Effects of magnetic fields on q ¯ q interactions [1607.08160] C.Bonati 1 , M.D’Elia 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 Astronomy, University of Southampton, UK Lattice 2016, 34th International Symposium on Lattice Field Theory 28 July 2016
table of contents Introduction 1 Effects of the magnetic field on the 2 static potential at T=0 What happens at finite temperatures? 3 ( T < T c ) Conclusions 4
intro physical conditions π ∼ 10 15 − 16 T QCD with strong magnetic fields eB ≃ m 2 Non-central heavy ion collisions with eB ∼ 10 15 T [Skokov et al. ’09] Possible production in early universe eB ∼ 10 16 T [Vachaspati ’91] In heavy ion collisions Expected eB ≃ 0 . 3 GeV 2 at LHC in Pb+Pb at √ s NN =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 1 Lattice 2016 A.Rucci
intro turning on the B field An external magnetic field B on the lattice is introduced through abelian parallel transports u µ ( n ) Abelian phases enter the Lagrangian by modifying the covariant derivative U µ ( n ) → U µ ( n ) u µ ( n ) External field is fixed: non-propagating fields, no kinetic term Periodic boundary conditions lead to the quantization condition 2 π b | q min | B = b ∈ Z a 2 N x N y 2 Lattice 2016 A.Rucci
intro static potential In the confining phase at low temperatures, the Q ¯ Q interaction is well described by the Cornell potential V C ( r ) = − α σ ≃ ( 440MeV ) 2 r + σ r + V 0 α ∼ 0 . 4 On the lattice: At T=0 it can be extracted from the Wilson loop � � W ( � � n , n t + 1 ) � aV ( a � n ) = − lim n t →∞ log � W ( � n , n t ) � For T>0 from Polyakov loop correlators F ( a � n , T ) ≃ − aN t log � Tr L † ( � r + � n ) Tr L ( � n ) � what about the effects of � B on the potential? (a first study: [Bonati et al. ’14]) 3 Lattice 2016 A.Rucci
T=0 setup and continuum results at B=0 ◆✉♠❡r✐❝❛❧ s❡t✉♣ - tr❡❡✲❧❡✈❡❧ ✐♠♣r♦✈❡❞ ❣❛✉❣❡ ❛❝t✐♦♥ - ◆ f ❂✷✰✶ r♦♦t❡❞ st❛❣❣❡r❡❞ ❢❡r♠✐♦♥s ✰ st♦✉t ✐♠♣r♦✈❡♠❡♥t - ❢♦✉r ❧❛tt✐❝❡s ✹✽ 3 × ✾✻✱ ✹✵ 4 ✱ ✸✷ 4 ❛♥❞ ✷✹ 4 - s♣❛❝✐♥❣ ❛ ≃ ✵✳✶ ❢♠ t♦ ❛ ≃ ✵✳✷✹ ❢♠ - s✐♠✉❧❛t✐♦♥s ❛t ♣❤②s✐❝❛❧ q✉❛r❦ ♠❛ss❡s 480 460 Parameters extracted 440 from the continuum limit 1/2 [MeV] at B = 0 420 400 α = 0 . 395 ( 22 ) σ √ σ = 448 ( 20 ) MeV 380 360 r 0 = 0 . 489 ( 20 ) fm 340 0 0.01 0.02 0.03 0.04 0.05 2 [fm 2 ] a 4 Lattice 2016 A.Rucci
T=0 angular dependence Turning on a constant uniform external field: residual rotation symmetry around � B survives. Our ansatz: V ( r , θ ) = − α ( θ, B ) + σ ( θ, B ) r + V 0 ( θ, B ) r with θ angle between quarks direction and � B . Angular dependence in Fourier expansion: � � O ( θ, B ) = ¯ � c O O ( B ) 1 − 2 n ( B ) cos ( 2 n θ ) O = α, σ, V 0 n = 1 General features: Assumption: V ( r , θ ) is in the Cornell form ∀ θ c 2 n + 1 terms vanish ( � B inversion θ → π − θ ) 5 Lattice 2016 A.Rucci
T=0 angular dependence 483 × 96 Some details: 1800 θ = 0° fixed | e | B ∼ 1 . 0 GeV 2 on two θ = 25° 1600 θ = 39° θ = 56° lattices aL ∼ 5 fm ( | � b | = 32) θ = 90° 1400 V(r) [MeV] Wilson loop averaged 1200 separately on orthogonal axes 1000 Access to 8 angles using 800 three � B orientations 600 0.3 0.45 0.6 0.75 0.9 1.05 1.2 r [fm] Results: a = 0.1535 fm a = 0.0989 fm 1 potential is anisotropic and σ ( σ(θ)−σ)/ c 2 V ( r , θ ) increases with θ 0 good description in terms of c 2 ’s only ( ∼ 0 . 2 − 0 . 3) ¯ O ( B ) compatible with values -1 at B = 0 0 0.125 0.25 0.375 0.5 θ 6 Lattice 2016 A.Rucci
T=0 anisotropy in the continuum Questions: Does the anisotropy survive when a → 0? Dependence to B ? Simplify the task: Angular dependence is All the informations accessed = ⇒ fully described by the by studying the potential lowest coefficients c 2 s along two directions only For each O = σ, α, V 0 we can study its anisotropy (with � B � ˆ z ) δ O ( B ) = O XY ( B ) − O Z ( B ) O XY ( B ) + O Z ( B ) then δ O ≃ c O 2 7 Lattice 2016 A.Rucci
T=0 anisotropy in the continuum Continuum extrapolation using 2 = A O ( 1 + C O a 2 )( | e | B ) D O ( 1 + E O a 2 ) c O O = σ, α, V 0 0.5 a = 0.2173 fm a = 0.1535 fm 0 0.4 a = 0.1249 fm a = 0.0989 fm -0.1 0.3 σ 2 α 2 -0.2 c c 0.2 -0.3 a = 0.2173 fm 0.1 a = 0.1535 fm -0.4 a = 0.1249 fm a = 0.0989 fm 0 -0.5 0.25 0.5 0.75 1 1.25 0.25 0.5 0.75 1 1.25 2 ] 2 ] eB [GeV eB [GeV Results: anisotropy c σ 2 of the string tension survives a → 0 2 and c V 0 c α 2 compatible with zero ¯ O ( B ) all compatible with values at B = 0 8 Lattice 2016 A.Rucci
T>0 effects on the free energy T=99.8 MeV what about (not so) high 1500 2 (XYZ) |e|B = 0.00 GeV 2 (XY) temperatures? |e|B = 0.76 GeV 2 (Z) |e|B = 0.76 GeV 1250 2 (XY) F QQ (r) [MeV] |e|B = 2.08 GeV 2 (Z) ❙❡t✉♣✿ |e|B = 2.08 GeV 1000 - ❋✐①❡❞ ❛❂✵✳✵✾✽✾ ❢♠ ♦♥ ❧❛tt✐❝❡s ✹✽ 3 × ◆ t ✇✐t❤ 750 ◆ t ❂✶✹✱✶✻✱✷✵ ✭❚ � ❚ c ✮ - ❙❡✈❡r❛❧ ♠❛❣♥❡t✐❝ q✉❛♥t❛ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 r [fm] ❜❂✵ t♦ ❜❂✻✹ ✇✐t❤ ❇✴✴③ 600 T = 99.8 MeV (XY) T = 99.8 MeV (Z) 500 T = 124.7 MeV (XY) T = 124.7 MeV (Z) Results: T = 142.5 MeV (XY) 400 1/2 [MeV] T = 142.5 MeV (Z) Anisotropy still visible but 300 disappears at large r σ 200 String tension σ decreases 100 Cornell form fits only at 0 small B 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 2 ] |e|B [GeV 9 Lattice 2016 A.Rucci
T>0 effects on the free energy From our results: Decrease of the free energy as B grows The effect is enhanced as T reaches T c This is compatible with a decrease of T c 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 2 (XYZ) 1500 |e|B = 0.00 GeV T = 99.8 MeV 0.3 2 (XY) |e|B = 0.52 GeV T = 124.7 MeV T = 142.5 MeV 2 (Z) |e|B = 0.52 GeV 2 (XY) |e|B = 1.04 GeV 0.2 F QQ (r) [MeV] 1250 2 (Z) |e|B = 1.04 GeV r ψ〉 _ 〈ψ 0.1 1000 0 -0.1 750 0 0.2 0.4 0.6 0.8 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 2 ] r [fm] |e|B [GeV 10 Lattice 2016 A.Rucci
conclusions and summary 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 1 − c σ 2 ( B ) cos 2 θ Anisotropy still visible at T > 0 Observations agree picture with deconfinement catalysis Possible implications: In meson production in heavy ion collisions [Guo et al. ’15] c and b ¯ Heavy meson spectrum c ¯ b [Alford and Strickland ’13, Bonati et al ’15] 11 Lattice 2016 A.Rucci
conclusions and summary 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 1 − c σ 2 ( B ) cos 2 θ Anisotropy still visible at T > 0 Observations agree picture with deconfinement catalysis Possible implications: In meson production in heavy ion collisions [Guo et al. ’15] c and b ¯ Heavy meson spectrum c ¯ b [Alford and Strickland ’13, Bonati et al ’15] THANK YOU 11 Lattice 2016 A.Rucci
backup magnetic field on the lattice With � B � ˆ z , a possible choice of the abelian links is i ; y = e ia 2 q f B z i x i ; x | i x = L x = e − ia 2 q f L x B z i y u f u f and all the other equal to 1. A general � B = ( B x , B y , B z ) : The quantization condition 2 π b | q min | B = b ∈ Z a 2 N x N y applies separately along each coordinate axis. If N x = N y = N z the condition is the same and hence � B ∝ � b = ( b x , b y , b z ) Phase in the fermion matrix is the product 12 Lattice 2016 A.Rucci
backup anisotropy at T=0 The O ( B ) values are accessible computing the quantities R O ( | e | B ) = O XY ( | e | B ) + O Z ( | e | B ) 2 O ( | e | B = 0 ) � � ¯ ¯ O ( | e | B ) O ( | e | B ) � c O = 1 − ≃ 2 n O ( | e | B = 0 ) O ( | e | B = 0 ) n ❡✈❡♥ and are compatible with those at B = 0 1.3 1.1 a = 0.2173 fm a = 0.1535 fm 1.2 a = 0.1249 fm 1.05 a = 0.0989 fm α (B)/ α (0) 1.1 σ (B)/ σ (0) 1 1 0.95 a = 0.2173 fm 0.9 a = 0.1535 fm 0.9 a = 0.1249 fm a = 0.0989 fm 0.8 0.25 0.5 0.75 1 1.25 0.25 0.5 0.75 1 1.25 2 ] 2 ] eB [GeV eB [GeV 13 Lattice 2016 A.Rucci
backup large B Extension to large fields (at a = 0 . 0989 fm on 48 3 × 96) longitudinal string tension seems to vanish for | e | B ∼ 4 GeV 2 problem: cut-off effects at | e | B ∼ 1 / a 2 ∼ 4 GeV 2 1.5 1 σ (B) / σ (0) 0.5 a = 0.0989 fm (XY) a = 0.0989 fm (Z) 0 0 1 2 3 4 2 ] eB [GeV 14 Lattice 2016 A.Rucci
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