The thermo-magnetic quark-gluon vertex María Elena Tejeda-Yeomans Departamento de Física, Universidad de Sonora, México in collaboration with A. Ayala (UNAM), J. Cobos-Martínez (UMICH), M. Loewe and R. Zamora (PUC) Workshop on Magnetic Fields in Hadron Physics 9-13 May 2016, ICTP-SAFIR M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 1 / 32
Magnetic fields in heavy-ion collisions M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 2 / 32
Magnetic fields in heavy-ion collisions A. Mocsy, BNL 2007 M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 3 / 32
Magnetic fields in peripheral HICs Y. Zhong, C.-B. Yang, X. Cai, S.-Q. Feng, Adv. High Energy Phys. 2014, 193039 (2014) snn = 62.4 GeV (a), 130 GeV (b), 200 GeV (c), 900 GeV (d) RHIC: (0.1 1)m 2 π, LHC: (10 15)m 2 π, m 2 π 10 19 G M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 4 / 32
Tc /condensate Inverse Magnetic Catalysis G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, S. Krieg, A. Schafer, K. K. Szabo, JHEP 02 (2012) 044 G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, A. Schafer, Phys. Rev. D 86, 071502 (2012) M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 5 / 32
IMC in NJL and/or QCD Magnetized effective QCD phase diagram A. Ayala, C.A. Dominguez, L.A. Hernandez, M. Loewe, R. Zamora Phys.Rev. D92 096011 (2015), Phys.Rev. D92 119905 (2015) Thermo-magnetic strong coupling in the local NJL model A. Ayala, C.A. Dominguez, L.A. Hernandez, M. Loewe, Alfredo Raya, J.C. Rojas, C. Villavicencio arxiv:1603.00833 [hep-ph] Finite temperature QGV with a magnetic field in HTL A. Ayala, J.J. Cobos-Martínez, M. Loewe, M. E. T-Y, R. Zamora Phys.Rev. D91 016007 (2015) IMC from QCD coupling in a magnetic field A. Ayala, C.A. Dominguez, L.A. Hernandez, M. Loewe, R. Zamora arxiv:1510.09134 [hep-ph] Study finite temperature and magnetic field dependence of the coupling constant through the QGV M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 6 / 32
Map M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 7 / 32
Map M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 8 / 32
Outline 1 QGV in HTL and weak B 2 Effective coupling from the QGV 3 Final remarks M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 9 / 32
QGV in HTL and weak B QGV at finite temperature with a weak magnetic field A. Ayala., M. Loewe, J. Cobos-Martinez, M. E. T-Y, R. Zamora, Phys. Rev. D 91, 016007 (2015) Set-up for leading B-field dependence of QGV at high T QCD matter in the presence of constant magnetic field B = Bẑ we work in the weak B-field limit for fermion propagator HTL approximation T 2 > qb Notation capital letters, four-momenta in Euclidean space K µ = (k 4, k) = ( ω, k) ω ω n = (2n + 1)πT, ω n = 2nπT the fermion/boson Matsubara frequencies Color factors C F = N2 1 2N, C A = N M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 10 / 32
QGV in HTL and weak B Diagrams at finite temperature A. Ayala., M. Loewe, J. Cobos-Martinez, M. E. T-Y, R. Zamora, Phys. Rev. D 91, 016007 (2015) M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 11 / 32
QGV in HTL and weak B Diagrams with magnetic field A. Ayala., M. Loewe, J. Cobos-Martinez, M. E. T-Y, R. Zamora, Phys. Rev. D 91, 016007 (2015) M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 12 / 32
QGV in HTL and weak B Charged fermion propagator in a medium The fermion propagator (Schwinger) where the phase factor is S(x, x ) = Φ(x, x ) Φ(x, x ) = exp d 4 p (2π) 4 e ip (x x ) S(k), { x iq dξ [A µ µ + 12 ]} F µν(ξ x ) ν x B-field breaks Lorentz invariance charged fermion prop (k, k ) M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 13 / 32
QGV in HTL and weak B Charged fermion propagator in a medium S(k) is given by S(k) = i 0 ds cos(qbs) eis(k2 k2 tan(qbs) m qbs 2 ) { [cos(qbs) + γ 1 γ 2 sin(qbs)] (m + k /) (Euclidean) finite T + weak B up to O(qB) k / } cos(qbs) S(K) = m K K 2 + m 2 iγ m K 1γ 2 (K 2 + m 2 ) 2 (qb). parallel and perpendicular components (a b) = a 0 b 0 a 3 b 3, (a b) = a 1 b 1 + a 2 b 2 M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 14 / 32
QGV in HTL and weak B QGV at finite T and weak qb: QED-like diagram Magnetic field dependent part of the QED-like diagram (factor-out gt a common to the bare and purely thermal) ( δγ (QED-like) µ = ig 2 C F C ) A (qb)t 2 n d 3 k (2π) 3 [ γ ν γ 1 γ 2 K γ µ K (P ] 2 K) + Kγ µ γ 1 γ 2 K (P1 K) (K) (P 2 K) (P 1 K) γ ν where (K) 1 ω 2 n + k 2 + m 2 (K) 1 ω 2 n + k 2 and ω n = (2n + 1)πT and ω n = 2nπT the fermion and boson Matsubara frequencies M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 15 / 32
QGV in HTL and weak B QGV at finite T and weak eb: pure-qcd diagram Magnetic field dependent part of the pure-qcd diagram (factor-out gt a common to the bare and purely thermal) ( ) δγ (pure-qcd) µ = 2ig 2 CA (qb)t d 3 k 2 (2π) 3 n [ Kγ 1 γ 2 K γ µ + 2γ ν γ 1 γ 2 K γ ν K µ γ µ γ 1 γ ] 2 K K (K) 2 (P 1 K) (P 2 K) where (K) 1 ω 2 n + k 2 + m 2 (K) 1 ω 2 n + k 2 ω n = (2n + 1)πT and ω n = 2nπT the fermion and boson Matsubara frequencies M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 16 / 32
QGV in HTL and weak B QGV at finite T and weak eb: tensorial structure γ 1 γ 2 K = γ 5 [(K b)u/ (K u)b/] medium s rest frame u µ = (1, 0, 0, 0) direction of the magnetic field b µ = (0, 0, 0, 1) M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 17 / 32
QGV in HTL and weak B QGV at finite T and weak eb: T 2 eb HTL approx: P 1 P 2 T [ γ 1 γ 2 K γ µ K (P ] 2 K) + Kγ µ γ 1 γ 2 K (P1 K) [ γ 1 γ 2 K γ µ K + Kγ µ γ 1 γ 2 K ] (P1 K) Tensorial structure for both diagrams: ( δγ (QED-like) µ = 2iγ 5 g 2 C F C A 2 δγ (pure-qcd) µ = 2iγ 5 g 2 ( CA 2 ) (qb)g (QED-like) µ (P 1, P 2 ) ) (qb)g (pure-qcd) µ (P 1, P 2 ) M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 18 / 32
QGV in HTL and weak B QGV at finite T and weak eb: leading T behaviour { G µ (QED-like) G (pure-qcd) µ } = 2T n d 3 k (2π) 3 {(K b) Ku µ (K u) Kb µ + [(K b)u/ (K u)b/] K µ } { (K) 2 (P 1 K) (P 2 K) 2 (K) (P 1 K) (P 2 K) } M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 19 / 32
QGV in HTL and weak B QGV at finite T and weak eb: leading T behaviour Bose-Einstein: f (E) Fermi-Dirac: f (E) G µ (QED-like) = G µ (pure-qcd) Adding the contributions from the two Feynman diagrams we get δγ µ = δγ (QED-like) µ + δγ (pure-qcd) µ = 2i g 2 (qb) C F γ 5 G µ (P 1, P 2 ) G µ (P 1, P 2 ) can be computed from the tensor J αi (α = 1,... 4, i = 3, 4) J αi = T n d 3 k (2π) 3 K αk i 2 (K) (P 1 K) (P 2 K) M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 20 / 32
QGV in HTL and weak B QGV at finite T and weak eb: J αi Leading temperature behaviour after frequency sums J αi = 1 dω 16π 2 [h ˆK α ˆK i 1(y) + f 1 (y)] 4π (P 1 ˆK)(P 2 ˆK) Using the high temperature expansions for h 1 (y) and f 1 (y) (Kapusta) h 1 (y) = π 2y + 1 ( y ) 2 ln + 1 4π 2 γ E +... f 1 (y) = 1 ( y ) 2 ln 1 π 2 γ E +... and keeping the leading terms, we get J αi = 1 [ 16π 2 ln(2) π ] T dω ˆK α ˆKi 2 m 4π (P 1 ˆK)(P 2 ˆK) M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 21 / 32
QGV in HTL and weak B QGV at finite T and weak eb: leading T behaviour Z 1 dω = 4i g CF M (T, m, qb) γ5 4π (P1 K )(P2 K ) n h i o (K b)6k uµ (K u)6k bµ + (K b)u/ (K u)b/ K µ 2 δγµ (P1, P2 ) 2 where qb πt M (T, m, qb) = ln(2) 16π 2 2m 2 M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 22 / 32
QGV in HTL and weak B QGV at finite T (HTL) and weak eb: QED-like WI In the presence of the magnetic field and provided the temperature is the largest of the energy scales, the thermo-magnetic correction to the quark-gluon vertex is gauge invariant (P1 P2 ) δγ(p1, P2 ) = Σ(P1 ) Σ(P2 ) M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 23 / 32
QGV in HTL and weak B Quark self-energy at finite T (HTL) and weak eb Σ(P) = 2i g 2 C F (qb) γ 5 T n { } (K b)u/ (K u)b/ = 2i g 2 C F M 2 (T, m, qb) γ 5 dω 4π d 3 k (2π) 3 (P K) 2 (K) [ ] ( ˆK b)u/ ( ˆK u)b/ (P ˆK) M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 24 / 32
g eff from QGV Thermo-magnetic QCD coupling: g q q scenario Thermal gluon decaying into a back-to-back q q pair p 1 and p 2 make a relative angle θ 12 = π What happens to the angular contribution? J αi (P 1, P 2 ) dω 4π ˆK α ˆK i (P 1 ˆK)(P 2 ˆK) M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 25 / 32
g eff from QGV Thermo-magnetic QCD coupling: CM g q q scenario Focus on J αi for α = i = 3, 4 J 33 J 44 1 iω 1 p 2 + iω 2 p 1 f (ω/p) iω 1 p 2 + iω 2 p 1 1 1 { ln { } p 1 dx f (x) iω 1 + p 1 x + p 2 iω 2 p 2 x ( ) ( )} iω1 + p 1 iω2 + p 2 + ln iω 1 p 1 iω 2 p 2 M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 26 / 32
g eff from QGV Thermo-magnetic QCD coupling: CM g q q scenario Analytic continuation to Minkowski space iω 1,2 p 01,02 [ ˆK ( 1, ˆk)] scenario where p 01 = p 02 p 0 and p 1 = p 2 p J 44 J 00 = 1 2p 0 p ln ( ) p0 + p, J 33 = 1 [ p 0 p p 2 1 p 0 2p ln static limit, quarks are almost at rest, p 0 p 0 J 00 1 p 0 p0 2, J 33 1 3p0 2 ( )] p0 + p p 0 p The rest of the components of J αi vanish Only the longitudinal components of the thermo-magnetic vertex are modified M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 27 / 32
geff from QGV Thermo-magnetic QCD coupling: CM g q q scenario The longitudinal QGV is modified as 2 ~ δγk (p0 ) = 4g 2 CF M 2 (T, m, qb) ~γk Σ3 3p02 % 1st order magnetic corr spin - qb intn where ~γk = (γ0, 0, 0, γ3 ) and Σ3 = iγ1 γ2 = 2i [γ1, γ2 ] quark mass m IR scale scale thermal quark mass. So choose p0 = T and m2 = mf2 = 18 g 2 T 2 CF M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 28 / 32
geff from QGV Thermo-magnetic QCD coupling: CM g q q scenario The purely thermal correction to QGV (LeBellac) δγtherm (P1, P2 ) = mf2 µ Z K µk 6 dω 4π (P1 K )(P2 K ) Extract geff looking at (using same configuration as before) mf2 mf2 γ g g (p ) = 1 δγtherm 0 0 0 therm T2 p02 The effective thermo-magnetic modification to the quark-gluon cou~ k (p0 ) as pling can be extracted from δγ m2 geff = g 1 f2 + T M.Tejeda-Yeomans (DF-USON, Mex) 8 3T 2 2 2 g CF M (T, mf, qb) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 29 / 32
g eff from QGV QGV effective coupling decrease more significant for larger α s 15% 25% smaller than thermal corr at qb T 2 1 M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 30 / 32
Final remarks Final remarks thermo-magnetic corrections to QGV for a weak B-field and in HTL T 2 >> qb QGV satisfies a QED-like Ward identity hints to gauge-independence thermo-magnetic correction only to QGV longitudinal component spin component in the direction of the B-field quark anomalous magnetic moment at high T and weak qb extract behaviour on B-field dependence of QCD coupling under conditions prevailing in a QGP: back-to-back static quarks whose energy T and with IR scale m thermal coupling decreases as the B-field strength increases result supports the idea that the decreasing of the coupling constant is an important ingredient to understand IMC in this regime M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 31 / 32
Final remarks Map M.Tejeda-Yeomans (DF-USON, Mex) The thermo-magnetic quark-gluon vertex WMFHP @ ICTP-SAFIR 2016 32 / 32