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Home , Diquark, Matter, Polarizability, Pseudoscalar meson, Rho meson, Scalar meson

D 97, 076008 (2018)

Neutral and charged scalar , pseudoscalar mesons, and diquarks in magnetic fields

Hao Liu,1,2,3 Xinyang Wang,1 Lang Yu,4 and Mei Huang1,2,5 1Institute of High , Chinese Academy of , Beijing 100049, People’s Republic of China 2School of Physics Sciences, University of Chinese Academy of Sciences, Beijing 100039, China 3Jinyuan Senior High School, Shanghai 200333, China 4Center of and College of Physics, Jilin University, Changchun 130012, People’s Republic of China 5Theoretical Physics Center for Facilities, Chinese Academy of Sciences, Beijing 100049, People’s Republic of China

(Received 16 January 2018; published 17 April 2018)

We investigate both (pseudo)scalar mesons and diquarks in the presence of external magnetic in the framework of the two-flavored Nambu–Jona-Lasinio (NJL) model, where mesons and diquarks are constructed by infinite sum of -loop chains by using random approximation. The function of the quark-loop is calculated to the leading order of 1=Nc expansion by taking the quark propagator in the Landau level representation. We systematically investigate the behaviors of scalar σ , neutral and charged as well as the scalar diquarks, with respect to the strength at finite and chemical potential. It is shown that the numerical results of both neutral and charged pions are consistent with the lattice QCD simulations. The of the charge neutral keeps almost a constant under the magnetic field, which is preserved by the remnant of QCD × QED in the . The mass of the charge neutral scalar σ is around two quark mass and increases with the magnetic field due to the magnetic catalysis effect, which is an typical example showing that the polarized internal quark structure cannot be neglected when we consider the meson properties under magnetic field. For the charged , the one quark-antiquark loop contribution to the charged π increases essentially with the increase of magnetic fields due to the magnetic catalysis of the polarized . However, the one quark-quark loop contribution to the scalar mass is negative comparing with the point- result and the loop effect is small.

DOI: 10.1103/PhysRevD.97.076008

I. INTRODUCTION to be on the order of 1018–1020 G [5]. Finally, in the non- The influence of an external magnetic field on QCD central heavy ion collisions, very strong but short-lived vacuum and has attracted great attention in the past magnetic fields can be generated, of which the strength can B ∼ 1018 few decades (see Refs. [1–3]), since there are at least three reach up to G at Relativistic Heavy Ion Collider B ∼ 1020 high-energy physical systems where strong magnetic fields (RHIC) and G at the Large Collider may play an important role. First, it is predicted by some (LHC) [6,7]. More importantly, heavy ion collisions cosmological models that extremely strong magnetic provides a controllable experimental platform to investigate fields as high as 1020–23 G might be produced during plenty of fascinating effects of strong magnetic fields on the electroweak in the early [4]. strongly interacting matter, for example, the chiral mag- Second, the magnetic fields on the surface of magnetars netic effect (CME) [8–10], the magnetic catalysis [11–13] could reach 1014–1015 G, while in the inner core of and inverse magnetic catalysis [14] effect, and the vacuum magnetars the magnitude of magnetic fields is expected [15,16]. Hadron properties at finite magnetic field have also attracted much interests and prompted many studies on low-lying in magnetic fields, including mesons Published by the American Physical Society under the terms of [15–49], heavy mesons [50–59], and [60–65]. the Creative Commons Attribution 4.0 International license. Especially, for a free pointlike charged particle in a static Further distribution of this work must maintain attribution to B the author(s) and the published article’s title, journal citation, uniform external magnetic field , its energy level has the 3 ε2 p p2 2n − 2 q s 1 qB m2 and DOI. Funded by SCOAP . form of n;sz ð zÞ¼ z þð signð Þ z þ Þj jþ

2470-0010=2018=97(7)=076008(12) 076008-1 Published by the American Physical Society LIU, WANG, YU, and HUANG PHYS. REV. D 97, 076008 (2018) with q the of the particle, n characterizing the magnetic field at finite temperature and chemical potential. Landau levels, sz the projection of particle’s on the In Sec. III we show our numerical results and . magnetic field axis z,andpz the particle’s momentum along Finally, the discussion and conclusion part is given the magnetic field. Forqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a pointlike charged ρ , in Sec. IV. M B m2 − eB its mass ρ ð Þ¼ ρ j j decreases linearly with the II. FORMALISM magnetic filed to zero at the critical magnetic field eBc ¼ m2 ≈ 0 6 2 A. The two flavor NJL model in an ρ . GeV [15], which indicates the instability of the ground state towards the of the charged ρ external magnetic field mesons in the vacuum. It is then checked in the NJL model, We investigate the (pseudo)scalar mesons and the by considering the quark-loop corrections, the charged ρ diquark in the scalar channel by using a low energy mass decreases to zero at a rather small critical magnetic field approximation of QCD effective model SU(2) NJL model 2 eBc ≈ 0.2 GeV [24], which is only 1=3 of the results from under an external magnetic field, the Lagrangian is given the point-particle approximation. The magnetic field strength by [70,71] dependence of the ρ meson mass has been widely inves- 0 tigated by different approaches [15–34], however, the exist- L ¼ ψ¯ ðiD − mˆ þ μγ Þψ ence of charged ρ meson condensation in strong magnetic 1 G ψψ¯ 2 ψ¯ iγ5τψ⃗ 2 − F Fμν: field is still under debate nowadays (see Refs. [20,66–68]). þ S½ð Þ þð Þ 4 μν ð1Þ One of our motivations in this work is to extend our method for charged vector mesons in [24–26] to less debated neutral Where ψ represents the quark field of two light flavors u and charged (pseudo)scalars, e.g., σ and π [20,30,35–49], and d, the current mass mˆ ¼ diagðmu;mdÞ, we can a 1 2 3 which has less debating result at finite magnetic field, and assume that mu ¼ md ¼ m0. τ ¼ðI;τ⃗Þ with τ⃗¼ðτ ;τ ;τ Þ analyze the contribution from the internal structure of corresponding to the Pauli matrix. GS is the mesons, due to the quark-loop corrections, by our method coupling constants corresponding to the (pseudo)scalar – ext in Refs. [24 26], since it causes a remarkable deviation from channel. The covariant derivative, Dμ ¼ ∂μ − iQeAμ , the results of the point-particle approximation for the critical couples quarks to an external magnetic field B ¼ð0;0;BÞ magnetic field of rho . along the positive z direction via a background field, for ext On the other hand, the and example, Aμ ¼ð0; 0; Bx; 0Þ and the field strength is restoration under a strong magnetic field is another sig- ext defined by Fμν ¼ ∂ μA , Q ¼ð−1=3; 2=3Þ is a diagonal nificant issue of QCD, which is deeply related to hadrons’ ½ ν matrix in the flavor which respects to the electric properties, such as the pion mass and the pion decay u d constant via the Gell-Mann-Oakes-Renner (GOR) relation charge of the quark field ( , ). [69]. It means that, exploring the modification of hadrons’ Semibosonizing the above Lagrangian and the Eq. (1) properties in the magnetized hot and/or dense medium, will can be rewritten as help to understand the effects of magnetic fields on the L ψ¯ x iγμD − mˆ μγ0 ψ x chiral phase transition at finite temperature and chemical sb ¼ ð Þð μ þ Þ ð Þ potential. Thus, the behaviors of the mass spectrum and ðσ2 þ π⃗2Þ B2 − ψ¯ ðσ þ iγ5τ⃗· π⃗Þψ − − ; ð2Þ weak decay constant of pions have then been extensively 4GS 2 studied recently [20,22,30,35–49]. However, most of them only focused on the neutral pion and/or sigma meson due to where the Euler-Lagrange equation of for the the difficulty of treating different charged quark propaga- auxiliary fields leads to the constraints as follows: tors under magnetic fields. Therefore, in this paper, we will study not only the pseudoscalar neutral pion and scalar σ σðxÞ¼−2GShψ¯ ðxÞψðxÞi; ð3Þ meson but also the charged pions in the two-flavored NJL model under magnetic fields. In addition, since the mass π⃗ðxÞ¼−2GShψ¯ ðxÞiγ5τψ⃗ ðxÞi: ð4Þ generation of is also an important feature of dynamical chiral symmetry breaking, we further explore For each flavor, by introducing the projector the scalar color 3¯ diquark channel in the magnetic field, 1þγ5 1−γ5 PL ¼ 2 and PR ¼ 2 , we have which will help to probe the properties of in the   magnetized medium in the future. uL u ¼ ; u¯ ¼ðu¯ R u¯ L Þ; ð5Þ This paper is organized as following: in the next section, u we give a general expression of the two-flavor NJL model R   including the scalar and pseudoscalar channels, and then d L ¯ ¯ ¯ derive the polarization functions of both (pseudo)scalar d ¼ ; d ¼ðdR dL Þ; ð6Þ meson channels and the scalar color 3¯ diquark channel in a dR

076008-2 NEUTRAL AND CHARGED SCALAR MESONS, … PHYS. REV. D 97, 076008 (2018) and for two-flavor spinor ψ¯ ¼ðu¯ d¯Þ, the auxiliary neutral σ2 B2 X q eB Xþ∞ Ω − 3 j f j α scalar and pseudoscalar σ and π0 can be written as follows: ¼ 4G þ 2 β p S 2 1 p 0 qf∈f ;− g ¼ Z 3 3 ¯ ¯ ∞ σ ∼ ψψ¯ ¼ u¯ RuL þ dRdL þ u¯ LuR þ dLdR; ð7Þ þ dp3 βE 1 e−βðEqþμÞ × 2 f q þ lnð þ Þ −∞ 4π 0 3 ¯ ¯ π ∼ ψ¯ iγ5τ ψ ¼ iðu¯ LuR − u¯ RuLÞ − iðdLdR − dRdLÞ: ð8Þ −β E −μ þ lnð1 þ e ð q ÞÞg; ð12Þ

It is obviously to find that the neutral scalar σ is symmetric 1 where β ¼ , αp ¼ 2 − δp;0 represents the spin degeneracy. in the flavor space, while the neutral π0 is antisymmetric in T the flavor space and keeps as a pseudo-Goldstone mode. B. The σ The charged π can be represented as: In the framework of the NJL model, the meson is qq¯ pffiffiffi þ − ¯ ¯ π ∼ ψ¯ iγ5τ ψ ¼ 2iðdLuR − dRuLÞ; ð9Þ or resonance, it can be obtained from the quark- antiquark scattering amplitude [72,73]. The meson is pffiffiffi − þ constructed by summing up infinite quark-loop chains in π ∼ ψ¯ iγ5τ ψ ¼ 2iðu¯ LdR − u¯ RdLÞ; ð10Þ the random phase approximation (RPA), the quark loop of σ 1ffiffi the meson polarization function is calculated to the with τ ¼ p ðτ1 iτ2Þ. 2 leading order of 1=Nc expansion, the one-loop polarization The quark-antiquark condensation which gives quark σ Π q ;q function of meson σð ⊥ jjÞ in the magnetic field takes the dynamical mass and the constituent quark mass is the form of [74] obtained as Z d4k Π q ;q −i S˜ k S˜ p ; M m − 2G ψψ¯ ; σð ⊥ kÞ¼ 4 Tr½ ð Þ ð Þ ð13Þ ¼ 0 Sh i ð11Þ ð2πÞ q 0;q ;q ; 0 ;q q ; 0; 0;q p k q We should minimize the effective potential in order to where ⊥ ¼ð 1 2 Þ k ¼ð 0 3Þ. ¼ þ σ obtain the dynamical quark mass, i.e., the σ condensation. corresponds to the momentum conservation and the Landau The one-loop effective potential in this model is given as level representation of the quark propagator S˜ðkÞ is given follows: by [13,75]

  k2 X∞ D QeB; k S˜ k i − ⊥ −1 n nð Þ ; Qð Þ¼ exp QeB ð Þ k2 − k2 − M2 − 2 QeB n ð14Þ j j n¼0 0 3 j j with      2 2 0 0 3 3 1 2 k⊥ 1 2 k⊥ DnðQeB; kÞ¼ðk γ − k γ þ MÞ ð1 − iγ γ signðQeBÞÞLn 2 − ð1 þ iγ γ signðQeBÞÞLn−1 2 jQeBj jQeBj   2 1 1 2 2 1 k⊥ þ 4ðk γ þ k γ ÞLn−1 2 : ð15Þ jQeBj

α 0 Here, Ln are the generalized Laguerre polynomials and Ln ¼ Ln. μ In the rest frame of σ meson, i.e., qσ ¼ðMσ; 0Þ, by using the quark propagator in the Eq. (14), the polarization function of σ is given as follows, Z   d4k X 2k2 X∞ 1 Π q ;q 3i − ⊥ −1 pþk σð ⊥ kÞ¼ 2π 4 exp q eB ð Þ p2 − k2 − M2 − 2 q eB p k2 − k2 − M2 − 2 q eB k ð Þ 2 1 j f j p;k 0 ð 0 3 j f j Þð 0 3 j f j Þ qf¼ ;− ¼  3 3          k2 k2 k2 k2 8 p k M2 L 2 ⊥ L 2 ⊥ L 2 ⊥ L 2 ⊥ × ð k · k þ Þ k p þ k−1 p−1 jqfeBj jqfeBj jqfeBj jqfeBj     2 2 2 1 k⊥ 1 k⊥ − 64k⊥Lk−1 2 Lp−1 2 jqfeBj jqfeBj Z X X∞ 2 dk dk p · k − 2jqfeBjk þ M 6i 0 3 k k q eB α : ¼ 2π 3 p2 − k2 − M2 − 2 q eB p k2 − k2 − M2 − 2 q eB k j f j k ð16Þ ð Þ 2 1 k 0 ð 0 3 j f j Þð 0 3 j f j Þ qf¼3;−3 ¼

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Here, we have used the relations Z       ∞ 2 2 2 2k⊥ k⊥ k⊥ jqfeBj k⊥dk⊥ exp − Lp 2 Lk 2 ¼ δp;k; 0 jqfeBj jqfeBj jqfeBj 4 Z       ∞ 2 2 2 q eB 2k 3 2k⊥ 1 k⊥ 1 k⊥ j f j k⊥dk⊥ exp − Lk−1 2 Lp−1 2 ¼ δp−1;k−1: ð17Þ 0 jqfeBj jqfeBj jqfeBj 8

Moreover, Z     dk X X∞ M2 Π q ;q 6i 3 I 2M2 − σ I q eB α ; σð ⊥ kÞ¼ 2π 2 1 þ 2 2 j f j k ð18Þ ð Þ 2 1 k 0 qf¼3;−3 ¼ where the functions I1 and I2 are given in Appendix A.

  σ Γ We can get the dispersion relation of meson by 1 − 2G Π M − i σ ; 0 0; solving [74] S σ σ 2 ¼ ð21Þ

which is in a complex form. In this paper, we focus on 1 − 2GSΠσðq⊥;q Þ¼0: ð19Þ k computing the pole mass of σ meson, and thus the imaginary part the inverse propagator is simply omitted. And especially, in the rest frame of σ meson, its mass can be In the following paragraph, the same method will be used to determined, determine the masses of π and the diquark.

C. The π 1 − 2GSΠσðMσ; 0Þ¼0: ð20Þ For neutral π0 meson, the one-loop polarization function is Z d4k However, when the meson mass exceeds two times quark 3 ˜ 3 ˜ Ππ0 ðq⊥;q Þ¼−i Tr½iγ5τ SðkÞiγ5τ SðpÞ: ð22Þ mass, the meson is not a stable bound state, but rather a jj ð2πÞ4 resonant state, since it can decay into qq¯ pairs. In this way, Γσ 0 by making a replacement Mσ → Mσ − i 2 [76], the reso- Similarly, p ¼ k þ qπ0 and in the rest frame of the π meson, μ M Γ q M 0 ; 0 Π 0 nant mass σ and its width σ are given by the relation i.e., π0 ¼ð π Þ, the π is written as Z   4 X 2 X∞ d k 2k⊥ p k 1 Π 0 q ;q −3i − −1 þ π ð ⊥ kÞ¼ 2π 4 exp q eB ð Þ p2 − k2 − M2 − 2 q eB p k2 − k2 − M2 − 2 q eB k ð Þ 1 2 j f j p;k 0 ð 0 3 j f j Þð 0 3 j f j Þ qf¼− ; ¼  3 3          k2 k2 k2 k2 8 −p k M2 L 2 ⊥ L 2 ⊥ L 2 ⊥ L 2 ⊥ × ð k · k þ Þ k p þ k−1 p−1 jqfeBj jqfeBj jqfeBj jqfeBj     2 2 2 1 k⊥ 1 k⊥ þ 64k⊥Lk−1 2 Lp−1 2 jqfeBj jqfeBj Z X X∞ 2 dk dk αkδp;kðp · k − 2jqfeBjk − M Þ 6i 0 3 q eB k k ¼ 2π 3 j f j p2 − k2 − M2 − 2 q eB p k2 − k2 − M2 − 2 q eB k ð Þ 1 2 k 0 ð 0 3 j f j Þð 0 3 j f j Þ qf¼−3;3 ¼ Z   dk X X∞ M2 6i 3 I − π0 I q eB α : ¼ 2π 2 1 2 2 j f j k ð23Þ ð Þ 2 1 k 0 qf¼3;−3 ¼

For charged π meson, the one-loop polarization function is (the notation is only for πþ meson, and there is a similar notation for π− meson) Z d4k − ˜ þ ˜ Ππþ ðq⊥;q Þ¼−i Tr½iγ5τ SðkÞiγ5τ SðpÞ; ð24Þ k ð2πÞ4

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þ where p ¼ k þ qπþ . In the rest frame of π meson, the polarization function is given by Z   d4k 9k2 X∞ 1 ⊥ pþk Ππþ q⊥;q −6i exp − −1 ð kÞ¼ 2π 4 2 eB ð Þ p2 − k2 − 4 eB p − M2 k2 − k2 − 2 eB k − M2 ð Þ j j k 0;p 0 ð 0 3 3 j j Þð 0 3 3 j j Þ    ¼ ¼       k2 k2 k2 k2 8 p k − M2 L 2 ⊥ L 2 ⊥ L 2 ⊥ L 2 ⊥ × ð k · k Þ p k−1 þ p−1 k jqueBj jqdeBj jqueBj jqdeBj     2 2 2 1 k⊥ 1 k⊥ þ 64k⊥Lp−1 2 Lk−1 2 jqueBj jqdeBj Z    d3k 9k2 X∞ −6i − ⊥ −1 pþk 4 I0 I00 4 2 q eB k 2 q eB p ¼ 2π 3 exp 2 eB ð Þ ½ ð 1 þ 1Þþ ð j d j þ j u j ð Þ j j k 0;p 0    ¼  ¼      k2 k2 k2 k2 − M2 I0 L 2 ⊥ L 2 ⊥ L 2 ⊥ L 2 ⊥ πþ Þ 2 p k−1 þ p−1 k jqueBj jqdeBj jqueBj jqdeBj     2 2 2 1 k⊥ 1 k⊥ þ 64k⊥Lp−1 2 Lk−1 2 : ð25Þ jqueBj jqdeBj

0 00 0 C T C T Here, the functions I1, I1 and I2 are represented in with ψ ¼ Cψ¯ , ψ¯ ¼ ψ C are charge-conjugate spinors, Appendix A. Because of electric charge difference, we C ¼ iγ2γ0 is the charge conjugation matrix (the superscript could not get a simple form as in Eq. (17) for σ meson, so T denotes the transposition operation), the quark field ψ ≡ we perform the numerical calculation for the integral of k⊥. ψ iα with i ¼ 1, 2 and α ¼ 1, 2, 3 is a flavor doublet and Also we will do the same operation in the diquark case. color triplet, as well as a four-component Dirac spinor, Similarly, we use the following dispersion relation to ðεÞik ≡ εik, ðϵbÞαβ ≡ ϵαβb are totally antisymmetric obtain the mass of π meson in the flavor and color . In this work, we choose 3 GD ¼ 4 GS. 1 − 2G Π 0 q ;q 0: S πþ=π ð ⊥ kÞ¼ ð26Þ Then we introduce the auxiliary diquark fields Δb and Δb

Δb ∼ iψ¯ Cεϵbγ ψ; Δb ∼ iψεϵ¯ bγ ψ C; D. The diquark in the scalar channel 5 5 ð28Þ For the diquark channel, the interaction term in the which are color antitriplet and (isoscalar) singlet under the Lagrangian L consists the term of the form I chiral SUð2ÞL × SUð2ÞR group. ψ¯ Aψ¯ T ψ TBψ ð Þð Þ and the A,B are matrices antisymmetric The dispersion relation for the diquark in the color in Dirac, isospin and color indices. Here we only consider antitriplet scalar channel is given by [77] the scalar channel in the color 3¯ channels, the interaction L – term I;D is given by [77 79] 1 2G Π b q ;q 0; þ D Δ ð ⊥ kÞ¼ ð29Þ L G iψ¯ Cεϵbγ ψ iψεϵ¯ bγ ψ C ; I;D ¼ D½ð 5 Þð 5 Þ ð27Þ where

Z 4 d k ˜ ˜ ΠΔb ðq⊥;q ÞδA0A ¼ −i × 3trΔb ½γ5Sð−d; kÞγ5Sðu; k þ qÞ þ ðu ⇔ dÞ k ð2πÞ4 Z     d4k k2 k2 i 3 − ⊥ − ⊥ ¼ 4 × exp exp ð2πÞ jqdeBj jqueBj X∞ 1 −1 pþk × ð Þ k2 − k2 − M2 − 2 q eB k p2 − k2 − M2 − 2 q eB p p;k 0 ð 0 3 j d j Þð 0 3 j u j Þ  ¼          k2 k2 k2 k2 8 −p k M2 L 2 ⊥ L 2 ⊥ L 2 ⊥ L 2 ⊥ × ð k · k þ Þ k p þ k−1 p−1 jqdeBj jqueBj jqdeBj jqueBj     2 2 2 1 k⊥ 1 k⊥ þ 64k⊥Lk−1 2 Lp−1 2 þðu ⇔ dÞ: ð30Þ jqdeBj jqueBj

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0 0 By introducing the functions I1;I2 defined in the Appendix, we can express the diquark loop polarization function as:     Z 3 2 2 d k k⊥ k⊥ Π b q ;q δ 24i − − Δ ð ⊥ kÞ A0A ¼¼ 3 exp exp ð2πÞ jqdeBj jqueBj     X∞ 1 1 −1 pþk − I0 I00 − q eB k q eB p − M2 I0 × ð Þ 2 ð 1 þ 1Þ j d j þj u j 2 Δb 2 p;k 0  ¼         2 2 2 2 k⊥ k⊥ k⊥ k⊥ × Lk 2 Lp 2 þ Lk−1 2 Lp−1 2 jqdeBj jqueBj jqdeBj jqueBj     2 2 2 1 k⊥ 1 k⊥ þ 64k⊥Lk−1 2 Lp−1 2 þðu ⇔ dÞ: ð31Þ jqdeBj jqueBj

μ p k q b q M b ; 0 Here, ¼ þ Δ and we assume Δb ¼ð Δ Þ (i.e., in the rest frame of diquark).

III. NUMERICAL RESULTS NJL model, there is no for the inverse magnetic catalysis around the critical temperature region, therefore, In the numerical calculation, we use the soft cutoff all results in this work are taken below the critical functions [80] temperature. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Λ10 σ π0 fΛ ¼ ; ð32Þ A. Numerical results for neutral and Λ10 þ k25 We first check the mass behavior of charge neutral σ and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π0 under magnetic fields. Λ10 fk ; The numerical result for the mass square of the σ meson Λ;eB ¼ 10 2 5 ð33Þ Λ þðk3 þ 2jQeBjkÞ is shown in Fig. 2, and the constant mass of the point- particle model is also listed for comparison. It is observed for zero magnetic field and nonzero magnetic field respec- that when the quark-loop polarization under magnetic fields tively. In order to reproduce the pion mass Mπ ¼ 140 MeV, is considered, similar to the behavior of the quark mass, the σ pion decay constant fπ ¼ 92.3 MeV, constituent quark mass mass of the meson increases with magnetic field for M ¼ 336 MeV in the vacuum, we choose the following different and chemical potentials. Comparing 2 σ model parameters: Λ ¼ 616 MeV, GSΛ ¼ 2.02 and the with the constant mass of meson as a point-particle, we current quark mass m0 is 5 MeV. find that the quark-loop contribution or quark polarization In Fig. 1, we show the magnetic field dependence of effect is very essential. On the other hand, at zero temper- quark mass with different fixed temperatures and chemical ature and below the critical chemical potential, we find that potentials. We can see that the quark mass increases when the chemical potential does not affect the mass of the σ the magnetic field strength increases, which is the magnetic meson, and for μ ¼ 0 MeV case, the mass of σ meson catalysis effect obviously. It is noticed that in the regular decreases when temperature increases at fixed magnetic field. 1.0 In Fig. 3, we show the magnetic field dependence of the mass square for neutral pion π0, and compare with the T 0.1 GeV, 0.1 GeV 0.8 result from the point-particle model. The neutral pion mass T 0.01 GeV, 0.25 GeV T 0.1 GeV, 0.25 GeV keeps as a constant as a function of the magnetic field in the 0.6 point-particle model. It is observed that in the vacuum when T ¼ 0, μ ¼ 0, the quark-loop contribution has tiny GeV

M 0.4 effect on neutral pion mass, which almost keeps a constant with the increasing of the magnetic field. This is preserved 0 0.2 by the remnant symmetry and π is the only pseudo- Goldstone in the system. But even it is a tiny effect, 0.0 we still can see that the π0 mass decreases a little first and 0.2 0.4 0.6 0.8 1.0 eB GeV2 then increases a little with the magnetic field, which is in agreement with the lattice result in Ref. [46]. At zero 0 FIG. 1. The eB dependence of the constituent quark mass M temperature when T ¼ 0 MeV, the mass of π meson does with fixed different temperature and chemical potential. not change with different chemical potentials in the chiral

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4 4 T 0 GeV, 0GeV T 0GeV, 0GeV T 0 GeV, 0.1 GeV 3 3 T 0.1 GeV, 0GeV T 0 GeV, 0.2 GeV T 0.16 GeV, 0GeV 2 T 0 GeV, 0.3 GeV 2 Point particle GeV 2 GeV 2 2 2 M M 1 1

0 0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 eB GeV2 eB GeV2

FIG. 2. The magnetic dependence of the mass square of σ meson. Left: The mass square of σ meson with fixed T ¼ 0 MeV with different μ. Right: The mass square of σ meson with fixed μ ¼ 0 MeV with different T.

0.05 0.05 T 0 GeV, 0GeV T 0GeV, 0GeV T 0 GeV, 0.1 GeV 0.04 0.04 T 0.1 GeV, 0GeV T 0 GeV, 0.2 GeV T 0.16 GeV, 0GeV 2 T 0 GeV, 0.3 GeV 2 0.03 0.03 Point particle

GeV Point particle GeV 0.02 2 2 0

0.02 0

M 0.01 M 0.01 0.00

0.00 0.01 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 eB GeV2 eB GeV2

FIG. 3. The magnetic field dependence of the mass square of neutral π0 meson. Left: The mass square of neutral π0 meson with fixed T ¼ 0 MeV with different μ. Right: The mass square of neutral π0 meson with fixed μ ¼ 0 MeV with different T.

symmetry breaking phase, and at zero , the chemical potential μ has no effect on the mass of the increasing temperature lowers the mass of neutral π0 from charged πþ with zero temperature in the chiral symmetry vacuum mass 140 MeV to almost 0 around the critical breaking phase, the increasing temperature shows tiny temperature. effect on the mass of the πþ meson. Obviously, the In the point-particle model, the mass of a neutral meson quark-loop contribution plays an important role by com- either for scalar σ or pseudoscalar π0 should keep a constant paring with the results of the one-loop polarization function as the function of the magnetic field. However, when the and the point-particle model. polarized quark-loop effect is considered, the results for We also investigate the mass square of scalar diquark in scalar σ or pseudoscalar π0 are quite different. Similar to the color anti-triplet channel in magnetic field with different quark mass, the mass of neutral scalar σ linearly rises as the temperature and chemical potential in Fig. 5. The mass of magnetic field and one has MσðeBÞ ≃ 2MðeBÞ. However, scalar diquark increases with the magnetic field. At zero as the only pseudo-Goldstone meson, the neutral π0 mass temperature, the chemical potential affects the mass of keeps as an almost constant value, which is preserved by diquark in the scalar channel slightly. Moreover, at zero the remnant symmetry of the system. chemical potential, the mass of diquark in the scalar channel decreases with the increasing temperature. B. Numerical results for charged π and Δb Generally, the one-loop contribution has important effect Then we analyze the mass behavior for charged π with on the mass of scalar diquark. However, it is important to b π electric charge 1 and Δ with electric charge 1=3 under notice that different from charged , including the quark- the magnetic field. loop contribution, the mass of the scalar diquark increases By solving the gap equation Eq. (26) numerically, the more slowly with magnetic fields comparing with the 2 mass square of charged πþ meson is given in Fig. 4 where point-particle model result when eB < 0.9 GeV , and only we also list the result of the point-particle model. The mass at high magnetic field when eB > 0.9 GeV2, the mass of of charged πþ increases with increasing magnetic field in scalar diquark increases faster with magnetic fields com- different temperature T and chemical potential μ. The paring with the result of the point-particle model.

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5 5 T 0 GeV, 0GeV T 0 GeV, 0 GeV 4 T 0 GeV, 0.1 GeV 4 T 0.1 GeV, 0 GeV T 0 GeV, 0.2 GeV 2 2 T 0.16 GeV, 0 GeV T 0 GeV, 0.3 GeV 3 3 Point particle Point particle GeV GeV 2 2 2 2 M M

1 1

0 0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 eB GeV2 eB GeV2

FIG. 4. The mass square of charged πþ meson in the magnetic field. Left: The mass square of charged πþ meson with fixed T ¼ 0 MeV with different μ. Right: The mass of charged πþ meson with fixed μ ¼ 0 MeV with different T.

FIG. 5. The mass square of diquark in the scalar channel in the magnetic field. Left: The mass of diquark in the scalar channel with fixed T ¼ 0 MeV for different μ. Right: The mass of diquark in the scalar channel with fixed μ ¼ 0 MeV for different T.

C. Quark-loop contribution one quark-loop contribution to σ, π and Δb as the function 0 T 0 μ 0 It is observed that except for the charge neutral π , which of the magnetic field at ¼ , ¼ in Fig. 6. The one σ is the only pseudo- of the system pre- quark-loop contribution to the charge neutral and charged served by the remnant symmetry of QCD × QED, for all π increases essentially with the increase of magnetic fields other scalar mesons including the charge neutral σ, 1 due to the magnetic catalysis of the polarized quarks. At eB > 0 5 2 charged pseudoscalar π, and 1=3 charged scalar diquark strong magnetic fields, when . GeV , the one σ Δb, their masses including the one quark-antiquark loop quark-loop contribution to the mass of charge neutral meson becomes more essential than the charged π, and at contribution under magnetic fields are quite different 2 from the point-particle results. We explicitly show the eB ¼ 1 GeV , the one quark-loop contribution to the mass of charge neutral σ meson can reach 8 times of the point- particle results. However, the one quark-quark loop con- tribution to the scalar diquark mass is negative and less than 50% of the point-particle result below eB < 0.9 GeV2.

IV. DISCUSSION AND CONCLUSION In this paper, we have studied the masses of charge neutral scalar σ and pseudoscalar π0, charged pseudoscalar meson π, and scalar diquark Δb in an external magnetic field in the NJL model, and the mesons and diquark are constructed by summing up infinite quark-loop chains in the random phase approximation (RPA), the quark-loop of mesons polarization function is calculated to the leading order of 1=Nc. 0 FIG. 6. The one quark-loop contribution to the mass of the σ, It is found that the mass of the charge neutral π πþ and the diquark Δb in the scalar channel. keeps almost constant under the magnetic field and the

076008-8 NEUTRAL AND CHARGED SCALAR MESONS, … PHYS. REV. D 97, 076008 (2018) one quark-loop contribution to π0 is tiny. This is because the translation variance part. Comparing the result in both that π0 is the only pseudo-Goldstone boson of the system papers, we can see the first-order expansion already gives a preserved by the remnant symmetry of QCD × QED. From very good result and physical quantities. the point-particle model, the mass of the charge neutral scalar σ should keep as a constant value under the magnetic ACKNOWLEDGMENTS field, however, considering the one quark-loop contribu- tion, the σ mass is around two times quark mass and We thank useful discussion with S. J. Mao. The work of increases with the magnetic field due to the magnetic M. H. is supported by the NSFC under Grants catalysis effect. This is a typical example showing that the No. 11725523, No. 11735007 and No. 11261130311 polarized quark structure cannot be neglected when we (CRC 110 by DFG and NSFC). L. Y. is supported by consider the meson properties under magnetic field, and the the NSFC under Grant No. 11605072 and the Seeds quark-loop contribution to σ mass plays a dominant role Funding of Jilin University. at strong magnetic field, e.g., the quark-loop contribution to σ mass can reach 8 times of the point-particle results at APPENDIX: INTEGRALS eB ¼ 1 GeV2. k For the charged particles, the contribution from the one We have introduced the integrals of 0 for different quark-loop are quite different to the masses of pseudoscalar channels, for the neutral meson, π Δb Z and scalar diquark under magnetic fields. The one dk 1 quark-antiquark loop contribution to the charged π I 0 ; 1 ¼ 2π k2 − ω2 increases essentially with the increase of magnetic fields Z 0 dk 1 due to the magnetic catalysis of the polarized quarks. I 0 ; 2 ¼ 2 2 2 2 ðA1Þ However, the one quark-quark loop contribution to the 2π ðk0 − ω Þððk0 þ Mπ0=σÞ − ω Þ scalar diquark mass is negative and less than 50% of the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eB < 0 9 2 point-particle result below . GeV . 2 2 þ with ω ¼ 2jqfeBjk þ k þ M . For the charged π As mentioned in the Introduction that one of the 3 motivations of this work is to check whether the method meson and the diquark in the scalar channel, we used to calculate the charged vector meson [24–26] is Z dk 1 correct. From the results of pseudoscalar π by using the I0 0 ; 1 ¼ 2π 2 2 ðA2Þ same method, we are confident that the results of charged k0 − ωu;p vector meson in [24–26] is correct at least in the framework Z dk 1 of the NJL model. First, we have reproduced the result for I00 0 ; 0 1 ¼ 2 2 ðA3Þ the charge neutral π , its mass is almost a constant under the 2π k0 − ωd;k magnetic fields, and this is preserved by the remnant Z 0 dk 1 symmetry and π is the only pseudo-Goldstone boson in I0 0 : 2 ¼ 2 2 2 2 ðA4Þ the system. Second, the charged pion mass increases with 2π ððk0 þ q0Þ − ωu;pÞðk0 − ωd;kÞ the magnetic field, which is in agreement with lattice results pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω k2 2 q eB p M2 ω in [45,46]. withpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu;p ¼ 3 þ j u j þ , d;k ¼ At last, we have to mention that when this paper was 2 2 k3 þ 2jqdeBjk þ M .FollowingtheRef.[73],theintegral almost finished, there was a similar paper published [81], k Rof 0 can be replaced by the matsubara sum, where the authors have pointed out that the method we are dk P 0 …: iT ∞ …: , and we can obtain using in [24–26] only take the translation invariance part of 2π ð Þ¼ m¼−∞ð Þ the Schwinger phase in quark propagator. In fact, we only   nfðω − μÞþnfðω þ μÞ − 1 keep the first order of local expansion for the charged field iI1 ¼ − ; ðA5Þ and the phase in our paper, which leads to a cancellation of 2ω

 n ω − μ 1 n −ω − μ 1 iI − fð Þ − fð Þ 2 ¼ 2 2 2 2 2ω ðω þ Mπ =σÞ − ω 2ω ð−ω þ Mπ0=σÞ − ω 0  n ω − μ 1 n −ω − μ 1 fð Þ − fð Þ ; þ 2 2 2 2 ðA6Þ 2ω ð−Mπ0=σ þ ωÞ − ω 2ω ð−Mπ0=σ − ωÞ − ω   nf ωu;p − μ nf ωu;p μ − 1 0 ð Þþ ð þ Þ iI1 ¼ − ; ðA7Þ 2ωu;p

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  nf ωd;k − μ nf ωd;k μ − 1 00 ð Þþ ð þ Þ iI1 ¼ − ; ðA8Þ 2ωd;k  nfðωd;k − μÞ 1 nfð−ωd;k − μÞ 1 iI0 − − 2 ¼ 2 2 2 2 2ωd;k ðωd;k þ Mπþ=Δb Þ − ωu;p 2ωd;k ð−ωd;k þ Mπþ=Δb Þ − ωu;p  n ω − μ 1 n −ω − μ 1 fð u;p Þ − fð u;p Þ ; þ 2 2 2 2 ðA9Þ 2ωu;p ð−Mπþ=Δb þ ωu;pÞ − ωd;k 2ωu;p ð−Mπþ=Δb − ωu;pÞ − ωd;k

1 with nfðxÞ¼ x . 1þeT

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