely correct). Nevertheless, the model considered in this note has the
considerable advantage of full calculability, providing one of the few examples in which all
the second order transport coefficients are determined. Moreover, the results in Table 1
hold for any theory with gravity dual, where conformality is broken at leading order by a
marginally (ir)relevant operator (dual to a scalar with the simplest possible potential (8)),
including the cascading plasma [13] and the D3D7 plasmas [14].
2
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