Quarks and gluons in a magnetic field

Transcription

1 Quarks and gluons in a magnetic field Peter Watson, Hugo Reinhardt Graz, November 2013 P.W. & H.Reinhardt, arxiv:

2 Outline of talk Brief introduction (magnetic catalysis) Landau levels (Dirac equation with a magnetic field) Ritus eigenfunction method for fermions in a constant magnetic field asymptotic nature of series (not good for QCD) Summation and Schwinger phase tree-level nonperturbatively Gap equation Results - chiral condensate, magnetic susceptibility Summary

3 Introduction magnetic catalysis: an increase of the fermion condensate due to the presence of an external magnetic field known for a long time - chiral symmetry breaking takes place for strong magnetic fields in the Gross-Neveu model and QED even at small coupling (strong fields - lowest Landau level approximation leads to a linear rise in the condensate) such behavior is relevant for the QCD phase diagram e.g., in heavy ion collisions (charged particles, high velocities) there may be an effect, critical temperature might change... Shovkovy, Lect.Notes Phys. 871 (2013) 13; Klimenko, Theor.Math.Phys. 89 (1991) 1161; Gusynin, Miransky, Shovkovy, Phys.Lett. B349 (1995) 477.

4 Introduction for quarks in QCD we have a problem - strong magnetic fields vs. strong interaction (similar scales are not good for approximations) estimated maximum magnetic fields at the LHC are not large in the context of QCD: also, we would like to know the magnetic susceptibility (calculated in the limit of vanishing field) - useful e.g., in normalizing chiral-odd transversity parton distribution functions eb 15m 2 0.3GeV 2 we want small and moderate fields too! Skokov, Illarionov, Toneev, Int.J.Mod.Phys.A24 (2009) 5925.

5 Landau levels consider: ~B = Bê 3, h = QB 0 choose a gauge: A 0 =0, ~ A = Bx1 ê 2 (minimal coupling) Dirac operator: D = ı@ µ µ h 2 x 1 for the energy levels: (D + m)(d m) (x) =0 Fourier transform (except the x-direction), noting the spin and Hermite eigenfunctions apple p 2 0 p 2 3 m " 2 f(") =0 p p " = hx 1 + p 2 / h, = ±1, f(") = n (") Landau (energy) levels: E 2 n = p m 2 + h(2n +1+ )

6 Landau levels constant magnetic field introduces Landau levels with Hermite functions as eigenfunctions the Landau levels get connected to the spin translational invariance is broken! ( = ±1) now we want the tree-level propagator and inverse: inverse propagator: propagator: (0) (x, y) =ı[d m] (x y) ı[d m]s (0) (x, y) = (x y)

7 Ritus eigenfunction method Ritus solution (replace momentum modes with the Hermite function basis): (0) (x, y) = Ritus matrices (orthonormal and complete) connect spin and Landau levels spin projectors: 1X n=0 Z d 3 p (2 ) 3 E(x; p, n) (0) (p, n)e(y; p, n) p µ =(p 0, 0,p 2,p 3 ), p µ =(p 0, 0, 0,p 3 ) E(x; p, n) =h 1/4 e ı p x n 1(") + + n (") ± = 1 1 ± ı notice that n=0 (lowest Landau level) is special! Ritus, Sov.Phys. JETP 48 (1978) 788.

8 Ritus eigenfunction method inverse propagator (function of eigenvalues) ı (0) (p, n) =p µ µ p 2nh 2 m propagator ıs (0) (p, n) = p µ µ p 2nh 2 + m p 2 2nh m 2 + ı0 + (Landau levels appear in the denominator) So, by using the Ritus matrices and the associated eigenvalues instead of momentum space, we can tackle the gap equation. Usually, only the lowest Landau level (n=0) is considered (works for large fields) and gives a linearly rising condensate. Gusynin, Miransky, Shovkovy, Phys.Lett. B349 (1995) 477.

9 Ritus eigenfunction method BUT the general form for the chiral condensate is hqqi = N c Tr d S(x, x) Z ( h d 2 p = N c 2 (2 ) 2 Tr d S(p, n =0)+ decomposing and projecting in terms of the Ritus matrices, a pre-factor of h appears in all loop integrals, regardless of the interaction (the propagator is a function of two momentum components and the Landau level: dimensions must be maintained)! we have an asymptotic expansion, not good for small magnetic fields where we know that the quark has a nontrivial condensate! we have to sum up the Landau levels... 1X n=1 S(p, n) )

10 Summation and Schwinger phase tree-level inverse propagator (0) (x, y) = 1X n=0 Z contains the integral d 3 p (2 ) 3 E(x; p, n) (0) (p, n)e(y; p, n) Z I = dp 2 e ıp 2(x 2 y 2 ) a(") b ( ) with " ( ) = p hx 1 (y 1 )+ p 2 p h it can be shown that ;-) this can be written (almost) in terms of transverse momenta and Laguerre polynomials I e ı Z d 2 p t e ı~p t (~x ~y ) f(~p t )exp p 2 t h L n 2 p2 t h Gorbar, Miransky, Shovkovy, Wang, PRD88 (2013) &

11 Summation and Schwinger phase it can be shown ;-) that this can be written (almost) in terms of transverse momenta where the Schwinger phase encodes the deviations from translational invariance (vanishes for h=0) I e ı... = h 2 (x 2 y 2 )(x 1 + y 1 ) the sums over the Laguerre polynomials are known, to give... Gorbar, Miransky, Shovkovy, Wang, PRD88 (2013) &

12 Summation and Schwinger phase the tree-level inverse propagator Z ı (0) (x, y) =e ı d 4 p (2 ) 4 e ıp (x y) [p µ µ m] and similarly the tree-level propagator (small h) ıs (0) (x, y) =e ı Z d 4 p ıp (x y) e (2 ) 4 ( [p µ µ + m] [p 2 m 2 + ı0 + ] + ıh 1 2 µ pµ + m ) [p 2 m 2 + ı0 + ] 2 reduction when magnetic field vanishes (unlike the Ritus decomposition), but no obvious relation between the two! up to the Schwinger phase, the momentum space expressions look promising... Gorbar, Miransky, Shovkovy, Wang, PRD88 (2013) & , Chodos, Everding, Owen, PRD42 (1990) 2881.

13 Nonperturbatively the strategy is to take a nonperturbative ansatz for the Ritus decomposed two-point functions (where the inverse can be found), with various spin components and see if we can sum to get similar expressions... ı (p, n) = + (p µ µ A B)+ (p µ µ C D) p 2nh 2 E (allow for different spin projections) sum isn t a problem for the inverse propagator: Z ı (x, y) =e ı d 4 p ıp (x y) e (2 ) 4 + (p µ µ A B)+ (p µ µ C D) ~p t ~ E functions of two variables, A = A(p 2,p 2 t ) reduction h! 0: (A, C, E)! A, (B,D)! B

14 Nonperturbatively corresponding propagator looks like... ıs(p, n) = + p µ µ 1C 2D = p 2 AC BD 2nhE 2, 2 = AD BC, = 2 1 p summed under approximation (suitable for small h) neglect n-dependence of functions (keep explicit n factors) expand denominator in (small h) 2 approximation retains the connection between spin structures in the end, the gap equation will determine the momentum dependence of the functions

15 Nonperturbatively approximated summed propagator Z ıs(x, y) =e ı d 4 p ıp (x y) e (2 ) 4 + (p µ µ C + D)+ (p µ µ A + B) ~p t ~ E [p 2 AC p 2 t E 2 BD + ı0 + ] + he 2 + (p µ µ C + D) (p µ µ A + B) [p 2 AC p 2 t E 2 BD + ı0 + ] 2 (AD BC) + (p µ µ D + p 2 C) (p µ µ B + p 2 A) [p 2 AC p 2 t E 2 BD + ı0 + ] 2 reduces to tree-level (all h), also to standard propagator in the absence of the magnetic field. )

16 Gap equation rainbow truncation (chiral quarks) (x, y) = (0) (x, y)+g 2 C F µ S(x, y) apple W appleµ (y, x) dressed (Landau gauge) gluon interaction ıw appleµ (y, x) = R g 2 G(q 2 ) q 2 =4 2 d exp d 4 q (2 ) e ıq (y x) t 4 appleµ (q) G(q2 ) q 2 n q 2! 2 o q 2 /! 2, I 1, II Schwinger phase factorizes, so we can work in momentum space - but with functions of two variables consider a range of widths and fix d from the condensate: hqqi h=0 =( 251 MeV) 3 Alkofer, Watson, Weigel, PRD65 (2002) , Aguilar, Papavassiliou, PRD83 (2011)

17 Results (dressing functions) I A(0,0) B(0,0) C(0,0) D(0,0) E(0,0) II A(0,0) B(0,0) C(0,0) D(0,0) E(0,0) h (GeV 2 ) I:! =0.5GeV,d=16GeV 2 II :! =0.5GeV,d=41GeV h (GeV 2 ) functions at zero momentum reduction for vanishing magnetic field (type I explicitly matches earlier results) similar patterns for both interactions, since the gluon is unaffected by the magnetic field (under truncation)

18 Results (magnetic catalysis) r(h), I, ω=0.4gev r(h), I, ω=0.5gev r(h), I, ω=0.6gev r(h), I, ω=0.7gev h (GeV 2 ) 0.8 r(h), II, ω=0.4gev r(h), II, ω=0.5gev r(h), II, ω=0.6gev h (GeV 2 ) relative increment: r(h) = hqqi h hqqi h=0 1 condensate rises quadratically for small h and linearly for large h qualitative agreement with lattice is good, even for large h! D Elia, Negro, Phys.Rev.D83 (2011) ; Bali et al., PRD86 (2012) ; Simonov, arxiv: BUT: Ilgenfritz et al., Phys.Rev.D85 (2012) ; Shushpanov, Smilga, Phys.Lett.B402 (1997) 351.

19 Results (comparison to lattice) small h quadratic behavior reproduced transition scale reproduced recall heavy ions: small h matters! val 0.25 r u ( eb ), lattice r u ( eb ), I, ω=0.7gev 0.2 D Elia, Negro, Phys.Rev.D83 (2011) eb (GeV 2 ) eb 15m 2 0.3GeV 2

20 Results (magnetic moment) mom, I, ω=0.4gev mom, I, ω=0.5gev mom, I, ω=0.6gev mom, I, ω=0.7gev mom, II, ω=0.4gev mom, II, ω=0.5gev mom, II, ω=0.6gev h (GeV 2 ) h (GeV 2 ) magnetic moment: hq 12 qi = N c Tr d ( + )S(x, x) (measures asymmetry between spin projected components) linear at small h

21 Results (magnetic susceptibility) χ(h), I, ω=0.4gev χ(h), I, ω=0.5gev χ(h), I, ω=0.6gev χ(h), I, ω=0.7gev h (GeV 2 ) χ(h), II, ω=0.4gev -1.8 χ(h), II, ω=0.5gev χ(h), II, ω=0.6gev h (GeV 2 ) magnetic susceptibility: lattice (quenched, chiral): lattice (unquenched, finite m): NJL: quark-meson model: (h) = hq 12 qi hhqqi, = 1.547(6)GeV 2 =lim h!0 (h) (2.08 ± 0.08)GeV 2 4.3GeV GeV 2 Buividovich et al., Nucl.Phys.B826 (2010) 313; Bali et al., Phys.Rev.D86 (2012) ; Frasca, Ruggieri, Phys.Rev.D83 (2011)

22 Summary quark gap equation (rainbow truncation) with a phenomenological gluon interaction in the presence of a constant magnetic field strong magnetic field vs. strong interaction - expansion in Landau levels is not suitable in this context we use an approximation to sum the Landau levels, ensuring the limit when the magnetic field vanishes in the presence of a constant magnetic field, the chiral quark condensate increases, quadratically for small field and linearly for large field magnetic susceptibility agreement with lattice (though more to be done) parameter dependence - sensitive probe of the interaction? small h is relative to QCD!

23 Results (large h, n=0 dominance) B(0,0)/A(0,0) h (GeV 2 ) I lowest Landau level approximation: h (GeV 2 ) and the mass function D/C dominates for large h! (because C decreases) D(0,0)/C(0,0) [ n 1 (") + + n (") ]! 0 (") I