We study how the static quark-antiquark potential for Nf = 2+1 QCD at the physical point gets modified by the presence of a constant and uniform magnetic field. We observe an anisotropy to appear in the potential: it gets steeper in the directions transverse to the magnetic field than in the longitudinal one. By comparing to the case with zero magnetic field, we show that the string tension increases

(decreases) in the perpendicular (parallel) direction, while the absolute value of the Coulomb coupling and the Sommer parameter show the opposite behavior

Bonati, C.

D'Elia, M.

Mariti, M.

Mesiti, M.

Negro, F.

Sanfilippo, F.

English

We study how the static quark-antiquark potential for Nf = 2+1 QCD at the physical point gets modified by the presence of a constant and uniform magnetic field. We observe an anisotropy to appear in the potential: it gets steeper in the directions transverse to the magnetic field than in the longitudinal one. By comparing to the case with zero magnetic field, we show that the string tension increases (decreases) in the perpendicular (parallel) direction, while the absolute value of the Coulomb coupling and the Sommer parameter show the opposite behavior

BONATI, CLAUDIO

D'ELIA, MASSIMO

MARITI, MARCO

MESITI, MICHELE

Archivio della Ricerca - Università di Pisa

DOI 10.1393/ncc/i2015-15011-8Colloquia: IFAE 2014IL NUOVO CIMENTO 38 C (2015) 11Anisotropy of the QQ potential in a magnetic fieldC. Bonati(1), M. D’Elia(1), M. Mariti(1), M. Mesiti(1), F. Negro(1) andF. Sanfilippo(2)(1) Dipartimento di Fisica dell’Università di Pisa and INFN, Sezione di Pisa,Largo Pontecorvo 3, I-56127 Pisa, Italy(2) School of Physics and Astronomy, University of Southampton, SO17 1BJ Southampton, UKreceived 7 January 2015Summary. — We study how the static quark-antiquark potential for Nf = 2 + 1QCD at the physical point gets modified by the presence of a constant and uniformmagnetic field. We observe an anisotropy to appear in the potential: it gets steeperin the directions transverse to the magnetic field than in the longitudinal one. Bycomparing to the case with zero magnetic field, we show that the string tension in-creases (decreases) in the perpendicular (parallel) direction, while the absolute valueof the Coulomb coupling and the Sommer parameter show the opposite behavior.PACS 11.15.Ha – Lattice gauge theory.PACS 12.38.Aw – General properties of QCD (dynamics, confinement, etc.).PACS 12.38.Gc – Lattice QCD calculations.PACS 12.38.Mh – Quark-gluon plasma.1. – Introduction and motivationsThe properties of strong interactions in the presence of an intense magnetic back-ground field have attracted much interest in the recent past [1]. Such properties areexpected to be relevant for the physics of compact astrophysical objects, of non-centralheavy ion collisions (HIC) [2] and of the early Universe. In such contexts the magnitudeof the field spans from 1010 Tesla up to 1016 Tesla (|e|B ∼ 1 GeV2). Gluon fields, thoughnot directly coupled to electromagnetic fields, may be subjected to significant modifica-tions, through effective QED-QCD interactions mediated by quark loop effects. In thispaper, we study in a nonperturbative way how the confining inter-quark potential, thatemerges as a property of the gauge fields only, depends on a magnetic field |e|B [3].2. – Numerical resultsWe adopted the lattice QCD approach [4] in order to obtain fully non-perturbativeresults. We made use of a state-of-art discretization of Nf = 2 + 1 QCD by consideringthe stout smearing improved fermionic action and the tree-level improved Symanzikgauge action (see [3] for the discretization details). A constant and uniform magneticCreative Commons Attribution 4.0 License (http://creativecommons.org/licenses/by/4.0) 12 C. BONATI ETC.3 4 5 6 7 8ns0.30.40.50.60.70.80.9aV(nsa)b=0 b=24   XYb=24   Z0 0.2 0.4 0.6 0.8 1 1.2eB [GeV2]0.911.11.2r 0(B)/r 0(B=0)L=24   XYL=32   XYL=40   XYL=24   ZL=32   ZL=40   ZFig. 1. – Left: QQ potential both for |e|B = 0 and for eB = 0.7 GeV2 on the finest 404 lattice.Right: The Sommer parameter ratio defined in eq. (3) as a function of |e|B.field enters the QCD Lagrangian by modifying the quark covariant derivatives. On thelattice, that amounts to adding proper U(1) phases to the usual SU(3) links entering theDirac operator. We included the electromagnetic Abelian field as in [5]. We consideredT = 0, three lattice spacings a = 0.2173 fm, 0.1535 fm, 0.1249 fm and a symmetric latticeof fixed physical volume V = (a L)4  (5 fm)4, L = 24, 32, 40. We varied the magneticfield, oriented along the Z-axis, going up to |e|B ∼ 1 GeV2.We evaluated the expectation values of rectangular Wilson loops W (an, ant) and de-termined from them the static potential aV (an) (see [4] for a description of the method):aV (an) = limnt→∞log(〈W (an, ant)〉〈W (an, a(nt + 1))〉),(1)where n and nt are the spatial and temporal sides of the loop in lattice units. In fig. 1,left, we show the QQ potential as a function of the quark separation both at zero andnonzero magnetic field on the 404 lattice. In the case of a nonzero field we observe aclear anisotropic behavior, with a remarkable separation of the values of the potentialmeasured along the Z or XY directions. A comparison with the |e|B = 0 case showsthat the potential increases in the transverse directions and decreases in the longitudinalone. This is observed for all the explored setups, starting from |e|B  0.2 GeV2. Wefitted the potential, for each value of |e|B, according to the Cornell parameterization:aV (and̂) = ĉd + σ̂dn +αdn,(2)where d̂ is a versor along the Z or XY directions, σ̂d is the string tension, αd theCoulomb coupling. We can keep track of the dependence of the fitted parameters on|e|B, by normalizing them to their values at zero field and at the same lattice spacing:ROd = Od(|e|B)/O(|e|B = 0).(3)These ratios are shown in figs. 1, right, 2, left and 2, right, respectively for the Sommerparameter (see ref. [4]) Od = r̂0d, the string tension σ̂d and the Coulomb term αd. Dueto the anisotropy of aV (an), we get significant splittings of the ratios, which are of theorder of 10–20%. In particular, the string tension increases (decreases), as a function ofeB, in the trasverse (longitudinal) direction, while r̂0 and α show an opposite behavior.A similar scenario has been recently observed by means of an holographic approach [6].ANISOTROPY OF THE QQ POTENTIAL IN A MAGNETIC FIELD 30 0.2 0.4 0.6 0.8 1 1.2eB [GeV2 ]0.60.811.21.41.6σ(B)/σ(B=0)L=24   XYL=32   XYL=40   XYL=24   ZL=32   ZL=40   Z0 0,2 0,4 0,6 0,8 1 1,2eB [GeV2]0,60,811,21,4α(B)/α(B=0)L=24 XYL=32 XYL=40 XYL=24 ZL=32 ZL=40 ZFig. 2. – The string tension (left) and the Coulomb term (right) ratios defined in eq. (3) as afunction of |e|B.Our results show a mild dependence on the lattice spacing; however our present accu-racy does not permit a proper continuum extrapolation. We fitted the L = 40 data withthe following ansatz for the |e|B−dependence of the ratios: ROd = 1 + AOd (|e|B)COd .The best fit results are reported in [3] and shown as solid lines in the figures.3. – DiscussionOur findings might be of phenomenological relevance for non-central events in HICexperiments. For example, it was recently observed that the spectrum of mesons getsmodified by the presence of a strong magnetic background [7]. Moreover, even slightchanges in the energy levels induce sizeable corrections to cross sections, production anddecay rates [8]. Our result is relevant in this context: we can expect the spectrum tobe furtherly modified by the anisotropy of the QQ potential. Anyhow, in order to havevalid predictions, it is necessary to extend the investigation to finite temperature.∗ ∗ ∗FN acknowledges financial support from the EU under project Hadron Physics 3(Grant Agreement n. 283286). This work was partially supported by the INFN SUMAproject. Simulations have been performed on the Blue Gene/Q Fermi at CINECA, basedon the agreement between INFN and CINECA (under INFN projects PI12 and NPQCD).REFERENCES[1] Kharzeev D., Landsteiner K., Schmitt A. and Yee H. U., Lect. Notes Phys., 871(2013).[2] Skokov V., Illarionov A. Y. and Toneev V., Int. J. Mod. Phys. A, 24 (2009) 5925.[3] Bonati C., D’Elia M., Mariti M., Mesiti M., Negro F. and Sanfilippo F., Phys.Rev. D, 89 (2014) 114502.[4] Gattringer C. and Lang C. B., Lect. Notes Phys., 788 (2010) 1.[5] D’Elia M., Mukherjee S. and Sanfilippo F., Phys. Rev. D, 82 (2010) 051501.[6] Rougemont R., Critelli R. and Noronha J., Phys. Rev. D, 91 (2015) 066001arXiv:1409.0556 [hep-th].[7] Alford J. and Strickland M., Phys. Rev. D, 88 (2013) 105017.[8] Machado C. S., Navarra F. S., de Oliveira E. G., Noronha J. and Strickland M.,Phys. Rev. D, 88 (2013) 034009; Filip P., PoS CPOD, 2013 (2013) 035.

Anisotropy of the QQ ̄ potential in a magnetic field

Scientific Open-access Literature Archive and Repository

DOI 10.1393/ncc/i2015-15011-8Colloquia: IFAE 2014IL NUOVO CIMENTO 38 C (2015) 11Anisotropy of the QQ potential in a magnetic fieldC. Bonati(1), M. D’Elia(1), M. Mariti(1), M. Mesiti(1), F. Negro(1) andF. Sanfilippo(2)(1) Dipartimento di Fisica dell’Universita` di Pisa and INFN, Sezione di Pisa,Largo Pontecorvo 3, I-56127 Pisa, Italy(2) School of Physics and Astronomy, University of Southampton, SO17 1BJ Southampton, UKreceived 7 January 2015Summary. — We study how the static quark-antiquark potential for Nf = 2 + 1QCD at the physical point gets modified by the presence of a constant and uniformmagnetic field. We observe an anisotropy to appear in the potential: it gets steeperin the directions transverse to the magnetic field than in the longitudinal one. Bycomparing to the case with zero magnetic field, we show that the string tension in-creases (decreases) in the perpendicular (parallel) direction, while the absolute valueof the Coulomb coupling and the Sommer parameter show the opposite behavior.PACS 11.15.Ha – Lattice gauge theory.PACS 12.38.Aw – General properties of QCD (dynamics, confinement, etc.).PACS 12.38.Gc – Lattice QCD calculations.PACS 12.38.Mh – Quark-gluon plasma.1. – Introduction and motivationsThe properties of strong interactions in the presence of an intense magnetic back-ground field have attracted much interest in the recent past [1]. Such properties areexpected to be relevant for the physics of compact astrophysical objects, of non-centralheavy ion collisions (HIC) [2] and of the early Universe. In such contexts the magnitudeof the field spans from 1010 Tesla up to 1016 Tesla (|e|B ∼ 1GeV2). Gluon fields, thoughnot directly coupled to electromagnetic fields, may be subjected to significant modifica-tions, through effective QED-QCD interactions mediated by quark loop effects. In thispaper, we study in a nonperturbative way how the confining inter-quark potential, thatemerges as a property of the gauge fields only, depends on a magnetic field |e|B [3].2. – Numerical resultsWe adopted the lattice QCD approach [4] in order to obtain fully non-perturbativeresults. We made use of a state-of-art discretization of Nf = 2 + 1 QCD by consideringthe stout smearing improved fermionic action and the tree-level improved Symanzikgauge action (see [3] for the discretization details). A constant and uniform magneticCreative Commons Attribution 4.0 License (http://creativecommons.org/licenses/by/4.0) 12 C. BONATI ETC.3 4 5 6 7 8ns0.30.40.50.60.70.80.9aV(n sa)b=0 b=24   XYb=24   Z0 0.2 0.4 0.6 0.8 1 1.2eB [GeV2]0.911.11.2r 0(B)/r0(B=0)L=24   XYL=32   XYL=40   XYL=24   ZL=32   ZL=40   ZFig. 1. – Left: QQ potential both for |e|B = 0 and for eB = 0.7GeV2 on the finest 404 lattice.Right: The Sommer parameter ratio defined in eq. (3) as a function of |e|B.field enters the QCD Lagrangian by modifying the quark covariant derivatives. On thelattice, that amounts to adding proper U(1) phases to the usual SU(3) links entering theDirac operator. We included the electromagnetic Abelian field as in [5]. We consideredT = 0, three lattice spacings a = 0.2173 fm, 0.1535 fm, 0.1249 fm and a symmetric latticeof fixed physical volume V = (a L)4  (5 fm)4, L = 24, 32, 40. We varied the magneticfield, oriented along the Z-axis, going up to |e|B ∼ 1GeV2.We evaluated the expectation values of rectangular Wilson loops W (an, ant) and de-termined from them the static potential aV (an) (see [4] for a description of the method):aV (an) = limnt→∞log( 〈W (an, ant)〉〈W (an, a(nt + 1))〉),(1)where n and nt are the spatial and temporal sides of the loop in lattice units. In fig. 1,left, we show the QQ potential as a function of the quark separation both at zero andnonzero magnetic field on the 404 lattice. In the case of a nonzero field we observe aclear anisotropic behavior, with a remarkable separation of the values of the potentialmeasured along the Z or XY directions. A comparison with the |e|B = 0 case showsthat the potential increases in the transverse directions and decreases in the longitudinalone. This is observed for all the explored setups, starting from |e|B  0.2GeV2. Wefitted the potential, for each value of |e|B, according to the Cornell parameterization:aV (andˆ) = cˆd + σˆdn +αdn,(2)where dˆ is a versor along the Z or XY directions, σˆd is the string tension, αd theCoulomb coupling. We can keep track of the dependence of the fitted parameters on|e|B, by normalizing them to their values at zero field and at the same lattice spacing:ROd = Od(|e|B)/O(|e|B = 0).(3)These ratios are shown in figs. 1, right, 2, left and 2, right, respectively for the Sommerparameter (see ref. [4]) Od = rˆ0d, the string tension σˆd and the Coulomb term αd. Dueto the anisotropy of aV (an), we get significant splittings of the ratios, which are of theorder of 10–20%. In particular, the string tension increases (decreases), as a function ofeB, in the trasverse (longitudinal) direction, while rˆ0 and α show an opposite behavior.A similar scenario has been recently observed by means of an holographic approach [6].ANISOTROPY OF THE QQ POTENTIAL IN A MAGNETIC FIELD 30 0.2 0.4 0.6 0.8 1 1.2eB [GeV2 ]0.60.811.21.41.6σ(B)/σ(B=0)L=24   XYL=32   XYL=40   XYL=24   ZL=32   ZL=40   Z0 0,2 0,4 0,6 0,8 1 1,2eB [GeV2]0,60,811,21,4α(B)/α(B=0)L=24 XYL=32 XYL=40 XYL=24 ZL=32 ZL=40 ZFig. 2. – The string tension (left) and the Coulomb term (right) ratios defined in eq. (3) as afunction of |e|B.Our results show a mild dependence on the lattice spacing; however our present accu-racy does not permit a proper continuum extrapolation. We fitted the L = 40 data withthe following ansatz for the |e|B−dependence of the ratios: ROd = 1 + AOd (|e|B)COd .The best fit results are reported in [3] and shown as solid lines in the figures.3. – DiscussionOur findings might be of phenomenological relevance for non-central events in HICexperiments. For example, it was recently observed that the spectrum of mesons getsmodified by the presence of a strong magnetic background [7]. Moreover, even slightchanges in the energy levels induce sizeable corrections to cross sections, production anddecay rates [8]. Our result is relevant in this context: we can expect the spectrum tobe furtherly modified by the anisotropy of the QQ potential. Anyhow, in order to havevalid predictions, it is necessary to extend the investigation to finite temperature.∗ ∗ ∗FN acknowledges financial support from the EU under project Hadron Physics 3(Grant Agreement n. 283286). This work was partially supported by the INFN SUMAproject. Simulations have been performed on the Blue Gene/Q Fermi at CINECA, basedon the agreement between INFN and CINECA (under INFN projects PI12 and NPQCD).REFERENCES[1] Kharzeev D., Landsteiner K., Schmitt A. and Yee H. U., Lect. Notes Phys., 871(2013).[2] Skokov V., Illarionov A. Y. and Toneev V., Int. J. Mod. Phys. A, 24 (2009) 5925.[3] Bonati C., D’Elia M., Mariti M., Mesiti M., Negro F. and Sanfilippo F., Phys.Rev. D, 89 (2014) 114502.[4] Gattringer C. and Lang C. B., Lect. Notes Phys., 788 (2010) 1.[5] D’Elia M., Mukherjee S. and Sanfilippo F., Phys. Rev. D, 82 (2010) 051501.[6] Rougemont R., Critelli R. and Noronha J., Phys. Rev. D, 91 (2015) 066001arXiv:1409.0556 [hep-th].[7] Alford J. and Strickland M., Phys. Rev. D, 88 (2013) 105017.[8] Machado C. S., Navarra F. S., de Oliveira E. G., Noronha J. and Strickland M.,Phys. Rev. D, 88 (2013) 034009; Filip P., PoS CPOD, 2013 (2013) 035.

Anisotropy of the QQ potential in a magnetic field

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Anisotropy of the QQ potential in a magnetic field

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