Magnetic Properties of Strongly Interacting Matter

Transcription

1 Magnetic Properties of Strongly Interacting Matter Francesco Negro 19 June 2014 XQCD 2014 Stony Brook In Collaboration with: C. Bonati, M. D Elia, M. Mariti, M. Mesiti, F. Sanfilippo. F. Negro XQCD / 26

2 Outline Introduction and phenomenological motivations Magnetizing Strongly Interacting Matter Magnetic Susceptibility at finite temperature [Bonati, D Elia, Mariti, N and Sanfilippo, PRL 111 (2013) ] [Bonati, D Elia, Mariti, N and Sanfilippo, PRD 89 (2014) ] Non-linearities of the magnetization Work in progress Anisotropy of the static quark-antiquark potential [Bonati, D Elia, Mariti, Mesiti, N, Sanfilippo, PRD 89 (2014) ] Conclusions and perspectives F. Negro XQCD / 26

3 X XL magnetic fields Electroweak corrections are usually small if compared to the strong interactions. But what happens if there is magnetic field large enough to be comparable with the scale Λ QCD? Astrophysics Large magnetic fields (eb Tesla) in a class of neutron stars called magnetars [Duncan and Thompson, 92] Cosmology Large magnetic fields (eb Tesla, eb 1.5 GeV) may have been produced at the cosmological electroweak phase transition [Vachaspati, 91; Grasso and Rubinstein, 01] Non-central heavy ion collisions (HIC) In non-central HIC the largest magnetic field ever created in a lab (eb up to Tesla 0.3 GeV 2 15mπ 2 at LHC) [Skokov, Illarionov and Toneev, 09] F. Negro XQCD / 26

4 Possible effects of the B field on QCD The presence of a large B field may have various impacts on QCD. Some examples and possible questions: Phase Diagram: - Shift of the deconfinement transition T C (or T PC ) decreases with B - New unconventional phases? Equation of State: - Magnetic contribution to the EoS - Is strongly interacting matter paramagnetic of diamagnetic? Vacuum Structure: - Magnetic catalysis Enhancement of chiral symmetry breaking - Influence on the topological properties of QCD? - Presence of anisotropies due to the B-field The Chiral Magnetic Effect [Vilenkin, 80; Kharzeev, McLerran, Warringa, Fukushima, Zhitnitsky,...] F. Negro XQCD / 26

5 Lattice QCD & magnetic field A background QED field enters the Lagrangian by modifying the covariant derivative: D µ = µ + iga a µt a µ + iga a µt a + iqa µ On the lattice: Gluon field A a µ(x)t a U µ(n), SU(3) link variables (integration variables) Photon field a µ(x) u µ(n), U(1) link variables (fixed) The lattice discrete derivative will read: D µψ 1 2a ( ) U µ(n)u µ(n)ψ(n + ˆµ) U µ(n ˆµ)u µ(n ˆµ)ψ(n ˆµ) Electric field det(m D [U, q, E]) becomes complex and sources a sign problem. Magnetic field det(m D [U, q, B]) is positive definite! For E-field one has to adopt an imaginary field E I = ie and exploit analytic continuation. F. Negro XQCD / 26

6 Lattice QCD & magnetic field IR Effect Finite volume periodic b.c. quantization of the B field (qb) = 2πb/(L xl y ) A simple choice of the lattice discretization is The elementary fluxes P ij are u y (n) =e ia2 qbn x u x (L x 1) chosen = e ia2 qbl x n y to be uniform otherwise u j (n) =1 The magnetic field on the lattice (2) An example for L x = L y = 4. The E.M. plaquettes are given by P ij = e (ia2qb) for (i, j) (3, 3) P ij = e ia2qb for (i, j) (3, 3) P 33 = exp(ia 2 qb + ia 2 qbl xl y ) P 33 =exp(ia 2 qb + ia 2 qbl x L y ) The same idea in the Dirac quantizationeverything of magnetic is ok monopoles. if a 2 qbl x L y =2πb An invisible with string b takes Z. Theideaisthesameas away all the exceeding the Dirac flux. quantization condition for monopoles (i.e. invisible string) P 03 P 13 P 23 P 33 P 02 P 12 P 22 P 32 P 01 P 11 P 21 P 31 P 00 P 10 P 20 P F. Negro XQCD / 26

7 Diamagnet or Paramagnet? Questions: Does strongly interacting matter behaves like a paramagnet or like a diamagnet? What are the magnetic properties of QCD at finite temperature? Magnetic susceptibility χ(t ): positive or negative? depends on T? How to answer (?): in principle one could evaluate directly the magnetization and higher order derivatives M = χ (1) = log Z (eb) ; χ(n) = n log Z (eb) n The free energy density (f = (T /V ) log Z) can be formally expandend in the vicinity of B = 0 f (T, B) = f (T, 0) T χ (n) eb=0 (eb) n V n! n=1 and the magnetic susceptibility would be χ (2) eb=0 χ. But On the lattice the magnetic field is quantized, hence the derivative / (eb) is not a well defined operation. F. Negro XQCD / 26

8 Our approach: free energy finite differences 1 We devise a way to compute f (T, b) = f (T, b + 1) f (T, b) 2 The determination of f can be used to extract χ (e.g. by means of suitable fitting procedure). The difficult part of the computation is part 1! Let s look closer at it: f = T ( ) ( Z(T, b + 1) V log = T ) DU e S YM [U] Z(T, b) V log det M[U, b + 1]. DU e S YM [U] det M[U, b] In principle this might be accessible via a sort of reweighting : f = T det M[U, b + 1] V log det M[U, b] Problems with this definition: Poor(?) overlap between the b and the b + 1 distributions Noisy estimators may introduce a bias b F. Negro XQCD / 26

9 Our approach: free energy finite differences The approach we devised (based a posteriori on a theorem by Jarzynski [Jarzynski, PRL 97]) can be summarized as differentiate and, then, integrate. The idea is then to find an appropriate path in parameter space that connects P (T, b) with Q (T, b + 1) and express f as f (P) f (Q) = f = T V Q P d p log Z. p The efficiency of the method is related to the shape of the path. This idea is well known in statistical physics and goes under the name of Thermodynamic Integration. [Kirkwood, J Chem Phys 35] Our choice is to go straight from b to b + 1 by introducing a real valued magnetic field: f = T V b+1 b db log Z b For the U(1) links we adopt the same definition as for the quantized case. The only difference is that the string is now visible. Warning: The integrand logz/ b is not the magnetization, it is the derivative of the interpolating free energy defined at real b. F. Negro XQCD / 26

10 Renormalization and QED quenching for χ Renormalization Quadratic in B divergences are still present in the finite difference f (T, B) = f (T, B) f (T, 0): they must properly subtracted. We are interested in the magnetic properties of the strongly interacting medium. Hence our renormalization prescription is to subtract the vacuum (T = 0) contribution f R (T, B) = f (T, B) f (0, B) No further divergences (depending on both B and T ) are present. This procedure has to be carried out at the same UV cutoff. QED quenching For a linear medium the magnetization is proportional to the field So the free energy changes into f R = M db = χ M = χb/µ 0; M = χh; H = B/µ 0 M B db = χ 2µ 0 B 2 = ˆχ 2 (eb)2 B is the total field felt by the medium no backreaction (QED is quenched) The determination of χ is not affected by quenching effects, however, in a real medium the backreaction would lead to an increase of f R by a factor 1/(1 χ) 2 F. Negro XQCD / 26

11 Numerical Setup The Lattice approach consists in a discretization of the Euclidean Feynman path integral: Z = DU Dψ Dψ e S YM [U] ψ f M D ψ f f = DU e S YM [U] (det Mf D [U]) 1/4 f We adopt a state-of-art discretization for QCD with N f = 2 + 1: Gauge sector: tree level improved Symanzik action [Weisz, Nucl Phys B 83; Curci, Menotti and Paffuti, Phys Lett B 83] Fermionic sector: rooted staggered fermions stout smearing improvement [Morningstar and Peardon, PRD 04] The bare parameters we adopted in our simulations have been taken from [Borsanyi, Endrodi, Fodor et al., JHEP 10]. They correspond to the physical line of constant physics (mπ LAT = mπ PHYS ). Simulations have been performed on the BlueGene/Q machine at CINECA, Italy. Lattices a = fm 24 4 and 24 3 (4, 6, 8, 10) a = fm 32 4 and 32 3 (4, 6, 8, 10, 12) a = fm 40 4 and 40 3 (4, 6, 8, 12, 16) We adopt the symmetric (T 40 MeV) lattices as a reference for T = 0 subtraction. F. Negro XQCD / 26

12 Pseudo-Magnetization In order to get f = b+1 db log Z/ b we have to compute b ) M Pseudo = M = log Z/ b = Tr ((M D ) 1 M D b b We divide the interval between each quanta in 16 parts 15 values of b. M b Results on the a = fm set. N t = 24 T 40 MeV N t = 4 T 227 MeV M(integer) = 0 Oscillating function Non-vanishing integral from b to b + 1 Depends on T Checked for various sources of ambiguities F. Negro XQCD / 26

13 Integrating the Pseudo-Magnetization We integrate M Pseudo between next quanta: not only the result, a 4 f, is different from zero, but the integral depends smoothly on b. For small b we can assume a 4 f (b) = c 2b 2 /2; hence a 4 f = f (b) f (b 1) = c 2(2b 1)/2 f(b)-f(b-1) By fitting f we can extract c 2 at various temperatures. We perform the renormalization c 2R = c 2(T ) c 2(0) Finally, we get b checked for finite volume effects χ = e 2 µ 0c 18 π 2 L4 s c 2R or, in natural units ˆχ = 1 18π 2 L4 s c 2R F. Negro XQCD / 26

14 Magnetic Susceptibility - Comparisons a= fm, L s =24 a= fm, L s =32 a= fm, L s =40 Levkova and DeTar Bali et al. χ ~ T [MeV] Other methods half - half method (or Taylor expansion method) [Levkova and DeTar] anisotropy method [Bali, Bruckmann, Endrodi et. al.] recently generalized integral method [Bali, Bruckmann, Endrodi et. al.] F. Negro XQCD / 26

15 Magnetic Susceptibility - Continuum Limit a= fm, L s =24 a= fm, L s =32 a= fm, L s =40 χ ~ Fit function for the continuum limit: T [MeV] χ(t ) = { A exp( M/T ) T T inspired by HRG A log(t /M ) T > T inspired by Perturbation Theory (1) Continuous and differentiable matching at T (5-2)=3 parameters We perform the continuum limit by letting either A = A 0 + a 2 A 2 or M = M 0 + a 2 M 2. Remarkably T = 160(10) MeV. F. Negro XQCD / 26

16 Flavours contributions 0,0035 χ ~ 0,003 0,0025 0,002 0,0015 0,001 0, u, a= fm d, a= fm s, a= fm u, a= fm d, a= fm s, a= fm u, a= fm d, a= fm s, a= fm χ ~ d / χ~ u χ ~ s / χ~ u -0, T [MeV] T [MeV] down contribution 4 times smaller than the up (because of an electric charge factor (q d /q u) 2 ) strange contribution: even smaller due to its large mass (same charge factor) What about the charm quark? (q c/q u) 2 = 1 no charge suppression m c is even larger suppression due to mass is stronger So we expect naively χ s χ c. F. Negro XQCD / 26

17 Magnetic contribution to the Pressure The pressure increases due to the presence of the magnetic field: P(B) = f R (B) P(B)/P(B=0) eb=0.2gev eb=0.1gev T [MeV] We take the pressure at B = 0, P(0), from [S. Borsanyi it et al, arxiv: ]. P(B)/P(0) is larger around the transition and 10 50% for the typical fields produced in HIC at LHC, eb = GeV 2. in the high T regime P(B)/P(0) 0, since P(0) T 4. F. Negro XQCD / 26

18 Comparison with the HRG model The contribution to the pressure P(B) (both as data and as continuum extrapolation) is plotted in the low T regime for eb = GeV 2. HRG model predicts a diamagnetic behaviour due to pions. Recently, (lattice) indications of a possible diamagnetic regime up to T 120 MeV. [Bali, Bruckmann, Endrodi et. al., arxiv: ] Our data are not enough precise to disinguish a possible (small) diamagnetic behaviour P [GeV 4 ] GeV GeV 2 a= a= a= HRG HRG T [MeV] F. Negro XQCD / 26

19 Nonlinearities Our method allows us, in a very simple way, to go to large B fields. We recall our expression for the free energy difference: f (T, b) = b+1 b db f b Even if we cannot reconstruct the whole free energy, finite free energy energy differences between adiacent quanta are enough to fit the free energy behavior with exepcted behaviors and ansatze. Hence, we can get informations about higher orders in B in the free energy expansion. Also in this case we have to renormalize the free energy density: Subtract only the quadratic vacuum term Subtract the whole vacuum term (with quartic, sextic,... terms) We plan to extract from the large B behaviour of f R the quartic term (and hopefully also the sextic one), and determine their dependence on the temperature. F. Negro XQCD / 26

20 Going to large fields Preliminary results at large magnetic fields at a = fm. Lattices explored 32 4, 32 3 x6, 32 3 x M(b) ; T=40 MeV 32 3 x4; T=320 MeV f = f(b) - f(b-1) T=40 MeV, 32 4 T=214 MeV, 32 3 x6 T=321 MeV, 32 3 x b b We can compute f between large quanta. The T = 0 set requires quadratic, quartic and sextic terms to be fitted. Also for T 0 sets the quadratic term is not enough. As a working choice, we fit data with f (b) = c2 2 b2 + c4 4! b4 + c6 6! b6 F. Negro XQCD / 26

21 paramagnetic QCD vacuum. The results for M r are also compared here to the gas (HRG) model Nonlinear terms at Tprediction = 0 [9], showing a remarkable agreement for eb < 0.4 G account the independence of A(Cf ) on the temperature, as discussed in subsec. 3.2, the magnetization remains roughly constant up to the transition region. This is al the HRG description of ref. [9]. Note that in ref. [19] we presented results on the magn For the T = 0 data, the quark only notrivial choice for the subtractionbehavior is to remove the of the spins, which indicated a diamagnetic of this contribution. H quadratic term. the total magnetization of the QCD vacuum, which besides the spin term also i We plot the renormalized magnetization M r as a function of the B-field. angular momentum contributions Mr(T=0,eB) Mr [GeV2] eb [GeV2] We are in agreement with [Bali, Bruckmann et al., JHEP 1304 (2013) 130]. Figure 6. Estimate of the renormalized magnetization of QCD. The positivity of M r indi vacuum is paramagnetic. Shown are lattice results on five different lattice spacings (colo hadron resonance gas curve [9] (solid line). F. Negro XQCD / 26

22 Anisotropies due to B The SU(3) gauge fields are affected by the U(1) only through quark loop effects. Anyhow, QCD is nonperturbative and, hence, they may (and actually are) be strongly affected. Moreover, the magnetic field direction can induce anisotropies. A partial list of literature: quark condensate = valence + sea [D Elia and N, PRD 83 (2011) ] inverse catalysis [Bruckmann, Endrodi and Kovacs, JHEP 1304 (2013) 112] anisotropy of the plaquettes [Ilgenfritz, Muller-Preussker et al., PRD 89 (2014); Bali, Bruckmann et al., JHEP 1304 (2013) 130] anisotropy of the fermionic part of the action [Bali, Bruckmann et al., JHEP 1304 (2013) 130] topological charge correlators (not anisotropic) [Bali, Bruckmann et al., JHEP 1304 (2013) 130] Polyakov loop dependence on (eb) Effective θ term induced by CP-odd e.m. fields [D Elia, Mariti and N, PRL 13] F. Negro XQCD / 26

23 ... what about the static potential? The static potential can be extracted from the Wilson Loop observable: av (an) = W (R, T + 1) lim T W (R, T ) Since we expect to see anisotropies, we cannot average all the possible Wilson Loops. We build two classes of Wilson Loops: W = (W (R x, T ) + W (R y, T ))/2 W = W (R z, T ) The potentials obtained from the two classes (V and V ) at eb 0 are different. We fitted the potentials separately, according to the standard Cornell potential: V (r) = c + α r + σ r; V (r) = c + α r See Marco Mariti s poster in the poster session for all the details + σ r F. Negro XQCD / 26

24 Anisotropic potential An example: 40 4 with a lattice spacing a = fm (eb) = 0.7 GeV b=0 b=24 XY b=24 XYZ b=24 Z av(n s a) n s If we forget to keep separated the XY and the Z directions, we almost loose any dependence of the potential on (eb). F. Negro XQCD / 26

25 String Tension and Coulomb Term We plot the ratios between each fitted parameter at B 0 divided by the same parameter at B = 0. α(b)/α(b=0) 1,4 1,2 1 0,8 L=24 XY L=32 XY L=40 XY L=24 Z L=32 Z L=40 Z σ(b)/σ(b=0) L=24 XY L=32 XY L=40 XY L=24 Z L=32 Z L=40 Z 0.8 0, ,2 0,4 0,6 0,8 1 1,2 eb [GeV 2 ] eb [GeV 2 ] α Z > α XY Influence on the heavy mesons mass spectrum! σ XY > σ Z Hence, possible influence on particle productions (e.g. charmonia and bottomonia) in non-central HIC. Does σ Z vanish at larger B fields? Dependence on the temperature? F. Negro XQCD / 26

26 Conclusions and perspectives Finite differences method Determination of the magnetic susceptibility at T 0 Diamagnetism below T c? Magnetic component of the EoS Preliminary results on the nonlinearities Further expand our b field range Determine higher orders coefficients Anisotropic QQ Potential Vanishing string tension at large field? Temperature dependence Impact on the heavy meson spectrum F. Negro XQCD / 26