Anisotropyof thequark-antiquarkpotential inamagneticfield

C. Bonati¹, M. D’Elia¹, M. Mariti¹†, M. Mesiti¹, F. Negro¹, F. Sanfilippo²¹Dipartimento di Fisica dell’Università di Pisa and INFN, Sezione di Pisa, Pisa, Italy

²School of physics and astronomy, University of Southampton, Southampton, United Kindgdom

XQCD 2014, Stony Brook, 19 - 21 June 2014

AbstractWe study the static quark-antiquark potential in the presence of an external magnetic field, using lattice QCD simulations. The external field breaks

the centrality of the potential, which has a different behavior in the directions parallel and perpendicular with respect to the magnetic field. In particularthe string tension is larger (smaller) in the perpendicular (parallel) direction. This effect might be relevant for the phenomenology of off-central heavyion collisions, leading to modifications of the heavy mesons properties, such as the mass spectrum. Phys. Rev. D 89, 114502 (2014)

QCD in the presence of strong magnetic fields◦ Electromagnetic background interacts only with quarks, however loop

effects can modify also gluon dynamics.◦ Magnetic fields at the strong scale (eB ' m2

π) can modify the QCDdynamics, leading to relevant non perturbative effects.

◦ Phenomenological contexts:. Class of neutron stars, the magnetars (eB ' 1010 Tesla).?

. Early stage of the Universe (eB ' 1016 Tesla).?

. Off-central heavy ion collisions (eB ' 1015 Tesla).?

◦ Useful to probes QCD properties→ Help to clarify many open questionsin the theory ( chiral and deconfinement transitions, θ-term, etc...).

Off-central heavy ion collisions◦ In off-central collisions strong mag-

netic fields are expected to be created:orthogonal to the reaction plane andalmost homogeneous.

◦ The nuclei contributions to the mag-netic field can be evaluate in simplemodels:?

eB ≈ aZαEMe−a′ζf(b, τ)

a, a′ constants, ζ nucleus rapidity,f(b, τ) function of the impact parame-ter (b) nucleus proper time (τ).

◦ RHIC→ Au-Au collisions with√s =

200 GeV per nucleon pair, expectedfields up to eB ' 0.02 GeV2.?

◦ LHC→ Pb-Pb collisions at center-of-mass energy

√s = 4.5 TeV per

nucleon pair, can reach fields up toeB ' 0.3 GeV2.?

◦ Higher√s gives stronger B fields,

b = 12 fmb = 8 fmb = 4 fm

τ(fm)

eB(M

eV2)

32.521.510.50

105

104

103

102

101

100

but shorter life time→ Electrical conductivity could play central roleto extend fields duration.?

Static q − q potential◦ Vqq confining potential: non perturbative property of QCD. For static

quarks, well described by the Cornell potential:

V (r) = c+α

r+ σr α Coulomb term, σ String tension

◦ From V (r) one qualitatively reproduces spectra of heavy quarkonia.◦ On the lattice, measure V (r) from Wilson loop, W (R, T ).

. Create a q − q pair at distance R

. Propagate it for a time interval T .

. Annihilate the pair.

〈W (R, T )〉 ' C exp (−TV (R))

◦ Extract the potential from large time limit:

T

R

τ

xi

V (r) = − limt→∞

W (r, t)

W (r, t+ 1)

◦ Static quark limit: Vqq due only to gauge fields → B fields modify Vqq?◦ B field breaks rotational invariance → Measure Vqq separating different

directions.◦ Fix B = Bz. Separate transverse (XY) and longitudinal (Z) directions

when calculating Wilson loops→ Construct the potential along differentdirections.

Simulation detailsL a(fm) β24 0.2173(4) 3.5532 0.1535(3) 3.6740 0.1249(3) 3.75

◦ Lattice QCD simulation, with Nf = 2 + 1flavous, physical quark masses and state ofart discretization.

◦ Exploratory study at T = 0.

◦ To implement B = Bz: addproper u(1) phases to gauge links:Uµ(n)→ Uµ(n)uµ(n).

◦ Smearing procedure to smoothgauge configurations:

. APE-smearing along spatiallinks.?

. HYP-smearing along tempo-ral links.?

0 5 10

n t

0.4

0.45

SM=0 XY

SM=8 XY

SM=24 XY

SM=0 Z

SM=8 Z

SM=24 Z

Results◦ For each B, fit the transverse and longitudi-

nal directions using:aV (

r

a; d) = cd +

αd( ra )

+ σdr

a

where d =XY, Z.◦ Extract the Sommer parameter r0 using:

αd = r20dσd − 1.65

◦ For each observable O, evaluate the ratiosfor each direction d:

ROd =Od(|e|B)

O(|e|B = 0)◦ Splitting of these ratios of the order 10 - 20 %◦ Fit L = 40 data, according to:

ROd = 1 +AOd(|e|B)COd

Oxy AOxy COxy χ2/dofr0xy -0.072(2) 0.79(5) 0.59σxy 0.29(2) 0.9(1) 1.14αxy -0.24(3) 0.7(2) 1.53

Oz AOz COz χ2/dofr0z 0.0161(6) 1.9(1) 1.28σz -0.34(1) 1.5(1) 0.92αz 0.24(3) 1.7(4) 0.32

3 4 5 6 7 8ns

0.3

0.4

0.5

0.6

0.7

0.8

0.9

aV

(nsa

)

eB = 0.0 GeV2

eB = 0.7 GeV2 XY

eB = 0.7 GeV2 XYZ

eB = 0.7 GeV2 Z

0 0.2 0.4 0.6 0.8 1 1.2

eB [GeV2]

0.9

1

1.1

1.2

r 0(B

)/r 0

(B=

0)

L=24 XYL=32 XYL=40 XYL=24 ZL=32 ZL=40 Z

0 0,2 0,4 0,6 0,8 1 1,2

eB [GeV2]

0,6

0,8

1

1,2

1,4

α(B

)/α

(B=

0)

L=24 XYL=32 XYL=40 XYL=24 ZL=32 ZL=40 Z

0 0.2 0.4 0.6 0.8 1 1.2

eB [GeV2

]

0.6

0.8

1

1.2

1.4

1.6

σ(B

)/σ

(B=

0)

L=24 XYL=32 XYL=40 XYL=24 ZL=32 ZL=40 Z

Conclusions• External B fields induce relevant anisotropy in the static Vqq, starting

from eB ' 0.2 GeV2.• With respect to B orientation, σ increase (decrease) in the transverse

(longitudinal) directions, r0 and α show an opposite behavior.◦ Investigate Vqq complete angular dependence with respect to B.◦ Finite temperature case → relevant for off-central heavy ion collisions.◦ Study of the flux tube profile along different directions.

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