Effects of magnetic fields on q q interactions [1607.08160] - - PowerPoint PPT Presentation
Effects of magnetic fields on q q interactions [1607.08160] C.Bonati 1 , M.DElia 1 , M.Mariti 1 , M.Mesiti 1 , F.Negro 1 , A.Rucci 1 , F.Sanfilippo 2 1 Department of Physics of University of Pisa and INFN Pisa, Italy 2 School of Physics and
[1607.08160]
C.Bonati1, M.D’Elia1, M.Mariti1, M.Mesiti1, F.Negro1, A.Rucci1, F.Sanfilippo2
1Department of Physics of University of Pisa and INFN Pisa, Italy 2School of Physics and Astronomy, University of Southampton, UK
Lattice 2016, 34th International Symposium
28 July 2016
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Introduction
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Effects of the magnetic field on the static potential at T=0
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What happens at finite temperatures? (T < Tc)
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Conclusions
QCD with strong magnetic fields eB ≃ m2
π ∼ 1015−16 T
Non-central heavy ion collisions with eB ∼ 1015T [Skokov et al. ’09] Possible production in early universe eB ∼ 1016T [Vachaspati ’91] In heavy ion collisions Expected eB ≃ 0.3 GeV2 at LHC in Pb+Pb at √sNN=4.5TeV and b=4fm Timescales depend on thermal medium properties (most pessimistic case: 0.1-0.5 fm/c) Spatial distribution of the field and lifetime are still debated
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An external magnetic field B on the lattice is introduced through abelian parallel transports uµ(n) Abelian phases enter the Lagrangian by modifying the covariant derivative Uµ(n) → Uµ(n)uµ(n) External field is fixed: non-propagating fields, no kinetic term Periodic boundary conditions lead to the quantization condition |qmin|B = 2πb a2NxNy b ∈ Z
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In the confining phase at low temperatures, the Q ¯ Q interaction is well described by the Cornell potential VC(r) = −α r + σr + V0 σ ≃ (440MeV)2 α ∼ 0.4 On the lattice: At T=0 it can be extracted from the Wilson loop aV(a n) = − lim
nt→∞ log
W( n, nt + 1) W( n, nt)
F(a n, T) ≃ −aNt logTrL†( r + n)TrL( n) what about the effects of B on the potential? (a first study: [Bonati et al. ’14])
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◆✉♠❡r✐❝❛❧ s❡t✉♣
Parameters extracted from the continuum limit at B = 0 α = 0.395(22) √σ = 448(20) MeV r0 = 0.489(20) fm
0.01 0.02 0.03 0.04 0.05
a
2 [fm 2]
340 360 380 400 420 440 460 480
σ
1/2 [MeV]
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Turning on a constant uniform external field: residual rotation symmetry around B survives. Our ansatz: V(r, θ) = −α(θ, B) r + σ(θ, B)r + V0(θ, B) with θ angle between quarks direction and B. Angular dependence in Fourier expansion: O(θ, B) = ¯ O(B)
cO
2n(B) cos(2nθ)
General features: Assumption: V(r, θ) is in the Cornell form ∀θ c2n+1 terms vanish ( B inversion θ → π − θ)
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Some details: fixed |e|B ∼ 1.0 GeV2 on two lattices aL ∼ 5 fm (| b| = 32) Wilson loop averaged separately on orthogonal axes Access to 8 angles using three B orientations Results: potential is anisotropic and V(r, θ) increases with θ good description in terms of c2’s only (∼ 0.2 − 0.3) ¯ O(B) compatible with values at B = 0
483 × 96 0.3 0.45 0.6 0.75 0.9 1.05 1.2 r [fm] 600 800 1000 1200 1400 1600 1800 V(r) [MeV] θ = 0° θ = 25° θ = 39° θ = 56° θ = 90° 0.125 0.25 0.375 0.5
θ
1
(σ(θ)−σ)/c2
σ
a = 0.1535 fm a = 0.0989 fm
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Questions: Does the anisotropy survive when a → 0? Dependence to B? Simplify the task: Angular dependence is fully described by the lowest coefficients c2s = ⇒ All the informations accessed by studying the potential along two directions only For each O = σ, α, V0 we can study its anisotropy (with B ˆ z) δO(B) = OXY(B) − OZ(B) OXY(B) + OZ(B) then δO ≃ cO
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Continuum extrapolation using cO
2 = AO(1 + COa2)(|e|B)DO(1+EOa2)
O = σ, α, V0
0.25 0.5 0.75 1 1.25
eB [GeV
2]
0.1 0.2 0.3 0.4 0.5
c
σ 2
a = 0.2173 fm a = 0.1535 fm a = 0.1249 fm a = 0.0989 fm
0.25 0.5 0.75 1 1.25
eB [GeV
2]
c
α 2
a = 0.2173 fm a = 0.1535 fm a = 0.1249 fm a = 0.0989 fm
Results: anisotropy cσ
2 of the string tension survives a → 0
cα
2 and cV0 2 compatible with zero
¯ O(B) all compatible with values at B = 0
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what about (not so) high temperatures? ❙❡t✉♣✿
❧❛tt✐❝❡s ✹✽3×◆t ✇✐t❤ ◆t❂✶✹✱✶✻✱✷✵ ✭❚ ❚c✮
❜❂✵ t♦ ❜❂✻✹ ✇✐t❤ ❇✴✴③ Results: Anisotropy still visible but disappears at large r String tension σ decreases Cornell form fits only at small B
T=99.8 MeV 0.2 0.3 0.4 0.5 0.6 0.7 0.8
r [fm]
750 1000 1250 1500
FQQ(r) [MeV] |e|B = 0.00 GeV
2 (XYZ)
|e|B = 0.76 GeV
2 (XY)
|e|B = 0.76 GeV
2 (Z)
|e|B = 2.08 GeV
2 (XY)
|e|B = 2.08 GeV
2 (Z)
0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
|e|B [GeV
2]
100 200 300 400 500 600
σ
1/2 [MeV]
T = 99.8 MeV (XY) T = 99.8 MeV (Z) T = 124.7 MeV (XY) T = 124.7 MeV (Z) T = 142.5 MeV (XY) T = 142.5 MeV (Z)
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From our results: Decrease of the free energy as B grows The effect is enhanced as T reaches Tc This is compatible with a decrease of Tc due to B [Bali et al.’12] Suppression of confining properties is evident before the appearance of inverse chiral magnetic catalysis Hence it seems to be the dominant phenomenon
T=124.7 MeV 0.2 0.3 0.4 0.5 0.6 0.7 0.8
r [fm]
750 1000 1250 1500
FQQ(r) [MeV] |e|B = 0.00 GeV
2 (XYZ)
|e|B = 0.52 GeV
2 (XY)
|e|B = 0.52 GeV
2 (Z)
|e|B = 1.04 GeV
2 (XY)
|e|B = 1.04 GeV
2 (Z) 0.2 0.4 0.6 0.8 1
|e|B [GeV
2]
0.1 0.2 0.3
〈ψ _ ψ〉
r
T = 99.8 MeV T = 124.7 MeV T = 142.5 MeV
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Investigation of the effect of B on the Q ¯ Q interaction [arXiv:1607.08160] The static potential becomes anisotropic V(r) → V(r, θ, B) Genuine effects in the continuum limit Modifications mostly due to the string tension σ → σ(B, θ) ≃ σ
2 (B) cos 2θ
Observations agree picture with deconfinement catalysis Possible implications: In meson production in heavy ion collisions [Guo et al. ’15] Heavy meson spectrum c¯ c and b¯ b [Alford and Strickland ’13, Bonati et al ’15]
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Investigation of the effect of B on the Q ¯ Q interaction [arXiv:1607.08160] The static potential becomes anisotropic V(r) → V(r, θ, B) Genuine effects in the continuum limit Modifications mostly due to the string tension σ → σ(B, θ) ≃ σ
2 (B) cos 2θ
Observations agree picture with deconfinement catalysis Possible implications: In meson production in heavy ion collisions [Guo et al. ’15] Heavy meson spectrum c¯ c and b¯ b [Alford and Strickland ’13, Bonati et al ’15]
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With B ˆ z, a possible choice of the abelian links is uf
i; y = eia2qf Bzix
uf
i; x|ix=Lx = e−ia2qf LxBziy
and all the other equal to 1. A general B = (Bx, By, Bz): The quantization condition |qmin|B = 2πb a2NxNy b ∈ Z applies separately along each coordinate axis. If Nx = Ny = Nz the condition is the same and hence
b = (bx, by, bz) Phase in the fermion matrix is the product
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The O(B) values are accessible computing the quantities RO(|e|B) = OXY(|e|B) + OZ(|e|B) 2O(|e|B = 0) = ¯ O(|e|B) O(|e|B = 0)
cO
2n
¯ O(|e|B) O(|e|B = 0) and are compatible with those at B = 0
0.25 0.5 0.75 1 1.25
eB [GeV
2]
0.8 0.9 1 1.1 1.2 1.3
α(B)/α(0) a = 0.2173 fm a = 0.1535 fm a = 0.1249 fm a = 0.0989 fm
0.25 0.5 0.75 1 1.25
eB [GeV
2]
0.9 0.95 1 1.05 1.1
σ(B)/σ(0) a = 0.2173 fm a = 0.1535 fm a = 0.1249 fm a = 0.0989 fm
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Extension to large fields (at a = 0.0989 fm on 483 × 96) longitudinal string tension seems to vanish for |e|B ∼ 4 GeV2 problem: cut-off effects at |e|B ∼ 1/a2 ∼ 4 GeV2
1 2 3 4
eB [GeV
2]
0.5 1 1.5
σ(B) / σ(0) a = 0.0989 fm (XY) a = 0.0989 fm (Z)
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