Inverse magnetic catalysis in the linear sigma model with quarks
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(2) ALEJANDRO AYALA, M. LOEWE, AND R. ZAMORA. PHYSICAL REVIEW D 91, 016002 (2015). λ LbI ¼ − ½ðσ 2 þ ðπ 0 Þ2 Þ2 4 þ 4π þ π − ðσ 2 þ ðπ 0 Þ2 þ π þ π − Þ;. temperature is a decreasing function of the magnetic field. We finally summarize and conclude in Sec. V.. LfI ¼ −gψ̄ðσ þ iγ 5 ~τ · π~ Þψ;. II. EFFECTIVE POTENTIAL. ð7Þ. The model is given by the Lagrangian 1 1 μ2 λ L ¼ ð∂ μ σÞ2 þ ðDμ π~ Þ2 þ ðσ 2 þ π~ 2 Þ − ðσ 2 þ π~ 2 Þ2 2 2 4 2 þ iψ̄γ μ Dμ ψ − gψ̄ðσ þ iγ 5 ~τ · π~ Þψ; ð1Þ. and represent the Lagrangian describing the interactions among the fields σ, π~ and ψ, after symmetry breaking. From Eq. (6) we see that the σ, the three pions and the quarks have masses given by. where ψ is an SU(2) isospin doublet, π~ ¼ ðπ 1 ; π 2 ; π 3 Þ is an isospin triplet and σ is an isospin singlet, with Dμ ¼ ∂ μ þ iqAμ. B ð0; −y; x; 0Þ; 2. m2π ¼ λv2 − μ2 ;. ð2Þ. as the covariant derivative. Aμ is the vector potential corresponding to an external magnetic field directed along the ẑ axis, Aμ ¼. m2σ ¼ 3λv2 − μ2 ;. ð3Þ. mf ¼ gv;. respectively. Using Schwinger’s proper-time method, the expression for the one-loop effective potential for one boson field with squared mass m2b and absolute value of its charge qb at finite temperature T in the presence of a constant magnetic field can be written as. and q is the particle’s electric charge. Aμ satisfies the gauge condition ∂ μ Aμ ¼ 0. Since A3 ¼ 0, the gauge field only couples to the charged pion combinations, namely 1 π ¼ pffiffiffi ðπ 1 ∓iπ 2 Þ: 2. σ → σ þ v;. ð1Þ. Vb ¼. ð4Þ. The neutral pion is taken as the third component of the pion isovector, π 0 ¼ π 3 . The gauge field is taken as classical and thus we do not consider loops involving the propagator of the gauge field in internal lines. The squared mass parameter μ2 and the self-coupling λ and g are taken to be positive. To allow for a spontaneous breaking of symmetry, we let the σ field to develop a vacuum expectation value v. TX 2 n ×e. ð1Þ. Vf ¼ −. where LbI and LfI are given by. ð6Þ. dm2b. Z. d3 k ð2πÞ3. Z. ∞. 0. tanhðqb BsÞ −sðω2n þk23 þk2⊥ q Bs þm2b Þ b. Z X XZ dm2f T. r¼1. ×e. 1 1 L ¼ − ½σð∂ μ þ iqAμ Þ2 σ − ð3λv2 − μ2 Þσ 2 2 2 1 1 μ2 − ½~π ð∂ μ þ iqAμ Þ2 π~ − ðλv2 − μ2 Þ~π 2 þ v2 2 2 2 λ 4 − v þ iψ̄γ μ Dμ ψ − gvψ̄ψ þ LbI þ LfI ; 4. Z. ds coshðqb BsÞ ð9Þ. ;. where ωn ¼ 2nπT are boson Matsubara frequencies. Similarly, the expression for the one-loop effective potential for one fermion field with mass mf and absolute value of its charge qf at finite temperature T in the presence of a constant magnetic field can be written as. ð5Þ. which can later be taken as the order parameter of the theory. After this shift, the Lagrangian can be rewritten as. ð8Þ. n. −sðω~ 2n þk23 þk2⊥. d3 k ð2πÞ3. Z. tanhðqf BsÞ 2 qf Bs þmf þrqf BÞ. 0. ;. ∞. ds coshðqf BsÞ ð10Þ. where ω~ n ¼ ð2n þ 1ÞπT are fermion Matsubara frequencies. The sum over the index r corresponds to the two possible spin orientations along the magnetic field direction. It has been shown in Ref. [19], by including the v-independent terms, choosing the renormalization scale as μ~ ¼ e−1=2 μ and after mass and charge renormalization, that the thermomagnetic effective potential in the small-tointermediate field regime in a high-temperature expansion can be written as. 016002-2.
(3) INVERSE MAGNETIC CATALYSIS IN THE LINEAR …. V ðeffÞ. X. PHYSICAL REVIEW D 91, 016002 (2015). μ 2 λ 4 ð4πTÞ2 π 2 T 4 m2i T 2 T 2 3=2 ðm þ ΠÞ ln − 2γ E þ 1 − ¼− v þ v þ þ − 4 12π i 2 90 24 64π 2 2μ2 i¼σ;π 0 X m4 ð4πTÞ2 π 2 T 4 m2i T 2 Tð2qBÞ3=2 1 1 m2i þ Π i ln − 2γ E þ 1 − þ þ ζ − ; þ þ 2 2 2qB 90 24 8π 64π 2 2μ2 i¼π þ ;π − ðqBÞ2 ð4πTÞ2 mi 2 3 mi 4 ln − 2γ ζð5Þ þ 1 þ ζð3Þ − − E 4 2πT 2πT 192π 2 2μ2 2 2 2 2 4 X m4f mf T ðqf BÞ2 ðπTÞ 7π T ðπTÞ2 ln − 2γ E þ 1 þ ln − 2γ E þ 1 ; − − þ 180 12 16π 2 2μ2 24π 2 2μ2 f¼u;d 2. m4i. where q is the absolute value of the charged pions’ charge (q ¼ 1), qu ¼ 2=3, qd ¼ 1=3 are the absolute values of the u and d quarks, respectively and γ E is Euler’s gamma. Though we take the quark masses as equal, the notation emphasizes that the effective potential is evaluated accounting for the different quark charges. We have introduced the leading temperature plasma screening effects for the boson’s mass squared, encoded in the boson’s self-energy Π. For the Hurwitz zeta function ζð−1=2; zÞ in Eq. (11) to be real, we need 2. −μ þ Π > qB;. The diagrams representing the bosons’ self-energies are depicted in Fig. 1. Each column corresponds to the diagrams contributing to the self-energy of a given boson. The total self-energy for any boson is identical to the other’s and thus we concentrate on computing the diagrams in column Fig. 1(a). The contribution from the individual diagrams require of the expressions [hereby capital letters are used to denote four-momenta in Euclidian space, e.g. K ≡ ðωn ; kÞ] Πa1 ðm2σ Þ. ð12Þ. a condition that comes from requiring that the second argument of the Hurwitz zeta function satisfies z > 0, even for the lowest value of m2b which is obtained for v ¼ 0. Furthermore, for the large T expansion to be valid, we also require that qB=T 2 < 1:. ð11Þ. ¼ λT. Πa2 ðm2π0 Þ ¼ λT Πa3 ðm2π Þ ¼ λT. ð13Þ. XZ n. d3 k DðK; m2σ Þ; ð2πÞ3. n. d3 k DðK; m2π0 Þ ð2πÞ3. XZ XZ n. Πa4 ðP; mf Þ ¼ −N f g2 T. d3 k DB ðK; m2π Þ; ð2πÞ3 X Z d3 k n. ð2πÞ3. × Tr SB ðK; mf ÞSB ðP − K; mf Þ;. ð14Þ. where N f is the number of fermions and the corresponding propagators are given by 1 ; K 2 þ m2i 2 2 2 tanhðqBsÞ 2 Z ∞ e−sðωn þk3 þk⊥ qBs þmi Þ 2 ds ; DB ðK; mi Þ ¼ coshðqBsÞ 0 DðK; m2i Þ ¼. (a). (b). Z. (c). SB ðK; mf Þ ¼. FIG. 1. Feynman diagrams contributing to the one-loop bosons’ self-energies. The dashed line denotes the charged pion, the continuous line is the sigma, the double line represents the neutral pion and the continuous line with arrows represents the fermions. Diagrams (a), (b) and (c) correspond to the selfenergy of the σ, π 0 and π , respectively.. 016002-3. ∞. 0. . −sðω~ 2n þk23 þk2⊥. ds. e. tanhðqf BsÞ 2 qf Bs þmf Þ. coshðqf BsÞ. × ðcoshðqf BsÞ − iγ 1 γ 2 sinhðqf BsÞÞ k⊥ × ðmf − k∥ Þ − ; coshðqf BsÞ. ð15Þ.
(4) ALEJANDRO AYALA, M. LOEWE, AND R. ZAMORA. PHYSICAL REVIEW D 91, 016002 (2015). and for the charged particle propagators we have used Schwinger’s proper-time representation. For the computation of Πa3 we work in the infrared limit, namely, P ¼ ð0; p → 0Þ and with the hierarchy of scales qB; m2i < T 2 . It has been shown [20] that this limit can be formally implemented by straightforwardly setting P ¼ 0 in the third line of Eqs. (14). The leading contribution at high temperature from each of these diagrams is T2 12 2 T Πa2 ðm2π 0 Þ ¼ λ 12 2 T Πa3 ðm2π Þ ¼ λ 12. on the diagrams in column Fig. 2(a). Each of the three diagrams involves two propagators of the same boson. For the first two diagrams the intermediate bosons are neutral and for the third one the intermediate bosons are charged. Therefore the expression for the diagrams can be obtained from IðPi ; m2i Þ. Πa1 ðm2σ Þ ¼ λ. Πa4 ð0; mf Þ ¼ N f g2. JðPi ; m2i Þ ¼ T. T2 ; 6. ð16Þ. and therefore, considering the permutation factors, the total self-energy is given by Π ¼ 3Πa1 ðm2σ Þ þ Πa2 ðm2π0 Þ þ 2Πa3 ðm2π Þ þ Πa4 ð0; mf Þ ¼λ. T2 T2 þ N f g2 : 2 6. ð17Þ. III. ONE-LOOP THERMOMAGNETIC COUPLINGS Let us now compute the one-loop correction to the coupling λ, including thermal and magnetic effects. Figure 2 shows the Feynman diagrams that contribute to this correction. The columns, Figs. 2(a), 2(b), 2(c), 2(d), 2(e) and 2(f), contribute to the correction to the σ 4 , ðπ 0 Þ4 , ðπ þ Þ2 ðπ − Þ2 , σ 2 π þ π − , ðπ 0 Þ2 π þ π − and σ 2 ðπ 0 Þ2 terms of the interaction Lagrangian in Eq. (7), respectively. Since each of these corrections lead to the same result, we concentrate. (a). (b). ¼T. XZ n. d3 k DðPi − KÞDðKÞ; ð2πÞ3. n. d3 k DB ðPi − KÞDB ðKÞ; ð2πÞ3. XZ. ð18Þ. where Pi is the total incoming four-momentum and D and DB are the boson propagators defined in Eqs. (15). Once again we work in the infrared limit, namely, Pi ¼ ð0; p → 0Þ and with the hierarchy of scales where qB; m2i < T 2 . It is known that in order to properly implement this hierarchy [21], it is necessary to separate the contribution from the Matsubara zero mode from the rest of the modes in Eq. (18). We therefore write Jð0; m2i Þ ≡ J n¼0 ð0; m2i Þ þ Jn≠0 ð0; m2i Þ X Z d3 k 2 DB ðωn ; kÞDB ðωn ; kÞ Jn≠0 ð0; mi Þ ¼ T ð2πÞ3 n≠0 Z d3 k DB ðkÞDB ðkÞ: Jn¼0 ð0; m2i Þ ¼ T ð2πÞ3. ð19Þ. Jn¼0 is straightforwardly computed with the result Jn¼0 ð0; m2i Þ. T 1 3 1 m2i þ Π ; ¼ ζ ; þ 16π ð2qBÞ1=2 2 2 2qB. ð20Þ. where we have also included the plasma screening effects for the boson’s mass squared [19]. The contribution from the rest of the modes is performed by resorting to the weak field limit of the boson propagator [22]. (d). (c). (e). (f). FIG. 2. One-loop Feynman diagrams that contribute to the thermal and magnetic correction to the coupling λ. The dashed line denotes the charged pion, the continuous line is the sigma and the double line represents the neutral pion. Diagrams (a), (b), (c), (d), (e), (f) correspond to the correction of the σ 4 , ðπ 0 Þ4 , ðπ þ Þ2 ðπ − Þ2 , σ 2 π þ π − , ðπ 0 Þ2 π þ π − and σ 2 ðπ 0 Þ2 vertices, respectively.. 016002-4.
(5) INVERSE MAGNETIC CATALYSIS IN THE LINEAR …. . The sum and integrals in Jn≠0 are performed by means of the Mellin summation technique [23] with the result 1 ð4πTÞ2 2 Jn≠0 ð0; mi Þ ¼ − ln þ 1 − 2γ E 16π 2 2μ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2i þ Π 2 þ ζð3Þ 2πT −. ðqBÞ2 ζð5Þ: 1024π 6 T 4. ð22Þ. We have also included the plasma screening effects for the boson’s mass squared and have carried out the mass renormalization introducing a counter term δm2 ¼ −1=ϵ þ γ E − lnð2πÞ. The function Jð0; m2i Þ is therefore explicitly obtained by adding up Eqs. (20) and (22). The function Ið0; m2i Þ is obtained as the limit when qB → 0 of the function function Jð0; m2i Þ, with the result Ið0; m2i Þ ¼. 0.1 g 0.1. 1 ðqBÞ 1− 2 2 2 2 ωn þ k þ mi ðωn þ k2 þ m2i Þ2 2ðqBÞ2 k2⊥ : ð21Þ þ 2 ðωn þ k2 þ m2i Þ3. T 1 2 8π ðmi þ ΠÞ1=2 1 ð4πTÞ2 ln þ 1 − 2γ E − 16π 2 2μ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2i þ Π 2 : þ ζð3Þ 2πT. 1.1 1.0 0.9. eff. DB ðωn≠0 ; kÞ ¼. PHYSICAL REVIEW D 91, 016002 (2015). 2. 0.8. t 5. 0.7. t 5.1. 0.6. t 5.2. 0.5 0.4 0.0. 0.2. 0.4. 0.6. 0.8. 1.0. b. FIG. 3 (color online). λeff ðv ¼ 0Þ as a function of b ¼ qB=μ2 for three different values of t ¼ T=μ.. Lagrangian of Eq. (7), respectively. Since each of these corrections lead to the same result, we concentrate on the diagrams in column Fig. 4(a). Note that from Eq. (8), for v ¼ 0 the masses of σ and π 0 become degenerate, and since the middle and bottom diagrams contribute with opposite sign, they cancel [incidentally, this also happens with the two bottom diagrams in columns Figs. 4(b) and 4(c)]. We thus concentrate on the top diagram in column Fig. 4(a). The expression for this diagram is written as LðPi ; vÞ ¼ 2Tg3. XZ n. d3 k SB ðK; mu Þ ð2πÞ3. × SB ðPi − K; md ÞDB ðPi − K; mπ Þ;. ð23Þ. ð25Þ. Considering the permutation factors and the contribution from the s, t and u channels, the correction to the selfcoupling λ to one-loop order is given by λeff ¼ λ½1 þ 24λð9Ið0; m2σ Þ þ Ið0; m2π Þ þ 4Jð0; m2π ÞÞ: ð24Þ Note that λeff depends on v through the dependence on the boson masses. Let us furthermore take the approximation where we evaluate λeff at v ¼ 0. The rationale is that we are pursuing the effect on the critical temperature which is the temperature where the curvature of the effective potential at v ¼ 0 vanishes. Figure 3 shows the behavior of λeff ðv ¼ 0Þ as a function of b ¼ qB=μ2 for three different values of t ¼ T=μ. Note that in all cases the effective coupling is a decreasing function of the magnetic field strength. We now turn to the calculation of the thermomagnetic correction of the coupling g. Figure 4 shows the Feynman diagrams that contribute to this correction. We are interested in also computing an effective value for this coupling, geff , for v ¼ 0, in the same manner we did for λeff . The columns Figs. 4(a), 4(b) and 4(c) contribute to the correction to the quark-σ, quark-π 0 and quark-π terms of the interaction. (a). (b). (c). FIG. 4. One-loop Feynman diagrams that contribute to the thermal and magnetic correction to the coupling g. The dashed line denotes the charged pion, the continuous line is the sigma, the double line represents the neutral pion and the continuous line with arrows represents the quarks. Diagrams (a), (b) and (c) correspond to the correction of the quark-σ, quark-π 0 and quarkπ vertices, respectively.. 016002-5.
(6) ALEJANDRO AYALA, M. LOEWE, AND R. ZAMORA. PHYSICAL REVIEW D 91, 016002 (2015). SB ðK; mf Þ ¼. ðmf − KÞ γ 1 γ 2 ðqBÞðmf − K ∥ Þ −i 2 2 K þ mf ðK 2 þ m2f Þ2 K ⊥ ðm2f þ K 2∥ Þ 2ðqBÞ2 K 2⊥ ðmf − K ∥ Þ þ : þ 2 ðK þ m2f Þ4 K 2⊥ ð26Þ. Using Eq. (26) in Eq. (25) and working in the high temperature limit, where we can neglect the incoming momentum in the numerator, we get up to order B2 X Z d3 k 3 LðPi ; vÞ ¼ 2Tg ð2πÞ3 n 1 × 2 2 ððPi − KÞ þ mf ÞððPi − KÞ2 þ m2π Þ ðqBÞ2 − 2 2 ðK þ mf ÞððPi − KÞ2 þ m2π Þ3 þ þ. 2ðqBÞ2 K 2⊥ 2 ððPi − KÞ þ m2f ÞððPi − KÞ2 þ m2π Þ. . 2ðqBÞ2 ðK 2∥ þ m2f Þ. 9ððPi − KÞ2 þ m2f Þ4 ððPi − KÞ2 þ m2π Þ. ; ð27Þ. where since we are pursuing an effective vertex correction, we have written Eq. (27) after averaging over the quark spins. Note that in order to handle high powers in the denominators, we can use the identity 1 ð−1Þn ∂ n 1 ¼ ; ð28Þ 2 2 nþ1 2 2 n! ðK þ mi Þ ∂mi ðK þ m2i Þ and thus, we only need to explicitly calculate the first term in Eq. (27). The sum over Matsubara frequencies can be carried out straightforwardly and the result is XZ n. d3 k. . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 þ m2f ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eπ ¼ k2 þ m2π :. Ef ¼. where we have introduced a counterterm δm2f ¼ −1=ϵ þ γ E − lnð2πÞ to take care of quark-mass renormalization. Note that the quark mass does not appear because, as can be seen from Eq. (8), this is proportional to v and we have computed the correction for v ¼ 0. Therefore the correction to g at one-loop order is given by geff ¼ g½1 þ g2 Lð0; 0Þ:. g 0.1 0.0999370. 0.0999365. 0.0999360. t 5. 0.0999361 0.0999361 0.0999361 0.0999361 0.0999361. 0.0999355. where we have taken the infrared limit, Pi → 0. nf stands for the Fermi-Dirac distribution and. ð32Þ. In order to use a set of values for the couplings λ and g appropriate for the description of the phase transition, note that the curvature of the effective potential vanishes for v ¼ 0. Since the boson thermal masses are proportional to this curvature, they also vanish at v ¼ 0. This observation provides a condition to obtain a relation between the model parameters at T c that can be supplemented with information from the physical masses of the pion and sigma in vacuum [26]. Figure 5 shows the behavior of geff as a. . 1 ð2πÞ3 ððPi − KÞ2 þ m2f ÞððPi − KÞ2 þ m2π Þ Z d3 k 1 − 2nf ðEπ =TÞ 1 − 2nf ðEf =TÞ ; ð29Þ − ¼ Eπ Ef ð2πÞ3. ð30Þ. Therefore, using Eqs. (28) and (29) and performing the integration by means of the method in Ref. [25] and evaluating at v ¼ 0, we get the expression for the effective thermomagnetic correction to the top diagram in column Fig. 4(a) in the infrared limit as 2 2 g3 T π Lð0; 0Þ ¼ 2 −2γ E þ 1 − ln 8π 2μ2 2 2 2 mπ mπ −7 ζð3Þ þ 31 ζð5Þ 2 2 8π T 8π 2 T 2 3410 ðqBÞ2 þ ζð5Þ ; ð31Þ 9 ð4πTÞ4. geff. where Pi represents the incoming momentum and mu;d are the u- and d-quark masses, respectively. We emphasize that the quark masses are taken as equal and that the notation makes reference to the fact that the propagators are evaluated accounting for the different quark charges. The explicit computation is carried out in the weak field limit, with the boson propagator given by Eq. (21) and the fermion propagator given by [24]. t 5.1 t 5.2. 0.0 0.2 0.4 0.6 0.8 1.0. 0.0. 0.2. 0.4. 0.6. 0.8. 1.0. b. FIG. 5 (color online). geff ðv ¼ 0Þ as a function of b ¼ qB=μ2 for three different values of t ¼ T=μ. geff is a growing function of the magnetic field strength, though the growing is rather mild; it can be better appreciated in the inset.. 016002-6.
(7) INVERSE MAGNETIC CATALYSIS IN THE LINEAR … 1.014. PHYSICAL REVIEW D 91, 016002 (2015) eff ,geff. 1.0004 1.0003. 1.012. 1.0002. g 0.8. 0.995. 1.010. 1.0001. 1.008. 1.0000 0.0. 0.5. 1.0. 1.5. 0.990. 2.0. 0. 0. ,g const.. 1.006 1.004. eff ,geff. tc b tc b. tc b tc b. t,b. 1.000. 0.985. 0.15 0.980. 1.002. 0.975. 1.000. 0.970 0.0. 0.5. 1.0. 1.5. 0.2. t,b 0. 0.225 2.5. 2.0. b. 0.0. 0.5. 1.0. 1.5. 2.0. b. FIG. 6 (color online). Effect of the couplings on the critical temperature. The solid curve corresponds to the case where the couplings are taken as constants λ ¼ 0.225 and g ¼ 0.3. The dashed curve corresponds to the calculation where geff , λeff ðT; qB ¼ 0; λ ¼ 0.9; g ¼ 0.3Þ. This last curve is also a growing function of b ¼ qB=μ2 , as can better be seen in the inset, though the growth is less strong than the case computed with λ and g taken as constants.. function of b ¼ qB=μ2 for three different values of t ¼ T=μ. Note that contrary to the behavior of λeff the effective boson-quark coupling geff is a growing function of the magnetic field strength, though the growing is rather mild. IV. CRITICAL TEMPERATURE Let us now study the effect that the thermomagnetic corrections to the couplings have on the critical temperature. We first look at the cases where we set the couplings to their tree-level values and where only thermal effects are included. Figure 6 shows the critical temperature in these. FIG. 8 (color online). Effect of the full thermomagnetic dependence of couplings on the critical temperature for a fixed value of the tree-level g ¼ 0.8 and different values of the treelevel λ as a function of b ¼ qB=μ2 . In all cases the critical temperature is a decreasing function of b.. cases, obtained from setting the second derivative of Eq. (11) equal to zero at v ¼ 0, normalized to the critical temperature for vanishing magnetic field. Note that in both cases the critical temperature is an increasing function of the field strength, though when the thermal effects on the couplings are included, the growth is tamed. Figures 7 and 8 show the critical temperature for the case where we consider the full thermomagnetic dependence of the couplings. Figure 7 shows the case when we set the treelevel coupling λ to a fixed value and vary the tree-level coupling g. Figure 8 shows the complementary case where we set the tree-level coupling g to a fixed value and vary the tree-level coupling λ. Note that in all cases the critical temperature is a decreasing function of the field strength. V. SUMMARY AND CONCLUSIONS. eff ,geff. t,b. 0.225. 1.00. tc b tc b. 0. 0.99. 0.98. g 0.2 g 0.4. 0.97. g 0.6 g 0.8. 0.96 0.0. 0.5. 1.0. 1.5. 2.0. b. FIG. 7 (color online). Effect of the full thermomagnetic dependence of couplings on the critical temperature for a fixed value of the tree-level λ ¼ 0.225 and different values of the treelevel g as a function of b ¼ qB=μ2 . In all cases the critical temperature is a decreasing function of b.. In summary, we have shown that when including the one-loop thermomagnetic effects for the couplings in the linear sigma model with fermions interacting with an external magnetic field, the critical temperature for the chiral transition is a decreasing function of the field strength. This behavior is a direct consequence of the decrease of the boson self-coupling with the field strength. The effect of the fermions is marginal and the main contribution comes from the charged pions. We emphasize that the thermomagnetic dependence of the couplings has been computed—as opposed to assumed—within the model itself. In order to have a better grasp about how we obtain results in this work different from earlier works that were also in the context of the linear sigma model (e.g., Ref. [8]), let us recall that for theories with spontaneous symmetry breaking the particle masses depend on the vacuum expectation value of the condensing boson, v ¼ v0 , and thus when working at T ≠ 0, the expectation value depends on T and so do the particle masses. Therefore v0 plays the role of. 016002-7.
(8) ALEJANDRO AYALA, M. LOEWE, AND R. ZAMORA. PHYSICAL REVIEW D 91, 016002 (2015). an order parameter, with the effective potential being a function of v and reaching its minimum at v ¼ v0 . To carry the analysis that leads to finding the change of v0 with T it is necessary to consider the effective potential evaluated in the full v domain and consequently the dependence of the particle masses on v. It happens that for some v values in this domain, the squares of the particle masses vanish or even become negative. This is the well-known signal for the requirement to go beyond mean field and consider the resummation of the ring diagrams. The approach has been successfully implemented since the pioneering work in Ref. [25] as well as in the context of the Standard Model (e.g. Ref. [27]). At T ≠ 0 and B ¼ 0 inclusion of these diagrams leads to terms that cancel the offending ones. They also produce that, for certain values of the model parameters and some temperatures, the effective potential becomes an expansion containing a cubic term in the order parameter signaling that the description goes beyond the mean field approximation (see Ref. [19] for details). The physics involved is the proper treatment of the plasma screening effects in the infrared, encoded in these ring diagrams. At T ≠ 0 and B ≠ 0 the infrared problems become more severe due to the effective dimensional reduction caused by the discretization of energy levels in the direction transverse to the magnetic field. Nevertheless, it has been shown in Ref. [19] that including the ring diagrams, computed in the presence of the magnetic field, also cures these problems, thereby canceling the new offending infrared terms. In summary, the difference between our approach and previous ones within the context of the linear sigma model is that we have let the particle masses carry its dependence on v and, upon inclusion of the plasma screening effects, have found the self-consistent thermomagnetic effective potential beyond mean field. which is then used to perform the analysis to include a thermomagnetic correction to the couplings. All together this leads to the inverse magnetic catalysis behavior. Though the linear sigma model is not equivalent to QCD, it captures one of its low-energy features, namely, chiral symmetry restoration at finite temperature, whose primary physical consequence is the change of the particle’s masses as a function of the order parameter. As follows from Eqs. (8), since the couplings enter as an ingredient into the boson’s masses, the modification of the couplings introduces a magnetic field dependence on these masses and thus on one of the main parameters describing symmetry restoration. Since the linear sigma model used in this calculation contains no variables that make reference to the deconfinement phase transition such as the Polyakov loop, the inverse magnetic catalysis we find is determined only by parameters associated to chiral symmetry. This finding reinforces the expectation that, since the deconfinement and chiral symmetry restoration transitions are intertwined, except for possible fine details, the description of either of the transitions in a model that emphasizes one or the other of these aspects must give very similar results for values of the physical parameters describing this transition, namely the critical temperature or eventually the location of the critical point.. [1] G. S. Bali, F. Bruckmann, G. Endrodi, Z. 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