PhysRevD 106 074002

PHYSICAL REVIEW D 106, 074002 (2022)
Neutral pseudoscalar and vector meson masses under strong magnetic fields
in an extended NJL model: Mixing effects
J. P. Carlomagno ,1,2 D. Gómez Dumm ,1,2 S. Noguera ,3 and N. N. Scoccola2,3,4
1
IFLP, CONICET—Departamento de Física, Facultad de Ciencias Exactas,
Universidad Nacional de La Plata, C.C. 67, (1900) La Plata, Argentina
2
CONICET, Rivadavia 1917, (1033) Buenos Aires, Argentina
3
Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC,
E-46100 Burjassot (Valencia), Spain
4
Physics Department, Comisión Nacional de Energía Atómica,
Av. Libertador 8250, (1429) Buenos Aires, Argentina
(Received 6 June 2022; accepted 9 September 2022; published 7 October 2022)
Mixing effects on the mass spectrum of light neutral pseudoscalar and vector mesons in the presence of
an external uniform magnetic field B ⃗ are studied in the framework of a two-flavor Nambu-Jona-Lasinio
(NJL)-like model. The model includes isoscalar and isovector couplings both in the scalar-pseudoscalar
and vector sectors, and also incorporates flavor mixing through a ’t Hooft-like term. Numerical results for
the B dependence of meson masses are compared with present lattice QCD results. In particular, it is shown
that the mixing between pseudoscalar and vector meson states leads to a significant reduction of the mass
of the lightest state. The role of chiral symmetry and the effect of the alignment of quark magnetic moments
in the presence of the magnetic field are discussed.
DOI: 10.1103/PhysRevD.106.074002
I. INTRODUCTION includes several interesting phenomena, such as the chiral

magnetic effect [10–12], which entails the generation of an
It is well known that the presence of a large background
⃗ has a significant impact on the physics of electric current induced by chirality imbalance, and the
magnetic field B
so-called magnetic catalysis [13,14] and inverse magnetic
strongly interacting particles, leading to important effects on
catalysis [15,16], which refer to the effect of the magnetic
both hadron properties and QCD phase transition features
[1–3]. By a “large” field it is understood here that the order field on the size of chiral quark-antiquark condensates and
of magnitude of B competes with the QCD confining on the restoration of chiral symmetry. Yet another interest-
ing issue is the possible existence of a phase transition of
scale ΛQCD squared, i.e., jeBj ≳ Λ2QCD , jBj ≳ 1019 G.
the cold vacuum into an electromagnetic superconducting
Such huge magnetic fields can be achieved in matter at
state. For a sufficiently large external magnetic field, this
extreme conditions, e.g., at the occurrence of the electro-
transition would be induced by the emergence of quark-
weak phase transition in the early Universe [4,5] or in the
antiquark vector condensates that carry the quantum
deep interior of compact stellar objects like magnetars [6,7].
numbers of electrically charged ρ mesons [17,18]. The
Moreover, it has been pointed out that values of jeBj ranging
presence of such a superconducting (anisotropic and
from m2π to 15 m2π (jBj ∼ 0.3 to 5 × 1019 G) can be reached
inhomogeneous) QCD vacuum state has been discussed
in noncentral collisions of relativistic heavy ions at RHIC
in the past few years and still remains as an open problem
and LHC experiments [8,9]. Though these large background
(see discussions in Refs. [19–24]).
fields are short lived, they should be strong enough to affect
It is clear that the study of the properties of light hadrons,
the hadronization process, offering the amazing possibility
in particular π and ρ mesons, comes up as a crucial
of recreating a highly magnetized QCD medium in the lab.
task toward the understanding of the above mentioned
From the theoretical point of view, the study of strong
phenomena. This represents a nontrivial problem, since
interactions in the presence of a large magnetic field
first-principle theoretical calculations require to deal in
general with QCD in a low energy nonperturbative
regime. Therefore, the corresponding theoretical analyses
Published by the American Physical Society under the terms of have been carried out using a variety of effective models
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to for strong interactions. The effect of intense external
the author(s) and the published article’s title, journal citation, magnetic fields on π meson properties has been studied
and DOI. Funded by SCOAP3. e.g., in the framework of Nambu-Jona-Lasinio (NJL)-like
2470-0010=2022=106(7)=074002(20) 074002-1 Published by the American Physical Society

J. P. CARLOMAGNO et al. PHYS. REV. D 106, 074002 (2022)
models [23,25–40], quark-meson models [41,42], chiral by isospin and angular momentum conservation, but they
perturbation theory (ChPT) [43–45], path integral arise (and may become important) in the presence of the
Hamiltonians [46,47], effective chiral confinement external magnetic field. In fact, our analysis shows that
Lagrangians [48,49] and QCD sum rules [50]. In addition, π 0 − η − ρ0 − ω mixing has a substantial effect on the B
several results for the π meson spectrum in the presence of dependence of the lowest mass state. As a additional
background magnetic fields have been obtained from lattice ingredient, we consider the case of B-dependent effective
QCD (LQCD) calculations [15,51–56]. Regarding the ρ coupling constants; this possibility—inspired by the mag-
meson sector, studies of magnetized ρ meson masses in the netic screening of the strong coupling constant occurring
framework of effective models and LQCD can be found in for large B [71]—has been previously explored in effective
Refs. [18,23,30,34,47,57–60] and Refs. [51,52,54,61,62], models [39,72–75] in order to reproduce the inverse
respectively. magnetic catalysis effect observed at finite temperature
In this work we study the mass spectrum of light neutral in LQCD calculations.
pseudoscalar and vector mesons in the presence of an In the case of the neutral vector mesons, we consider
⃗ considering a two-flavor
external uniform magnetic field B, both states with quantum numbers Sz ¼ 0 and Sz ¼ 1,
NJL-like model [63–65]. In general, in this type of model where Sz is the spin projection in the direction of the
the calculations involving quark loops for nonzero B magnetic field (it is worth noticing that only Sz ¼ 0 states
include the so-called Schwinger phases [66], which are can mix with pseudoscalar states). Most LQCD results and
responsible for the breakdown of translational invariance effective model calculations agree in the finding that the
of quark propagators. However, in the particular case of masses of Sz ¼ 1 states get monotonically enhanced with
neutral mesons these phases cancel out, and one is free to the magnetic field, while results for Sz ¼ 0 mesons are still
take the usual momentum basis to diagonalize the corre- not conclusive [34,47,51,52,54,60,61]. In our framework,
sponding polarization functions [25–29]. One also has to which lacks a description of confinement, for large mag-
care about the regularization procedure, since the presence netic fields the masses of some of the Sz ¼ 0 states are
of the external field can lead to spurious results, such as found to grow beyond the qq̄ pair production threshold;
unphysical oscillations of various observables [67,68]. We therefore our results in this region should be taken just as
consider here a magnetic field independent regularization qualitative ones.
(MFIR) method [27,28,35,69], which has been shown to be The paper is organized as follows. In Sec. II we introduce
free from these effects and reduces the dependence of the the theoretical formalism used to obtain neutral pseudo-
results on model parameters. In addition, in our work we scalar and vector meson masses. Then, in Sec. III we
consider two mixing effects that have been mostly present and discuss our numerical results, while in Sec. IV
neglected in previous analyses. The first one is flavor we provide a summary of our work, together with our main
mixing in the spin zero sector; while we restrict to a two- conclusions. We also include Appendices A–C to provide
flavor model (keeping a reduced number of free parame- some technical details of our calculations.
ters, and assuming that strangeness does not play an
essential role), we consider quark-antiquark interactions II. THEORETICAL FORMALISM
both in I ¼ 1 and I ¼ 0 scalar and pseudoscalar channels,
introducing a ’t Hooft-like effective interaction [70]. The A. Effective Lagrangian and mean field properties
second one is the mixing between pseudoscalar and vector Let us start by considering the Euclidean action for an
mesons, which arises naturally in the context of the NJL extended NJL two-flavor model in the presence of an
model. These mixing contributions are usually forbidden electromagnetic field. We have
Z X
3
SE ¼ d4 x ψ̄ðxÞð−i=
D þ mc ÞψðxÞ − gs ½ðψ̄ðxÞτa ψðxÞÞ2 þ ðψ̄ðxÞiγ 5 τa ψðxÞÞ2
a¼0

2 2
− gv3 ðψ̄ðxÞγ μ τ⃗ ψðxÞÞ − gv0 ðψ̄ðxÞγ μ ψðxÞÞ þ 2gd ðdþ þ d− Þ ; ð1Þ
where ψ ¼ ðudÞT , τa ¼ ð1; τ⃗ Þ, τ⃗ being the usual Pauli- d ¼ det½ψ̄ðxÞð1 γ 5 ÞψðxÞ. The interaction between the
matrix vector, and mc is the current quark mass, which fermions and the electromagnetic field Aμ is driven by the
is assumed to be equal for u and d quarks. The model covariant derivative
includes isoscalar and isovector vector couplings, and also
a ’t Hooft-like flavor-mixing term where we have defined Dμ ¼ ∂μ − iQ̂Aμ ; ð2Þ
074002-2
NEUTRAL PSEUDOSCALAR AND VECTOR MESON MASSES … PHYS. REV. D 106, 074002 (2022)
where Q̂ ¼ diagðQu ;Qd Þ, with Qu ¼ 2e=3 and Qd ¼ −e=3, values given by τa σ̄ a ¼ diagðσ̄ u ; σ̄ d Þ, while vacuum expect-
e being the proton electric charge. In Euclidean space we ation values of other bosonic fields are zero; thus, we write
use the conventions γ 4 ¼ iγ 0 , x4 ¼ it, A4 ¼ iA0 , hence
⃗ We consider the particular case in which Dx;x0 ¼ DMF
x;x0 þ δDx;x :
0 ð6Þ
∂ ¼ γ 4 ∂4 þ γ⃗ · ∇.
=
one has a homogenous stationary magnetic field B ⃗ ori-
The MF piece is diagonal in flavor space. One has
entated along the 3, or z, axis. Then, choosing the Landau
gauge, we have Aμ ¼ Bx1 δμ2 .
x;x0 ¼ diagðDx;x0 ; Dx;x0 Þ;
DMF MF;u MF;d
ð7Þ
Since we are interested in studying meson properties, it is
convenient to bosonize the fermionic theory, introducing
with
scalar, pseudoscalar and vector fields σ a ðxÞ, π a ðxÞ and
ρaμ ðxÞ, with a ¼ 0, 1, 2, 3, and integrating out the fermion ð4Þ 0
DMF;f
x;x0 ¼ δ ðx − x Þð−i=
∂ − Qf Bx1 γ 2 þ M f Þ; ð8Þ
fields. The bosonized Euclidean action can be written as
Z where M f ¼ mc þ σ̄ f is the quark effective mass for each
1
Sbos ¼ − ln det D þ d4 x½σ 0 ðxÞσ 0 ðxÞ þ ⃗πðxÞ · ⃗πðxÞ flavor f.
4g
Z The MF action per unit volume is given by
1
þ d4 x½ ⃗σðxÞ · ⃗σðxÞ þ π 0 ðxÞπ 0 ðxÞ
4gð1 − 2αÞ SMF ð1 − αÞðσ̄ 2u þ σ̄ 2d Þ − 2ασ̄ u σ̄ d
Z Z bos
¼
1 4 1 V ð4Þ 8gð1 − 2αÞ
þ d x ⃗ρμ ðxÞ · ⃗ρμ ðxÞ þ d4 xρ0μ ðxÞρ0μ ðxÞ; Z
4gv3 4gv0 Nc X −1
− ð4Þ d4 xd4 x0 trD ln ðS MF;f
x;x0 Þ ; ð9Þ
ð3Þ V f¼u;d
with where trD stands for the trace in Dirac space, and S MF;f x;x0 ¼
MF;f −1
ðDx;x0 Þ is the MF quark propagator in the presence of the
Dx;x0 ¼ δð4Þ ðx − x0 Þ½−i=
D þ m0 þ τa ðσ a ðxÞ
magnetic field. As is well known, the explicit form of the
þ iγ 5 π a ðxÞ þ γ μ ρaμ ðxÞÞ; ð4Þ propagators can be written in different ways [2,3]. For
convenience we take the form in which S MF;f x;x0 is given by a
where a direct product to an identity matrix in color space is
product of a phase factor and a translational invariant
understood. Note that for convenience we have introduced
function, namely
the combinations
Z
iΦf ðx;x0 Þ 0
g ¼ gs þ gd ; α ¼ gd =ðgs þ gd Þ; ð5Þ
MF;f
S x;x0 ¼ e eipðx−x Þ S̃fp ; ð10Þ
p
so that the flavor mixing in the scalar-pseudoscalar sector is

regulated by the constant α. For α ¼ 0 quark flavors u where Φf ðx; x0 Þ ¼ qf Bðx1 þ x01 Þðx2 − x02 Þ=2 is the so-
and d get decoupled, while for α ¼ 0.5 one has maximum called Schwinger phase. We have introduced here the
flavor mixing, as in the case of the standard version of the shorthand notation
NJL model. Z Z
We proceed by expanding the bosonized action in d4 p
≡ : ð11Þ
powers of the fluctuations of the bosonic fields around p ð2πÞ4
the corresponding mean field (MF) values. We assume that
the fields σ a ðxÞ have nontrivial translational invariant MF Now S̃fp can be expressed in the Schwinger form [2,3]
Z
∞
2 2 2
tanhðτBf Þ p⊥ · γ ⊥
S̃fp ¼ dτ exp −τ M f þ pk þ p⊥ − iϵ ðMf − pk · γ k Þ½1 þ isf γ 1 γ 2 tanhðτBf Þ − ; ð12Þ
0 τBf cosh2 ðτBf Þ
where we have used the following definitions. The Other definitions in Eq. (12) are sf ¼ signðQf BÞ and
perpendicular and parallel gamma matrices are collected Bf ¼ jQf Bj. The limit ϵ → 0 is implicitly understood.
in vectors γ ⊥ ¼ ðγ 1 ; γ 2 Þ and γ k ¼ ðγ 3 ; γ 4 Þ, and, similarly, we The corresponding gap equations can be obtained from
have defined p⊥ ¼ ðp1 ; p2 Þ and pk ¼ ðp3 ; p4 Þ. Note that minimization of the mean field action SMF bos with respect
we are working in Euclidean space, where fγ μ ; γ ν g ¼ −2δμν . to σ̄ f . One obtains in this way
074002-3
Mu ¼ mc − 4g½ð1 − αÞϕu þ αϕd ; B. Neutral meson system

Md ¼ mc − 4g½ð1 − αÞϕd þ αϕu ; ð13Þ As expected from charge conservation, it is easy to
see that the contributions to the bosonic action that are
quadratic in the fluctuations of charged and neutral mesons
where
decouple from each other. In this work we concentrate on
Z Z 2
the neutral meson sector. For notational convenience
N c Mf ∞ e−τMf τBf we will denote isospin states by M ¼ σ a ; π a ; ρaμ, with
ϕf ¼ −N c trD S̃fp ¼− dτ 2 :
p 4π 2 0 τ tanhðτBf Þ a ¼ 0, 3. Here σ 0 , π 0 and ρ0 correspond to the isoscalar
states σ, η and ω, while σ 3 , π 3 and ρ3 stand for the neutral
ð14Þ
components of the isovector triplets a⃗ 0 , π⃗ and ρ⃗ , respec-
tively. Thus, the corresponding quadratic piece of the
Notice that, as anticipated, Eqs. (13) get decoupled for bosonized action can be written as
α ¼ 0. On the other hand, for α ¼ 0.5 the right-hand sides
Z X
of these equations become identical, thus in that case one 1
gets M u ¼ Md .
quad;neutral
Sbos ¼ d4 xd4 x0 δMðxÞGMM0 ðx; x0 ÞδM0 ðx0 Þ:
2 M;M 0
The integral in Eq. (14) is divergent and has to be
properly regularized. As stated in the Introduction, we use ð18Þ
here the magnetic field independent regularization (MFIR)
scheme: for a given unregularized quantity, the correspond- The functions GMM0 ðx; x0 Þ can be separated in two terms,
ing (divergent) B → 0 limit is subtracted and then it is namely
added in a regularized form. Thus, the quantities can be
separated into a (finite) “B ¼ 0” part and a “magnetic” 1
GMM0 ðx; x0 Þ ¼ δ 0 δð4Þ ðx − x0 Þ þ J MM0 ðx; x0 Þ; ð19Þ
piece. Notice that, in general, the “B ¼ 0” part still depends 2gM MM
implicitly on B (e.g., through the values of the dressed
quark masses Mf ), hence it should not be confused with where δMM0 is an obvious generalization of the Kronecker
the value of the studied quantity at vanishing external field. δ, and the constants gM are given by
The divergence in the “B ¼ 0” terms are treated here using 8
> g for M ¼ σ 0 ; π 3
a 3D cutoff regularization scheme. >
>
Following this procedure, the expression in Eq. (14) is < gð1 − 2αÞ for M ¼ σ 3 ; π 0
regularized as gM ¼ : ð20Þ
>
> gv for M ¼ ρ3μ
>
: 3
0;reg gv0 for M ¼ ρ0μ
ϕreg
f ¼ ϕf þ ϕmag
f ; ð15Þ
The polarization functions J MM0 ðx; x0 Þ can be separated
where into u and d quark pieces,
ϕ0;reg ¼ −N c M f I 1f ; ϕmag ¼ −N c M f I mag ð16Þ J MM0 ðx; x0 Þ ¼ F uMM0 ðx0 ; xÞ þ εM εM0 F dMM0 ðx0 ; xÞ: ð21Þ
f f 1f :
Here εM ¼ 1 for the isoscalars M ¼ σ 0 ; π 0 ; ρ0μ and εM ¼

The form of I 1f for the 3D cutoff regularization is given
−1 for M ¼ σ 3 ; π 3 ; ρ3μ, while the functions F fMM0 ðx0 ; xÞ are
by Eq. (A3) of Appendix A [64], while the function I mag
1f , found to be
which depends explicitly on B, reads [13,67]
0
Z F fMM0 ðx0 ; xÞ ¼ N c trD ½S MF;f M MF;f M

x;x0 Γ S x0 ;x Γ ; ð22Þ
−τM2f
1 ∞ τBf e
I mag
1f ¼ dτ 2 −1
4π 2
0 τ tanhðτB fÞ with
8
Bf 1 ln2π
¼ 2 lnΓðxf Þ − xf − lnxf þ xf − ð17Þ >
<1
> for M ¼ σ 0 ; σ 3
;
2π 2 2
Γ ¼ iγ 5
M for M ¼ π 0 ; π 3 : ð23Þ
>
>
where xf ¼ M 2f =ð2Bf Þ. : γμ for M ¼ ρ0μ ; ρ3μ
It is easy to see that ϕ0;regu and ϕ0;reg
d are in fact the
regularized expressions for the quark-antiquark conden- As stated, since we are dealing with neutral mesons, the
sates, which can be obtained from the mean field action by contributions of Schwinger phases associated with the
partial derivation with respect to the current quark masses quark propagators in Eq. (10) cancel out, and the polari-
(i.e., ϕf ¼ hψ̄ f ψ f i, f ¼ u; d). zation functions depend only on the difference x − x0 , i.e.,
074002-4
they are translationally invariant. After a Fourier trans- polarization function is evaluated at the meson rest frame.
formation, the conservation of momentum implies that the In this way, the scalar meson sector gets decoupled at
polarization functions turn out to be diagonal in the this level; we will not take into account these mesons in
momentum basis. Thus, in this basis the neutral meson what follows. It can also be shown that Ĵρaμ ρbν , with a,
contribution to the quadratic action can be written as
b ¼ 0, 3, vanish for μ ≠ ν, while the functions Ĵπa ρbμ , with
Z
1X a, b ¼ 0, 3, turn out to be proportional to δμ3 .
Squad;neutral ¼ δMð−qÞGMM0 ðqÞδM 0 ðqÞ: ð24Þ
bos
2 M;M0 q It is found that all nonvanishing polarization functions are
in general divergent. As done at the MF level, we consider
Now we have the magnetic field independent regularization scheme, in
which we subtract the corresponding “B ¼ 0” contributions
1 and then we add them in a regularized form. Thus, for a
GMM0 ðqÞ ¼ δ 0 þ JMM0 ðqÞ; ð25Þ
2gM MM generic polarization function ĴMM0 we have
0;reg
and the associated polarization functions are given by Ĵ reg mag
MM0 ¼ Ĵ MM0 þ Ĵ MM0 : ð28Þ
JMM0 ðqÞ ¼ FuMM0 ðqÞ þ εM εM0 FdMM0 ðqÞ: ð26Þ The regularized “B ¼ 0” pieces Ĵ 0;reg MM0 are given in
Appendix A; it is easy to see that all nondiagonal polari-
The functions FfMM0 ðqÞ read zation functions Ĵ0;reg 0
MM0 , M ≠ M , are equal to zero. In the case
Z of the “magnetic” contributions Ĵmag MM 0 , after a rather long
0
FfMM0 ðqÞ ¼ N c trD ½S̃fpþ ΓM S̃fp− ΓM ; ð27Þ calculation it is found that they can be expressed in the form
p given by Eq. (26), viz.
where we have defined p ¼ p q=2, and the quark Ĵmag u;mag d;mag
MM0 ¼ F̂MM0 þ εM εM F̂MM 0 ; ð29Þ
0
propagators S̃fp in the presence of the magnetic field have
been given in Eq. (12). where the functions F̂f;mag
MM0 are given by
It is relatively easy to see that the functions Jσ a πb ðqÞ are
zero for either a or b equal to 0 or 3. However, the F̂f;mag mag 2 mag 2
π a π b ¼ −N c ½I 1f − m I 2f ð−m Þ;
remaining polarization functions do not vanish in general.
2
Since we are interested in the determination of meson F̂f;mag f;mag mag
π a ρbμ ¼ −F̂ρaμ π b ¼ iN c I 3f ð−m Þδμ3 ;
masses, we consider here the particular case in which 2 ⊥ 2 mag 2
mesons are at rest, i.e., we take q⃗ ¼ 0, q24 ¼ −m2 , where m F̂f;mag mag
ρaμ ρbν ¼ N c ½I 4f ð−m Þ1μν þ m I 5f ð−m Þδμ3 δν3 ; ð30Þ
stands for the corresponding meson mass. In that situation
the nondiagonal polarization functions that mix the neutral with 1⊥ ¼ diagð1; 1; 0; 0Þ. The expression for I mag
1f has
mag
scalar and vector mesons also vanish, i.e., for a, b ¼ 0, 3 been given in Eq. (17), whereas the integrals I nf for
one has Ĵσa ρbμ ¼ 0, where the notation Ĵ indicates that the n ¼ 2; …; 5 read
Z 1
2 1 1
I mag
2f ð−m Þ ¼ 2 dv ψðx̄f Þ þ − ln x̄f ;
8π 0 2x̄f
Z
2
sf M f Bf 1
I mag
3f ð−m Þ ¼ dvðv2 þ 4M2f =m2 − 1Þ−1 ;
π2m 0
Z Z
mag 2 mag m2 1
2 1X 1 2
I 4f ð−m Þ ¼ −I 1f − dvðv þ γÞ ln x̄f − dvðv þ sv=λ þ γÞψðx̄f þ ð1 þ svÞ=2Þ ;
16π 2 0 2 s¼1 0
Z 1
mag 2 1 2 1
I 5f ð−m Þ ¼ 2 dvð1 − v Þ ψðx̄f Þ þ − ln x̄f ; ð31Þ
8π 0 2x̄f
where λ ¼ m2 =ð4Bf Þ, γ ¼ 1 þ 4M2f =m2 and x̄f ¼ ½M2f − ð1 − v2 Þm2 =4=ð2Bf Þ. For m < 2Mf these integrals are well
2
defined. In fact, in the case of I mag
3f ð−m Þ one can even get the analytic result
0 1
2
sf B f B m C
I mag
3f ð−m Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi arctan@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA; m < 2M f : ð32Þ
2π 2 1 − m2 =ð4M 2f Þ 2Mf 1 − m2 =ð4M 2f Þ
074002-5
2
In the case of I mag
4f ð−m Þ, it is worth noticing that in the
antiparallel to the magnetic field—from those with polari-
2 ð0Þ
limit m → 0 the second term on the rhs of the correspond- zation ϵμ , which have spin projections Sz ¼ 0—spin
ing expression in Eqs. (31) is found to be equal to I mag
1f .
perpendicular to the magnetic field. From Eqs. (30) it is
mag
Thus, in this limit one has I 4f → 0, as it is required in seen that for nonzero B pseudoscalar mesons get coupled
order to avoid a nonzero contribution to the photon mass only to neutral vector mesons with spin projection Sz ¼ 0,
coming from the “magnetic piece” of the polarization which we call ρ0⊥ and ρ3⊥ . In fact, this is expected from
function. On the other hand, for m ≥ 2M f (i.e., beyond the invariance under rotations around the direction of B.⃗
the qq̄ production threshold) the integrals are divergent. In this way, taking into account Eq. (25), one can
To obtain finite results we perform in this case analytic define a 4 × 4 matrix G⊥ with elements GMM0 , where
extensions. The corresponding expressions, as well as some M; M 0 ¼ π 0 ; π 3 ; ρ0⊥ ; ρ3⊥ . The pole masses of Sz ¼ 0
ðkÞ
technical details, are given in Appendix B. physical mesons, m⊥ (with k ¼ 1; …; 4), will be given
The vector fields ρ0μ and ρ3μ can be written in a by the solutions of
polarization vector basis. Since we assume that the mesons
ðS Þ
are at rest, we can choose polarization vectors ϵμ z det G⊥ ¼ 0: ð34Þ
associated to spin projections Sz ¼ 0; 1, namely
For Sz ¼ 1 states we call our vector states ρ0k and ρ3k . In
ð0Þ ð1Þ 1 this case, we get two identical 2 × 2 matrices Gk with
ϵμ ¼ ð0; 0; 1; 0Þ; ϵμ ¼ pffiffiffi ð1; i; 0; 0Þ;
2 elements GMM0 , where M; M0 ¼ ρ0k ; ρ3k . The pole masses
1 ðkÞ
ϵμ
ð−1Þ
¼ pffiffiffi ð1; −i; 0; 0Þ ð33Þ of Sz ¼ 1 physical mesons, mk (with k ¼ 1, 2), will be
2 given by the solutions of
Notice that the fourth components of these vectors, which
are given in Euclidean space, are related to the temporal det Gk ¼ 0: ð35Þ
components of the polarization vectors in Minkowski
space. We find it convenient to distinguish between vector Once the masses are determined, the spin-isospin com-
ð1Þ
states with polarization ϵμ , which have spin projections position of the physical meson states jki is given by the
Sz ¼ þ1 or Sz ¼ −1—i.e., the spin is parallel or corresponding eigenvectors cðkÞ . Thus, one has
ðkÞ ðkÞ ðkÞ ðkÞ

jki ¼ cπ0 jπ 0 i þ cπ3 jπ 3 i þ icρ0⊥ jρ0⊥ i þ icρ3⊥ jρ3⊥ i; k ¼ 1; …; 4 for Sz ¼ 0 states;
ðkÞ ðkÞ
ð36Þ
jki ¼ cρ3k jρ0k i þ cρ3k jρ3k i; k ¼ 1; 2 for Sz ¼ 1 states:
It is also useful to consider the flavor basis π f , ρf , where 1 4N

Gfk ð−m2 Þ ¼ þ c ½ð2M2f þ m2 ÞI 2f ð−m2 Þ
f ¼ u, d. Isospin states can be written in terms of flavor 2gv 3
states using the relations
− 2M2f I 2f ð0Þ þ 2N c I mag 2
4f ð−m Þ; ð38Þ
1 1
jπ 0 i ¼ pffiffiffi ðjπ u i þ jπ d iÞ; jπ 3 i ¼ pffiffiffi ðjπ u i − jπ d iÞ; where the expression for I 2f ðq2 Þ can be found in
2 2
Appendix A, and I mag 2
1 1 4f ð−m Þ has been given in Eqs. (31).
jρ0⊥ i ¼ pffiffiffi ðjρu⊥ i þ jρd⊥ iÞ; jρ3⊥ i ¼ pffiffiffi ðjρu⊥ i − jρd⊥ iÞ: A similar situation occurs in the Sz ¼ 0 sector if one has
2 2
α ¼ 0. In this particular case there is no flavor mixing either
ð37Þ in the pseudoscalar or vector meson sectors, hence the
4 × 4 matrix G⊥ can be written as a direct sum of 2 × 2
In the Sz ¼ 1 sector, where there is no mixing between flavor matrices Gu⊥ and Gd⊥ . Moreover, for a given value
pseudoscalar and vector mesons, the states jρuk i and jρdk i of B, the meson masses of, e.g., u-like mesons (solutions of
turn out to be the mass eigenstates that diagonalize Gk . the equation det Gu⊥ ¼ 0) can be obtained from those of
This can be easily understood noticing that the external d-like mesons for B0 ¼ 2B, since jQu j ¼ 2jQd j and
magnetic field distinguishes between quarks that carry det Gf⊥ depends on Qf and B only through the combina-
different electric charges, and this is what breaks the tion Bf ¼ jQf Bj (this also holds for the implicit depend-
u − d flavor degeneracy. In the flavor basis one has ence on Qf and B through the quark effective masses Mf ).
Gk ¼ diagðGuk ; Gdk Þ, where If one has α ≠ 0 this relation is no longer valid, and in
074002-6
0.9
general G⊥ cannot be separated into flavor pieces. In fact,
as we discuss below, in the pseudoscalar sector it is seen 0.8
that chiral symmetry largely dominates over flavor sym- 0.7

metry; for the range of values of B considered in this work,
0.6
we find that even for α ≪ 1 the lightest Sz ¼ 0 mass
Mf [ GeV ]
eigenstates are very close to isospin states π 3 and π 0 , 0.5
instead of approximating to flavor states π f . 0.4

Mu
0.3 Md
III. NUMERICAL RESULTS 0.2 α = 0.0
α = 0.1
A. Model parametrization and mean field results 0.1 α = 0.5
To obtain numerical results for the dependence of meson 0.0
masses on the external magnetic field, one first has to fix 0.0 0.2 0.4 0.6 0.8 1.0
the parameters of the model. Here we take the parameter eB [ GeV2 ]

set mc ¼ 5.833 MeV, Λ ¼ 587.9 MeV and gΛ2 ¼ 2.44,
which—for vanishing external field—lead to effective FIG. 1. Effective quark masses M u (red upper band), M d (blue
lower band) as functions of eB. The extremes of the bands
quark masses M f ¼ 400 MeV and quark-antiquark con-
correspond to α ¼ 0 (dashed lines) and α ¼ 0.5 (full line). The
densates ϕ0f ¼ ð−241 MeVÞ3 , for f ¼ u, d. This para- dotted lines correspond to α ¼ 0.1.
metrization properly reproduces the empirical values of
the pion mass and decay constant in vacuum, namely
mπ ¼ 138 MeV and f π ¼ 92.4 MeV. Regarding the vector where ΔΣf ¼−2mc ½ϕf ðBÞ−ϕ0f =D4 , D¼ð135×86Þ1=2 MeV
couplings, we take gv3 ¼ 2.651=Λ2 , which leads to mρ ¼ being a phenomenological normalization constant. In the
770 MeV at B ¼ 0, and gv0 ¼ gv3 , which is consistent with left and right panels of Fig. 2 we plot the values of ΔΣ̄
the fact that mρ ≃ mω at vanishing external field. For these and Σ− , respectively, as functions of eB. The gray bands
correspond to LQCD values taken from Ref. [16], whereas
constants we use from now on the notation gv ≡ gv0 ¼ gv3 .
the red bands cover our results for the range α ¼ 0 to
Finally, as stated in Sec. II A, the amount of flavor mixing α ¼ 0.5. We observe from this figure that the model
induced by the ’t Hooft-like interaction is controlled by the reproduces properly the zero-temperature magnetic cata-
parameter α. In this work we choose to take as a reference lysis found in LQCD calculations. Moreover, it is seen that
value α ¼ 0.1, since it leads (at B ¼ 0) to an approximate η the dependence on the flavor mixing parameter α is
meson mass mη ≃ 520 MeV, in reasonable agreement with rather mild.
the physical value mphys
η ¼ 548 MeV. In fact, this mass is
very sensitive to minor changes in α. An alternative B. Pseudoscalar and Sz = 0 vector meson sector
estimate for this parameter can be obtained from the
In this subsection we present and discuss the results
η − η0 mass splitting within the 3-flavor NJL model [76],
associated with the coupled system composed by neutral
which leads to α ≃ 0.2 [77]. In any case, to obtain a full
pseudoscalar mesons and Sz ¼ 0 neutral vector mesons. As
understanding of the effects of flavor mixing we will also ðkÞ
consider the values α ¼ 0 and α ¼ 0.5, corresponding to discussed in Sec. II B, the corresponding masses m⊥ ,
the situation in which flavors are decoupled and in which k ¼ 1; …; 4, can be obtained from Eq. (34). The depend-
there is full flavor mixing, respectively. It is easily seen ence of these masses with the magnetic field for the
that for α ¼ 0 the π and η mesons have equal (finite) reference value α ¼ 0.1 are shown in Fig. 3. As discussed
masses, while when α approaches 0.5 the mass of the pion below, the spin-isospin compositions of the associated
stays finite and that of the η meson becomes increas- states do not coincide in general with those of the usual
ingly large. B ¼ 0 states π 0 , η, ρ0 and ω. For this reason, we use for
In Fig. 1 we show the numerical results obtained for the these states the notation M̃, where in each case M is the
ðkÞ
magnetic field dependence of the dynamical quark masses state that has the larger weight cM in the spin-isospin
Mu and M d . Both masses are found to get increased with B, decomposition given by Eq. (36) (see Table I). In Fig. 3
and it is seen that for M u (Md ) the slope becomes larger we also show the qq̄ production thresholds m ¼ 2M d and
(smaller) as α decreases from α ¼ 0.5—where both masses m ¼ 2Mu (dotted and short-dotted lines, respectively),
coincide—to α ¼ 0. Next, in Fig. 2, we show the depend- beyond which some of the matrix elements of G⊥ get
ence of normalized light quark-antiquark condensates on B. absorptive parts. The presence of these absorptive parts
Following Ref. [16], we introduce the definitions implies that for the states ρ̃ and ω̃ there are certain values of
the magnetic field above which the associated particles are
ΔΣu þ ΔΣd unstable with respect to an unphysical decay into a qq̄ pair.
ΔΣ̄ ¼ ; Σ− ¼ ΔΣu − ΔΣd ; ð39Þ
2 In fact, the existence of such decays is a well known feature
074002-7
1.2 1.2
α=0
α = 0.1
1.0 1.0
α = 0.5
LQCD [15]
0.8 This work 0.8
ΔΣ
Σ−
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
2 2
eB [ GeV ] eB [ GeV ]
FIG. 2. Normalized average condensate (left) and normalized condensate difference (right) as functions of eB, for values of α from
0 to 1 (see text for definitions). LQCD results from Ref. [16] (gray bands) are added for comparison.
of the NJL model, even in the absence of an external field to the charges of the quark components. In the case α ¼ 0,
[63–65]; it arises as a consequence of the lack of a although there is no flavor mixing, flavor degeneracy gets
confinement mechanism, which is a characteristic of this broken due to the magnetic field. Therefore, mass eigen-
type of model. In the presence of the magnetic field, one states turn out to be separated into particles with pure u or d
also has to deal with new poles that may arise from the quark content. If we use the labels k ¼ 1, 3 and k ¼ 2, 4 for
thresholds related to the Landau level decomposition of the u- and d-like states respectively, we get [see Eqs. (37)]
intermediate quark propagators. As customary, we will
assume that the widths associated to these unphysical
decays are small. Then, to determine the values of the 1.2 ω
ρ
corresponding masses, we consider an extremum condition
for the meson propagators, similar to the method discussed, 1.0
2 Mu
e.g., in Ref. [78]. It has to be kept in mind, however,
2 Md
that these predictions for the meson masses are less reliable
mM [ GeV ]
0.8
in comparison to those obtained for the states lying below
the quark pair production threshold, and should be taken
0.6
just as qualitative results. For this reason, in Fig. 3 we use
dashed lines to plot ρ̃ and ω̃ masses above the 2Md η
threshold. It can be seen that for eB ≃ 0.15 GeV2 there 0.4
is a small bump in the curve for the ρ̃ mass. This can be

attributed to the mixing between vector meson states, since 0.2
π
in this region the ω̃ becomes unstable. Some similar
behavior has been found in Ref. [79], related with the 0.0
exchange of dominant scalar and pseudoscalar components 0.0 0.2 0.4 0.6 0.8 1.0
of mass eigenstates. eB [ GeV ]2
It is interesting at this stage to discuss the spin-isospin
composition of mass states and their variation with the FIG. 3. Masses of Sz ¼ 0 mesons as functions of eB,
external field. As mentioned at the end of Sec. II B, for α ¼ 0.1. The dotted lines indicate uū and dd̄ production
the magnetic field tends to separate the states according thresholds.
074002-8

ðkÞ ðkÞ cπ cπ cρ cρ
jki ¼ cπu jπ u i þ icρu⊥ jρu⊥ i ¼ puffiffiffi jπ 0 i þ puffiffiffi jπ 3 i þ i pu⊥ffiffiffi jρ0⊥ i þ i pu⊥ffiffiffi jρ3⊥ i; k ¼ 1; 3;
2 2 2 2
ðkÞ ðkÞ cπ cπ cρ cρ
jki ¼ cπd jπ d i þ icρd⊥ jρd⊥ i ¼ pdffiffiffi jπ 0 i − pdffiffiffi jπ 3 i þ i pd⊥ffiffiffi jρ0⊥ i − i pd⊥ffiffiffi jρ3⊥ i; k ¼ 2; 4: ð40Þ
2 2 2 2
ð1Þ ð3Þ ð2Þ ð4Þ

For definiteness, let us take m⊥ < m⊥ , m⊥ < m⊥ . Since though α is relatively small, the effect of flavor mixing is
for α ¼ 0 and B ¼ 0 the Lagrangian shows an approximate already very strong; the spin-isospin composition is clearly
symmetry under SUð2ÞA ⊗ Uð1ÞA chiral transformations, dominated by the π 3 component, which is given by an
spontaneous symmetry breaking leads to four pseudo- antisymmetric equal-weight combination of u and d quark
Goldstone bosons, viz. the three pions and the η meson. flavors. Thus, the mass states are far from satisfying the
In the presence of the magnetic field, chiral symmetry flavor disentanglement expected for the case α ¼ 0 [see
is explicitly broken from SUð2ÞA ⊗ Uð1ÞA down to Eqs. (40)], in which one has two approximate Goldstone
Uð1ÞT 3 ;A ⊗ Uð1ÞA ; thus, one still has two neutral mesons— bosons. In fact, once α is turned on, explicitly breaking the
combinations of the neutral pion and the η—that remain as Uð1ÞA symmetry, π 3 is the only state that remains being a
pseudo-Goldstone bosons. Moreover, according to the pseudo-Goldstone boson; this forces the lowest-mass state
previous discussion, the latter must be pure u and d-states. π̃ to be dominated by the π 3 component. As discussed
Since they should be approximate mass eigenstates, above, the presence of the magnetic field distinguishes
ð1Þ ð1Þ ð2Þ ð2Þ between flavor components π u and π d instead of isospin
one expects to find ðcπu ; cρu⊥ Þ ≈ ðcπd ; cρd⊥ Þ ≈ ð1; 0Þ and
ð3Þ ð3Þ ð4Þ ð4Þ states. However, it is found that even for values of α as
ðcπu ; cρu⊥ Þ ≈ ðcπd ; cρd⊥ Þ ≈ ð0; 1Þ. On the other hand, for small as 0.01 the mass state π̃ is still dominated by the π 3
α ≠ 0 the presence of the ’t Hooft term introduces flavor ð1Þ
component (jcπ3 j2 ≳ 0.9) for the full range of values of eB
mixing at the level of scalar and pseudoscalar four-quark
considered here. In other words, extremely large magnetic
interactions, breaking the Uð1ÞA symmetry. Thus, the spin-
fields would be required in order to rule the composition of
isospin decomposition gets the more general form given in
light mass eigenstates, which is otherwise dictated by the
Eq. (36), where the lightest state can still be identified as an
invariance under Uð1ÞT 3 ;A transformations. Coming back to
approximate Goldstone boson. When α approaches 0.5, the η̃
the case α ¼ 0.1, we see that, although relatively small, the
mass goes to infinity and, accordingly, the jπ 0 i component in
effect of the magnetic field on the composition of the π̃ state
Eq. (36) disappears from the remaining states.
can be observed from the values in Table I. When eB gets
In Table I we quote the composition of the mass
increased, it is found that there is a slight decrease of the
eigenstates M̃ described in Fig. 3, for some representative
component π 3 in favor of the others. In addition, a larger
values of the magnetic field. For completeness, the coef-
weight is gained by the u-flavor components, as one can see
ficients corresponding to both spin-isospin and spin-flavor
by looking at the entries corresponding to the spin-flavor
basis are included. We note that while the mass eigenvalues ð1Þ
do not depend on whether B is positive or negative, the states (last four columns of Table I): one has jcπu j2 þ
ð1Þ 2
corresponding eingenvectors do. The relative signs in jcρu⊥ j ¼ 0.50ð0.66Þ for eB ¼ 0.05ð1.0Þ GeV2. This can
Table I correspond to the choice B > 0. be understood noticing that the magnetic field is known to
Let us first discuss the composition of the π̃ state (k ¼ 1), reduce the mass of the lowest neutral meson state
which is the one that has the lowest mass. We see that even [51,54,55]. Thus, for large eB it is expected that π̃ will
TABLE I. Composition of the Sz ¼ 0 meson mass eigenstates for some selected values of eB. Results correspond to α ¼ 0.1. Relative
signs hold for the choice B > 0.
Spin-isospin composition Spin-flavor composition

ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ
State eB ½GeV2 cπ 0 cπ 3 cρ0⊥ cρ3⊥ cπ u cπ d cρu⊥ cρd⊥
π̃ðk ¼ 1Þ 0.05 0.0037 0.9998 −0.0203 −0.0068 0.7096 −0.7043 −0.0192 −0.0095
0.5 0.1019 0.9910 −0.0822 −0.0285 0.7728 −0.6287 −0.0783 −0.038
1.0 0.1566 0.9841 −0.0797 −0.0274 0.8066 −0.5851 −0.0757 −0.037
η̃ðk ¼ 2Þ 0.05 0.9899 −0.0413 −0.0381 −0.1301 0.6708 0.7292 −0.1189 0.0651
0.5 0.8661 −0.3246 0.0582 −0.3757 0.3829 0.8420 −0.2245 0.3068
1.0 0.8353 −0.3445 0.1048 −0.4154 0.3470 0.8342 −0.2196 0.3678
ω̃ðk ¼ 3Þ 0.05 −0.1979 0.2693 0.7601 −0.5572 0.0505 −0.3304 0.1435 0.9315
ρ̃ðk ¼ 4Þ 0.05 0.4925 0.3312 0.4685 0.6544 0.5824 0.1141 0.7940 −0.1315
074002-9
have a larger component of the quark flavor that couples interactions in the Lagrangian in Eq. (1), the small
strongly to the magnetic field (i.e., the u quark). deviation of ρ̃ and ω̃ from pure flavor states can be
Concerning the vector meson components of the π̃ state, attributed to the mixing with the pseudoscalar sector, where
it is seen that they are completely negligible at low values isospin states are dominant. Notice that although the vector
ð1Þ ð1Þ
of eB, reaching a contribution jcρu⊥ j2 þ jcρd⊥ j2 ≃ 0.01 (i.e., components are larger than the pseudoscalar ones, the
about 1%) at eB ¼ 1 GeV . 2 weight of the latter is not negligible, specially for the ρ̃ state
Turning now to the composition of the η̃ state (k ¼ 2) (which is the one with a larger mass, as shown in Fig. 3),
ð4Þ ð4Þ
in Table I, we see that, as expected from the above with jcπu j2 þ jcπd j2 ≃ 0.35. This can be understood from
discussion, it is dominated by the π 0 (I ¼ 0) component an analysis similar to the one performed for the meson mass
for values of eB up to 1 GeV2 . Regarding the thresholds in terms of the quark spins. A larger content of
flavor composition, in this case the d-quark content is the dðSz ¼ − 12Þd̄ðSz ¼ þ 12Þ component has to be expected
ð2Þ ð2Þ
the one that increases as eB does, with jcπd j2 þ jcρd⊥ j2 ¼ in the case of the ω̃, while there should be a larger content
0.54ð0.70Þ for eB ¼ 0.05ð1.0Þ GeV2. Now the weight of the uðSz ¼ þ 12ÞūðSz ¼ − 12Þ one in the case of
of the vector components is larger than in the case of the ρ̃. From Table I it is seen that these combinations
ð3Þ ð3Þ pffiffiffi
ð1Þ ð1Þ correspond to ðcπd − cρd⊥ Þ= 2 ¼ −0.89 for the ω̃ and
the π̃ state, jcρu⊥ j2 þ jcρd⊥ j2 ranging from 0.02 at eB ¼ pffiffiffi
ð4Þ ð4Þ
0.05 GeV2 to 0.17 at eB ¼ 1.0 GeV2 . This is probably due ðcπ u þ cρu⊥ Þ= 2 ¼ 0.97 for the ρ̃, under a magnetic field
to the fact that for α ¼ 0.1 the η̃ mass is closer to vector as low as eB ¼ 0.05 GeV2 —and this effect should be more
meson masses. significant for larger values of eB.
Finally, let us comment on the composition of the ω̃ and We analyze in what follows the impact of both flavor
ρ̃ states (k ¼ 3 and k ¼ 4, respectively). As mentioned mixing and pseudoscalar-vector mixing on the masses of
above, the masses of these states reach the threshold for qq̄ the lightest states. In fact, this is one of the main issues of
decay for rather low values of the magnetic field, hence our this work. In Fig. 4 we show the B dependence of light
predictions for these quantities should be taken as quali- meson masses with (dashed lines) and without (dotted
tative ones for a major part of the eB range considered here. lines) pseudoscalar-vector mixing, considering three
It is worth noticing that there is a multiple number of representative values of the flavor-mixing constant α.
thresholds, which get successively opened each time the The results without pseudoscalar-vector mixing are
meson mass is sufficiently large so that the quark and obtained just by setting to zero the off-diagonal polari-
antiquark meson components can populate a new Landau zation functions Ĵmag mag
π a ρbμ and Ĵ ρaμ π b in Eq. (28). Let us focus
level. The first thresholds in the ūu and the d̄d sectors are on mπ̃ , considering first the effect of varying α; as can be
reached at meson masses equal to 2M u and 2Md , respec- seen from Fig. 4, this effect is rather independent of
tively. It is important to realize that they do not correspond whether pseudoscalar states mix with vectors or not. We
to a free quark together with a free antiquark, but observe that for α ¼ 0 (no flavor mixing) there are two
to the quark and antiquark in their lowest Landau levels. light mesons having similar masses; as stated above,
Taking B > 0, if both the quark and the antiquark have these are pure flavor states and can be identified as
vanishing z component of the momentum, the correspond- approximate Goldstone bosons. For α ≠ 0, the mass of the
ing spin configurations are uðSz ¼ þ 12ÞūðSz ¼ − 12Þ and π̃ state is still protected owing to its pseudo-Goldstone
dðSz ¼ − 12Þd̄ðSz ¼ þ 12Þ. In both cases, the magnetic dipole boson character, whereas the η̃ state becomes heavier
moments of the quark and the antiquark are parallel to the when α gets increased, and disappears from the spectrum
magnetic field; the difference between both configurations in the limit α ¼ 0.5.
arises from the opposite signs of the quark electric charges. From Fig. 4 it is also seen that, for all values of α, the
We only quote in Table I the ω̃ and ρ̃ compositions in mixing between pseudoscalar and vector meson states
the presence of a low magnetic field eB ¼ 0.05 GeV2 , produces a significant decrease in the mass of the lightest
for which the masses of both states are below the 2Mf state. This might be surprising, since—as shown above—
ðkÞ the vector meson components of the π̃ state are found to be
threshold and the values of the coefficients cM should be
very small even for large values of eB. The explanation of
more reliable. Interestingly, we note that even at this low
this puzzle is discussed in detail in Appendix C, where it is
value of the magnetic field the composition of the vector
shown that these two facts are indeed consistent. Moreover,
meson mass states is clearly flavor-dominated: from Table I
ð3Þ ð3Þ ð4Þ ð4Þ
for α ¼ 0.5 it is shown that if the pseudoscalar-vector
one has jcπd j2 þ jcρd⊥ j2 ¼ 0.98, jcπu j2 þ jcρu⊥ j2 ¼ 0.97. meson mixing is treated perturbatively, one can derive a
Thus, whereas for no external field one usually identifies simple formula for the B dependence of the π̃ mass, viz.
the (approximately degenerate) mass states as isospin
eigenstates ρ0 and ω, in the presence of the magnetic field m̄π̃
mπ̃ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi; ð41Þ
the states ρ̃ and ω̃ are closer to a ρu⊥ and a ρd⊥ , rather than a
1 þ κðm̄π̃ eBÞ2 =Mf
ρ3⊥ and a ρ0⊥ . In fact, given the symmetry of the vectorlike
074002-10
0.7
0.6
α=0 α = 0.1 η α = 0.5
0.5
0.4 η
mM [ GeV ]
0.3
0.14 πd
π π
0.12 πu
0.10
πd π
0.08
π
0.06 πu
0.04
0.02
0.00
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
2 2 2
eB [ GeV ] eB [ GeV ] eB [ GeV ]
FIG. 4. Masses of the lightest Sz ¼ 0 mesons as functions of eB, for various values of α. Dashed (dotted) lines correspond to the case
in which the mixing between pseudoscalar and vector states is (is not) included. The dash-dotted line in the right panel is obtained from
the approximate expression in Eq. (41).
where κ ¼ 5N 2c ggv =ð18π 4 mc Þ, f is either u or d, and m̄π̃ staggered quark action that uses mπ ðB ¼ 0Þ ¼ 220 MeV,
stands for the π̃ mass when no mixing is considered. Taking while the calculations in Refs. [15,54,80] take the physical
into account that m̄π̃ is very weakly dependent on B (see value of mπ within a staggered simulation setup. Anyway,
dotted lines in Fig. 4), it follows that mπ̃ basically depends in our model we see that when the pseudoscalar-vector
on the magnetic field through the ratio ðeBÞ2 =Mf . Notice meson mixing is included, the values for the π̃ meson mass
that the B dependence of Mf for α ¼ 0.5 is represented by
the solid line in Fig. 1. The numerical results for mπ̃ from 1.2
Eq. (41), within the approximation m̄π̃ ¼ mπ ðB ¼ 0Þ [see
Eq. (C9)] are indicated by the black dash-dotted line in the
1.0
right panel (corresponding to α ¼ 0.5) of Fig. 4. It can be
seen that they are in excellent agreement with those
obtained from the full calculation. 0.8
mπ (B) / mπ (0)
To conclude this subsection, in Fig. 5 we compare our

results for the mass of the π̃ state with those obtained in 0.6
LQCD calculations, reported in Ref. [54] (quenched
Wilson fermions), Ref. [55] (improved staggered quarks) 0.4 Mixing with vectors
and Refs. [15,54,80] (dynamical staggered quarks). We first No mixing with vectors
note that in those calculations the authors neglect discon- LQCD (impr. staggered)
0.2
nected diagrams as well as the associated mixing, and work LQCD (quenched Wilson)
with the individual flavor states instead. In our calculation LQCD (dyn. staggered)
this can be achieved by setting α ¼ 0. In any case, as seen 0.0

0.0 0.2 0.4 0.6 0.8 1.0
from the above analysis, the mass of the lightest meson is
2
approximately independent of the value of α; therefore, it is eB [ GeV ]
reasonable to compare the mentioned LQCD results with
FIG. 5. Normalized mass of the π̃ meson (lightest state of the
those obtained using the reference value α ¼ 0.1 that leads
Sz ¼ 0 sector) as a function of eB, compared with LQCD results
to an acceptable value for the η meson mass at vanishing quoted in Ref. [54](quenched Wilson fermions), Ref. [55]
external field. We also note that LQCD results have been (improved staggered quarks) and Refs. [15,54,80] (dynamical
obtained using different methods and values of the pion staggered quarks). Solid and dotted lines correspond to NJL
mass at B ¼ 0. In particular, the most recent ones (i.e., results with and without pseudoscalar-vector meson mixing,
those in Ref. [55]) are based on a highly improved respectively.
074002-11
lie in general below LQCD predictions. We have checked the values of m−f for f ¼ u or d are not surpassed by the
that this general result is quite insensitive to a reasonable corresponding meson masses mρfk in the studied region,
variation of the model parameters. In addition, we have and consequently these masses are found to be smooth real
verified that the situation does not change significantly if functions of eB, as shown in Fig. 6. We stress that the mass
the B ¼ 0 expressions are regularized using the Pauli- values m ¼ 2Mu and m ¼ 2M d are not actual thresholds in
Villars scheme, as proposed, e.g., in Ref. [60]. this case, since—as discussed above—they correspond
to lowest Landau level quark configurations that lead to
C. Sz = 1 vector meson sector Sz ¼ 0 meson states. The absence of these thresholds can
In this subsection we present the numerical results be formally shown by looking at the expression in Eq. (38);
associated with the coupled system composed by the it can be seen that although the functions I 2f ð−m2 Þ and
2
neutral vector mesons with jSz j ¼ 1. As discussed in I mag
4f ð−m Þ become complex for m > 2M f , imaginary parts
Sec. II B, for any value of α the mass eigenstates can be cancel each other and one ends up with a vanishing
identified according to their flavor content, jρuk i and jρdk i. absorptive contribution.
The corresponding masses can be obtained by solving the It should be pointed out that even though there is no
equations Gfk ð−m2ρfk Þ ¼ 0, for f ¼ u, d, with Gfk ð−m2 Þ direct flavor mixing in this sector, ρuk and ρdk meson
given by Eq. (38). masses still depend on α. This is due to the fact that the
The numerical results for the meson masses as functions values of M u and M d obtained at the MF level get modified
of the magnetic field for the case α ¼ 0.1 are shown in by flavor mixing. We recall that M u ðeBÞ ¼ Md ð2eBÞ for
Fig. 6, where it is seen that both mρuk and mρdk get increased α ¼ 0, while for α ¼ 0.5 one has Mu ðeBÞ ¼ Md ðeBÞ. The
effect of flavor mixing is illustrated in Fig. 7, where we
with B. The enhancement is larger in the case of the ρuk
show the B dependence of ρuk and ρdk meson masses for
mass; this can be understood from the larger (absolute)
α ¼ 0, 0.1 and 0.5. As expected from the aforementioned
value of the u-quark charge, which measures the coupling
relations between M u and M d , it is seen that the curves for
with the magnetic field. As in the case of Sz ¼ 0 mesons,
both masses tend to become more similar as α increases.
there are multiple mass thresholds for qq̄ pair production
However, the overall effect is found to be relatively weak.
[see Eqs. (B4) and q (B7)]. The lowest one, reached at
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi As a reference we also plot (full black line) the situation
mρfk ¼ m−f ¼ M f þ M 2f þ 2Bf , corresponds now to the in which the mixing between Sz ¼ 1 vector states is
situation in which both the spins of the quark and the neglected, and, therefore, the masses of both states
antiquark components of the ρfk are aligned (or antia- coincide. We see that even in the case α ¼ 0.5 there is a
ligned) with the magnetic field. Notice that in this case one certain non-negligible mass splitting between states when
of the fermions lies in its lowest Landau level, while the the mixing term is turned on.
other one is in the first excited Landau level; whether both It is also interesting at this stage to analyze the impact
particle spins are aligned or antialigned with the magnetic of the regularization procedure on the predictions of the
field depends on the signs of Sz and B. It can be seen that model. In Fig. 8 we show our results for the ρ3k mass
together with those obtained in Ref. [22] and Ref. [60]. To
1.2 carry out a proper comparison, in our model we have taken
α ¼ 0.5 and have set to zero the ρ0k − ρ3k mixing con-
tributions, as done in those works (in which the ρ0k state is
1.1
not included). Notice that this case corresponds to the
ρu solid line in the right panel of Fig. 7. In Ref. [22], divergent
ρd integrals are regularized through the introduction of
mM [ GeV ]
1.0
Lorenztian-like form factors, both for vacuum and B-
dependent contributions. On the other hand, in Ref. [60]
0.9 the regularization is carried out using the MFIR method, as
mu− in the present work. However, to deal with vacuumlike
terms the authors of Ref. [60] choose a Pauli-Villars
0.8
md− regularization, instead of the 3D-cutoff scheme considered
here. From Fig. 8 it is seen that our results for mρ3k (black
0.7
0.0 0.2 0.4 0.6 0.8 1.0
solid line) are quite similar to those found in Ref. [60]
(red dotted line), indicating that they are not too much
eB [ GeV2 ]
sensitive to the prescription used for the regularization of
FIG. 6. Masses of the Sz ¼ 1 vector meson states as functions vacuumlike terms, once the MFIR method is implemented.
of eB. Dotted and short-dotted lines indicate m−d and m−u quark- Meanwhile, the ρ3k mass obtained by means of a form
antiquark production thresholds, respectively. factor regularization (blue dashed line) shows a much
074002-12
1.6
α=0 α = 0.1 α = 0.5
1.5
mρSz = ±1(B) / mρ(0) 1.4
1.3
1.2
1.1 ρd
ρu
1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
eB [ GeV2 ] eB [ GeV2 ] eB [ GeV2 ]
FIG. 7. Masses of Sz ¼ 1 vector mesons as functions of eB, for various values of α. The full black lines correspond to the case in
which the mixing between pseudoscalar and vector meson states is not considered.
stronger dependence on the magnetic field, specially for the current quark masses leading to such a large value of
large values of eB. These results are consistent with those mπ . Considering the large error bars, from the figure one
found in Ref. [68] for the regularization scheme depend- observes that LQCD results seem to indicate an enhance-
ence of the condensates in the presence of the magnetic ment of mρuk when the magnetic field is increased, in
field. agreement with the predictions from the NJL model. This
Finally, in Fig. 9 we compare our results for the case qualitative behavior has been also found in previous LQCD
α ¼ 0.1 (dashed and dotted lines in the central panel of studies [47,51,52,61].
Fig. 7) with those quoted in Ref. [54] for the ρuk mass using
LQCD calculations. In fact, these lattice results are
D. B-dependent four-fermion couplings
obtained for a large vacuum pion mass of about 400 MeV;
the comparison still makes sense, however, since we have As mentioned in the Introduction, while local NJL-like
checked that our results are rather robust under changes in models are able to reproduce the magnetic catalysis (MC)
effect at vanishing temperature, they fail to describe the so-
2.0
called inverse magnetic catalysis (IMC) observed in lattice
1.8
1.6
mρ3Sz = ± 1(B) / mρ (0)
ρu
1.6
ρd
mρSz = ± 1(B) / mρ (0)
1.4
1.4
1.2
1.2
MFIR 3D (α = 0.5)
MFIR PV [56]
1.0 Lorentzian ff [21] 1.0
ρu (LQCD)
0.0 0.2 0.4 0.6 0.8 1.0
eB [ GeV2 ] 0.8
0.0 0.2 0.4 0.6 0.8 1.0
FIG. 8. Mass of the ρ meson with Sz ¼ 1 for the case in which eB [ GeV2 ]
and α ¼ 0.5 and there is no mixing between pseudoscalar and
vector meson states. Results quoted in the literature using other FIG. 9. Masses of Sz ¼ 1 vector meson states for α ¼ 0.1,
regularization methods are also shown. compared with LQCD results given in Ref. [54].
074002-13
1.2
ρ 2 Mu
2 Md ρu
1.0
ρd
0.8
ω
mM [ GeV ]
0.6
0.4
η
0.2
π
0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
2 2
eB [ GeV ] eB [ GeV ]
FIG. 10. Left (right) panel: masses of the Sz ¼ 0 (Sz ¼ 1Þ meson states as functions of eB, for B-dependent couplings gðeBÞ=g ¼
gv ðeBÞ=gv ¼ F ðeBÞ [see Eq. (43)]. The results correspond to the case α ¼ 0.1.
QCD. Among the possible ways to deal with this problem, assumptions, we have obtained numerical results for the
one of the simplest approaches is to allow the model behavior of meson masses with the magnetic field. The
coupling constants to depend on the magnetic field. With curves for the case α ¼ 0.1 are given in Fig. 10, where we
this motivation, we explore in this subsection the possibility also show the qq̄ production thresholds (dotted lines).
of considering a magnetic field dependent coupling gðeBÞ. By comparison with the results in Figs. 3 and 6, it can be
For definiteness, we adopt for this function the form observed that the B dependence of the couplings has a
proposed in Ref. [28], namely significant qualitative effect only in the case of the ω̃ state.
It is found that the mass of this state follows quite closely
gðeBÞ ¼ gF ðeBÞ; ð42Þ the position of the lowest qq̄ production threshold, 2Md ,
which—as stated—does get affected by the B dependence
where of g. The behavior of the masses of the other mesons do not
change qualitatively with respect to the case g ¼ constant,
2
F ðeBÞ ¼ κ 1 þ ð1 − κ 1 Þe−κ2 ðeBÞ ; ð43Þ and something similar happens with their composition and
their dependence on α. In particular, the results for the ratio
with κ 1 ¼ 0.321, κ 2 ¼ 1.31 GeV−2 . Assuming this form rπ ¼ mπ̃ ðeBÞ=mπ ð0Þ are almost identical to those obtained
for gðeBÞ, the effective quark masses are found to be less in Sec. III B (solid line in Fig. 5).
affected by the presence of the magnetic field than in the Given the fact that gv ðeBÞ is not so well constrained as
case of a constant g. In fact, they show a nonmonotonous in the case of the scalar coupling, one can, in principle,
behavior for increasing B, resembling the results found in introduce a new function F v ðeBÞ, different from F ðeBÞ.
Refs. [40,75]. It should be stressed that in spite of the rather The freedom in the election of this function can be used to
different behavior of the dynamical quark masses, a similar reproduce the results for the ratio rπ obtained through
zero-temperature magnetic catalysis effect is obtained both LQCD calculations. It can be seen, however, that in this
for a constant g and for a variation with B of the form given case the masses of the Sz ¼ 1 vector mesons increase
by Eq. (43). even faster than in the case in which B-independent
Regarding the vector meson sector, one has to choose couplings are used.
some assumption for the B dependence of the vector
coupling constant. One possibility is to suppose that,
IV. CONCLUSIONS
due to their common gluonic origin, the vector couplings
are affected by the magnetic field in the same way as the In this work we have studied the mass spectrum of light
scalar and pseudoscalar ones. That is to say, one could take neutral pseudoscalar and vector mesons in the presence of
gv ðeBÞ ¼ gv F v ðeBÞ, with F v ðeBÞ ¼ F ðeBÞ. Under these ⃗ For this purpose we
an external uniform magnetic field B.
074002-14
have considered a two-flavor NJL-like model in the Landau we can conclude that even for magnetic field values as large
gauge. This model includes isoscalar and isovector cou- as eB ¼ 1 GeV2 , the π̃ state is mostly a pseudoscalar
plings in the scalar-pseudoscalar sector and in the vector isovector (third component) and η̃ is mostly a pseudoscalar
sector. A flavor mixing term in the scalar-pseudoscalar isoscalar. Increasing the magnetic field intensity from a low
sector, regulated by a constant α, has also been included. value of eB ¼ 0.05 GeV2 to eB ¼ 1 GeV2 we observe that
For α ¼ 0 there is not flavor mixing, but flavor degeneracy the u content of the π̃ and the d content of the η̃ get
gets broken by the magnetic field and Mu ≠ M d , while for enhanced.
α ¼ 0.5 one has maximum flavor mixing, as in the case of On the other hand, regarding the quark structure
the standard version of the NJL model, and in this case of the two heaviest mesons, which we call ω̃ and ρ̃, it is
Mu ¼ M d . To account for the usual divergences of the NJL found that even for a low value of the magnetic field,
model, we have considered here the magnetic field inde- eB ¼ 0.05 GeV2 , the mass eigenstates turn out to be
pendent regularization (MFIR) method, which has been clearly dominated by the quark flavor content and spin
shown to reduce the dependence of the results on the model orientation. This is what we could expect, since the
parameters. It should be stressed that for neutral mesons the magnetic field tends to separate quarks according to their
contributions to the polarization functions arising from electric charges, and favors that their magnetic moments be
Schwinger phases in quark propagators get canceled; as a orientated parallel to the field direction.
consequence, the polarization functions turn out to be The lack of confinement in the NJL model implies that
diagonal in the usual momentum basis. the polarization functions get absorptive contributions,
It is important to note that the presence of an electro- related with qq̄ pair production, beyond certain thresholds.
magnetic field allows for isospin mixing. In addition, the In the presence of the magnetic field, the position of each
axial character of the magnetic field together with the loss threshold is flavor and spin dependent, in such a way that
of rotational invariance lead to pseudoscalar-vector mixing. for Sz ¼ 0 we have thresholds for meson mass values
These mixing contributions are usually forbidden by mf ¼ 2M f , while for Sz ¼ 1 the thresholds rise to higher
isospin and angular momentum conservation. However, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
they arise and may become important in the presence of the values mf ¼ M f þ M 2f þ 2Bf . As a consequence, we
external magnetic field. Although full rotational invariance find that ω̃ and ρ̃ states with Sz ¼ 0 enter into the
is broken, invariance under rotations around the magnetic continuum for values of the magnetic field around
field direction survives. Therefore, the projection of the eB ∼ 0.1 GeV2 , whereas Sz ¼ 1 meson masses always
vector meson spin in the field direction, Sz , is the lie under qq̄ production thresholds for the considered range
observable that organizes the obtained results. Our analysis of values of eB. A common result for all these states is that
shows that for the determination of the masses (i.e., if their masses show an appreciable growth when the mag-
particles are taken at rest), the scalar mesons, which in our netic field varies from zero to eB ¼ 1 GeV2 . In the case of
case include the f 0 (or σ) and a00 states, mix with each other Sz ¼ 1, the model reproduces reasonably well present
but decouple from other mesons. Thus, they can be LQCD results for ρuk , taking into account the uncertainties
disregarded in the analysis of the pseudoscalar and vector in LQCD simulations.
meson masses. The remaining meson space can be sepa- We have observed that the mass of the lightest state, π̃,
rated into three subspaces: pseudoscalar and vector mesons gets reduced as the magnetic field increases. This behavior
with Sz ¼ 0, including π 0 , η, ρ0 and ω, which mix with reproduces the trend of existing LQCD results. However,
each other; vector mesons with Sz ¼ þ1, including ρ0 and our results overestimate the mass reduction as compared to
ω mesons; same as before, with Sz ¼ −1. the one found in LQCD simulations. It is seen that this
Regarding the Sz ¼ 0 sector, we observe two different reduction is significantly affected by the mixing between
behaviors for the meson masses. The masses of the two pseudoscalar and vector components, a fact that turns out to
lightest mesons, which we have called π̃ and η̃, are be independent of the value of the flavor mixing parameter
determined by the underlying symmetries and their break- α. From an analytical perturbative analysis, we have care-
ing pattern. In the presence of the magnetic field, with fully studied how a small value of the vector components in
α ¼ 0, one has a “residual” Uð1ÞT 3 ⊗ Uð1ÞT 3 ;A ⊗ Uð1ÞA the π̃ state can lead to a significant reduction of its mass. It
chiral symmetry, explicitly broken only by a (small) is seen that both the mixture of the π channel with the ω and
current mass term, mc ≠ 0, which guarantees the ρ channels contribute to this mass shrinkage.
pseudo-Goldstone character of these two states. We have While local NJL-like models are able to reproduce the
shown that flavor degeneracy gets broken by the magnetic magnetic catalysis effect at vanishing temperature, they fail
field and mass eigenstates are separated into particles with to lead to the so-called inverse magnetic catalysis. One of
pure u or d quark content. For α ¼ 0.1, which leads to a the simplest ways to deal with this problem is to allow that
reasonable value for the η mass in the absence of the the model coupling constants depend on the magnetic field.
magnetic field, the Uð1ÞA symmetry is broken and only one With this motivation, we have explored the possibility of
pseudo-Goldstone boson, π̃, survives. From our results, considering magnetic field dependent couplings gðeBÞ and
074002-15
gv ðeBÞ. For definiteness we take the same dependence on B Ĵ0;reg

MM0 ðqÞ, defined within the MFIR scheme. As stated, it
for both couplings; in that case, our results show that, for can be easily seen that these are zero for M ≠ M0, while for
any value of α, the mass of the ω̃ state with Sz ¼ 0 is the M ¼ M0 one has
only one that becomes significantly modified with respect
to the case in which g and gv do not depend on the magnetic X
field. In particular, the B-dependence of the ratio rπ ¼ Ĵ0;reg
π a π a ðqÞ ¼ −N c ½I 1f þ q2 I 2f ðq2 Þ;
mπ̃ ðeBÞ=mπ ð0Þ is almost identical to that obtained when f
the couplings g and gv are kept constant. If one allows for 2N c X
different B dependences for g and gv it is possible to Ĵ0;reg
ρaμ ρaν ðqÞ ¼ ½ð2M2f − q2 ÞI 2f ðq2 Þ − 2M 2f I 2f ð0Þ
3 f
improve on the agreement with LQCD results for this ratio.
However, this implies a rather strong enhancement in the qμ qν
masses of Sz ¼ 1 vector meson states, leading to a rather × δμν − 2 : ðA1Þ
q
large discrepancy with LQCD results in Ref. [54].
For simplicity, in the present work we have not taken into
account the axial vector interactions. The influence of these Here, the integrals I 1f and I 2f ðq2 Þ are defined as
degrees of freedom in the magnetic field dependence of
light neutral meson masses, and, in particular, on the ratio Z
1
rπ , is certainly an issue that deserves further investigation. I 1f ¼ 4 2 2
;
It would be also interesting to study the effect of the p M f þp
Z
inclusion of quark anomalous magnetic moments. We 1
I 2f ðq2 Þ ¼ −2 2
; ðA2Þ
expect to report on these issues in future publications. p ðM f þ p2þ ÞðM 2f þ p2− Þ
ACKNOWLEDGMENTS
with p ¼ p q=2. Within the 3D-cutoff regularization
We are grateful to M. F. Izzo Villafañe for helpful
scheme used in this work, the first of these integrals is
discussions at the early stages of this paper. This work
given by
has been partially funded by CONICET (Argentina) under
Grant No. PIP17-700, by ANPCyT (Argentina) under

Grants No. PICT17-03-0571 and No. PICT19-0792, by 1 2 2
Mf
the National University of La Plata (Argentina), Project I 1f ¼ 2 Λ rΛf þ Mf ln ; ðA3Þ
2π Λð1 þ rΛf Þ
No. X284, by Ministerio de Ciencia e Innovación and
Agencia Estatal de Investigación (Spain) MCIN/AEI and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
European Regional Development Fund Grant No. where we have defined rΛf ≡ 1 þ M 2f =Λ2 . In the case of
PID2019- 105439 GB-C21, by EU Horizon 2020 Grant
No. 824093 (STRONG-2020), and by Conselleria de I 2f ðq2 Þ, we note that in order to determine the meson
Innovación, Universidades, Ciencia y Sociedad Digital, masses, the external momentum q has to be extended to
Generalitat Valenciana, GVA PROMETEO/2021/083. the region q2 < 0. Hence, we find it convenient to write
N. N. S. would like to thank the Department of q2 ¼ −m2 , where m is a positive real number. Then, within
Theoretical Physics of the University of Valencia, where the 3D-cutoff regularization scheme, the regularized real
part of this work was carried out, for their hospitality within part of I 2f ð−m2 Þ can be written as
the visiting professor program of the University of Valencia.

APPENDIX A: REGULARIZED B = 0 2 1 Λ
Re½I 2f ð−m Þ ¼ − 2 arcsinh − Ff ; ðA4Þ
POLARIZATION FUNCTIONS 4π Mf
In this appendix we give the expressions for the
regularized B ¼ 0 pieces of the polarization functions, where
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
> 2 2
4Mf =m − 1 arctan pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
if m2 < 4M2f
>
>
>
> rΛf 4M2f =m2 −1
>
>
< qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ff ¼ 2
1 − 4Mf =m arccoth 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
if 4M 2f < m2 < 4ðM2f þ Λ2 Þ : ðA5Þ
>
> rΛf 1−4M2f =m2
>
>
>
> qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
> 2 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
: 1 − 4Mf =m arctanh 2 2
if m2 > 4ðM2f þ Λ2 Þ
rΛf 1−4Mf =m
074002-16
For the regularized imaginary part we get

( qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
− 8π 1 − 4M 2f =m2 if 4M 2f < m2 < 4ðM 2f þ Λ2 Þ
Im½I 2f ð−m2 Þ ¼ : ðA6Þ
0 otherwise
APPENDIX B: INTEGRALS Imag

nf ð − m Þ
2
replacement M2f → M2f − iϵ (note that this implies the
FOR m > 2M f replacement x̄f → x̄f − iϵ). Once this is done, one can
2 proceed by using the digamma recurrence relation
The expressions for the integrals I mag nf ð−m Þ for
n ¼ 2; …; 5 given in Eqs. (31) are only valid when
m < 2M f . For m > 2M f , it happens that the corresponding Xn
1
integrands can become divergent at some points within the ψðxÞ ¼ ψðx þ n þ 1Þ − ; ðB1Þ
j¼0
xþj
integration domain, leading to divergent integrals.
However, we can get finite results by considering the
analytical extension of the functions in Eqs. (31). For this and taking ϵ → 0þ through a generalized version of the
purpose it is worth taking into account that the Feynman Sokhotski-Plemelj formula [see, e.g., Eq. (A8) of
quark propagators originally contain iϵ terms, which can be Ref. [40]]. In this way, we find that for m > 2Mf the
integrals I mag 2
easily recovered in the integrands of Eqs. (31) through the nf ð−m Þ, n ¼ 2; …; 5, can be extended to
Z
2 1 1 4Bf X
N
αn i 2Bf X
N
αn
I mag
2f ð−m Þ ¼ 2 dv ψðx̄f þ N þ 1Þ − ln xf þ 2 − 2β0 arctanh β0 þ 2 arctanhβn þ β − ;
8π 0 m n¼0 βn 8π 0 m2 n¼0 βn
ðB2Þ

mag 2
Qf M f arctanhβ0 π
I 3f ð−m Þ ¼ − 2 −i ; ðB3Þ
π m β0 2β0

2 þ 2 − 2 m2 1 2 7 2 4 2 38
I mag ð−m Þ ¼ −I mag
þ T ð−m Þ þ T ð−m Þ − 4β 0 1 − β arctanh β 0 þ − β ln x þ β −
4f 1f f f
16π 2 3 0 3 0 f
3 0 9

im2 1 2 −
4Bf XN−
ð2λ − 2n − 1Þ
þ β 1 − β0 − θðm − mf Þ 2 ; ðB4Þ
8π 0 3 m n¼0 rn
Z
mag 2 1 1
2 2 8 2 2
I 5f ð−m Þ ¼ 2 dvð1 − v Þψðx̄f þ N þ 1Þ − ln xf − þ β0 þ β0 ð3 − β0 Þarctanh β0
8π 0 3 3

4Bf X αnN
i 1 2Bf X N
αn
þ 2 ½βn þ ð1 − β2n Þarctanh βn þ β0 1 − β20 − 2 ð1 − β2n Þ : ðB5Þ
m n¼0 βn 8π 3 m n¼0 βn
Here, we have used the definition αn ¼ 2 − δ0n , together with

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4ðM 2f þ 2nBf Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
βn ¼ 1 − ; r ¼ 1 − 4λð2n þ 1Þ þ 4λ2 β20 ; N ¼ Floor½λβ20 =2;
m2
n
where λ ¼ m2 =ð4Bf Þ. In the expression of I mag 2 mag

4f ð−m Þ the integral I 1f is that given by Eq. (17), and we have introduced the
functions T 2
f ð−m Þ given by
Z
m2 1
T 2
f ð−m Þ ¼ dvðv2 v=λ þ γÞ ψðx̄f þ ð1 vÞ=2 þ θðm − m
f Þð1 þ N ÞÞ
32π 2 0
N
X
Bf ð2λ − 2n − 1Þ ð2λ − rn 1Þjrn 1j
− θðm − m Þ 1 − ln ; ðB6Þ
4π 2 f
n¼0
rn ð2λ þ rn 1Þðrn ∓ 1Þ
074002-17
with γ ¼ 2 − β20 ¼ 1 þ 4M2f =m2 , and consider the π 3 —ρ0⊥ system (see, however, discussion at
the end of this appendix). To check whether we are
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
capturing the main effect of pseudoscalar-vector meson
m−f ¼ M f þ M2f þ 2Bf ; mþ ¼ 2 M 2f þ Bf : ðB7Þ
f mixing on the π̃ mass it is useful to consider the ratio
rπ ¼ mπ̃ ðeBÞ=mπ ð0Þ. Assuming that g and gv do not
The integers N have been defined as depend on the magnetic field, and taking α ¼ 0.1, for eB ¼
1 GeV2 we get rπ ¼ 0.32 for the full π 0 −π 3 −ρ0⊥ −ρ3⊥
N − ¼ Floor½r20 =ð8λÞ; N þ ¼ Floor½ðλβ20 − 1Þ=2: ðB8Þ system, to be compared with the values rπ ¼ 0.38, obtained
when we consider only the π 3 −ρ0⊥ system, and rπ ¼ 0.92,
obtained for the case in which there is no mixing at all.
APPENDIX C: A SIMPLIFIED MODEL FOR THE These values clearly support our approximation of the
LOWEST STATE OF THE Sz = 0 SECTOR
full system by the much simpler π 3 ρ0⊥ one. It should be
In this appendix we present a simplified model to stressed that even in this simplified situation the
analyze the mass and composition of the lowest lowest mass state is still found to be strongly dominated
state of the Sz ¼ 0 meson sector. As seen in Sec. III B by the π 3 contribution. In fact, for eB ¼ 1 GeV2 we get
(see the discussion concerning Fig. 4), the mass of this ð1Þ
cρ0⊥ ¼ −0.083, close to the value −0.0797 obtained for the
state, while almost independent of the value of α, is full system (see Table I). Defining a mixing angle θ by
significantly affected by the existence of a mixing between ð1Þ ð1Þ
pseudoscalar and vector meson states. Thus, to simplify the tan θ ¼ cρ0⊥ =cπ3 , this implies θ ≃ −50 .
analysis we consider the case α ¼ 0.5, in which the relevant The strong dominance of the π 3 contribution to the π̃
basis is only composed by the states π 3 , ρ3⊥ and ρ0⊥ . state suggests that one should be able to determine the
In addition, one has Mu ¼ M d ≡ M for any value mixing effect on mπ̃ using first order perturbation theory.
of eB. Assuming as in the main text gv0 ¼ gv3 ¼ gv , On the other hand, this appears to be in contradiction with
it is easy to see that the ratio between the off-diagonal the aforementioned significant reduction of the π̃ mass. To
π 3 ρ3⊥ and π 3 ρ0⊥ mixing matrix elements is given by get a better understanding of the situation, it is convenient
G⊥π3 ρ3 =G⊥π3 ρ0 ¼ ðBu − Bd Þ=ðBu þ Bd Þ ¼ 1=3. Hence, to to carry out some further approximations. The relevant
mixing matrix elements to be considered are
simplify the problem even further, in what follows we only
1 X
2 2 2
G⊥π3 π3 ¼ − N c ½ðI 1f þ I mag mag
1f Þ − m ðI 2f ð−m Þ þ I 2f ð−m ÞÞ;
2g f
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
iN c sB Be arctanð1= 4M 2 =m2 − 1Þ
G⊥π3 ρ0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
2π 2 1 − m2 =4M2
1 X2
G⊥ρ0 ρ0 ¼ þ Nc ½ð2M2 þ m2 ÞI 2f ð−m2 Þ − 2M2 I 2f ð0Þ þ m2 I mag
5f ð−m 2
Þ ; ðC1Þ
2gv f
3
where we have denoted Be ¼ jeBj and sB ¼ signðBÞ. For mc N c sB Be 1

a¼ ; c¼ ; d¼ ;
the π̃ state, we have m2 =ð4M 2 Þ ≪ 1 (for our parametriza- 2gM 4π 2 M 2gv
tion we find m2 =ð4M 2 Þ ≈ 0.03 at vanishing magnetic field, 2b N Λ3
and even a smaller value at eB ¼ 1 GeV2 ). Thus, we can b0 ¼ þ c2 2 ðC3Þ
3 18π ðΛ þ M2 Þ3=2
obtain a good approximation to these matrix elements by
expanding up to Oðm2 =4M 2 Þ. In this way we get a mixing
matrix of the form and

Nc Λ 4Λ B 9M4
b ¼ 2 4 arcsinh − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi − e2 þ ln 2
8π M Λ2 þ M 2 M 8Be
ðπ ρ0 Þ a − bm2 icm 2 2
G⊥ 3 ¼ ; ðC2Þ 3M 3M
−icm d − b0 m 2 −ψ −ψ : ðC4Þ
4Be 2Be
We note that here the gap equation has been used to get
where the expression for a. Given the model parameters, these
074002-18
coefficients can be easily computed for a given value ð1Þ

This expression, together with the one for cρ0⊥ in
of Be . Eq. (C7), are the relations that one would obtain from
Keeping terms up to the leading order in m2 , one gets in a first-order perturbation analysis of the system
this way described by the matrix in Eq. (C2) when
d ≫ a þ ðb0 − bÞm2 , a condition that is always well
ad
m2π̃ ¼ : ðC5Þ satisfied in our case. One can observe that the some-
ab0 þ bd þ c2 what unexpectedly “large” value of the mass shift arises
In addition, it can be seen that a=d ¼ ðgv0 =gÞðmc =MÞ ≪ 1, from the small value of the coefficient b, which is
and consequently ab0 ≪ bd. Using this approximation found to be about 0.034 for eB ¼ 1 GeV2 (assuming
we obtain B-independent couplings). In a conventional eigenvalue
problem, one would have b ¼ 1.
m̄π̃ Finally, we note that the effect on this game of the ρ3⊥
mπ̃ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðC6Þ meson, so far neglected, can be easily taken into account at
1 þ c2 =ðbdÞ
this stage. Since, as shown above, the G⊥π3 ρ0 matrix
pffiffiffiffiffiffiffiffi element can be treated perturbatively, and G⊥π3 ρ3 is even
where m̄π̃ ¼ a=b is the mass of the lightest state if there
is no mixing at all. Within the same approximation, the 3 times smaller, one can account for the ρ3⊥ meson just
coefficient of the ρ0⊥ piece of the lightest state is given by replacing the factor c2 in Eq. (C6) by 10=9c2. The resulting
expression for mπ̃ can be rewritten as
ð1Þ mπ̃ c
cρ0⊥ ¼ − : ðC7Þ
d
m̄π̃
The numerical values for the above quantities can be mπ̃ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi; ðC9Þ
calculated from Eqs. (C3) and (C4). For a large magnetic 1 þ κ m̄2π̃ B2e =M
field eB ¼ 1 GeV2 , assuming that the coupling constants
ð1Þ
are independent of B, we get rπ ¼ 0.39 and cρ0⊥ ¼ −0.084,
in excellent agreement with the results quoted above for the where κ ¼ 5N 2c ggv =ð18π 4 mc Þ. To obtain this expression we
π 3 ρ0⊥ system. This confirms the validity of the approx- have made use of Eq. (C3) together with the rela-
imations made so far. tion b ¼ a=m̄2π̃ ¼ mc =ð2gMm̄2π̃ Þ.
It is also interesting to note that the expression for mπ̃ It should be emphasized that although the numerical
given in Eq. (C6) implies values quoted in this appendix correspond to the case in
which the couplings g and gv are kept fixed, Eq. (C9) can be
m2π̃ c2 shown to be approximatively valid also when they depend
bðm2π̃ − m̄2π̃ Þ ¼ − : ðC8Þ
d on B as considered in Sec. III D.
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PhysRevD 106 074002

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PHYSICAL REVIEW D 106, 074002 (2022)

(Received 6 June 2022; accepted 9 September 2022; published 7 October 2022)

I. INTRODUCTION includes several interesting phenomena, such as the chiral

2470-0010=2022=106(7)=074002(20) 074002-1 Published by the American Physical Society

so that the flavor mixing in the scalar-pseudoscalar sector is

Mu ¼ mc − 4g½ð1 − αÞϕu þ αϕd ; B. Neutral meson system

Here εM ¼ 1 for the isoscalars M ¼ σ 0 ; π 0 ; ρ0μ and εM ¼

Z  F fMM0 ðx0 ; xÞ ¼ N c trD ½S MF;f M MF;f M

ðkÞ ðkÞ ðkÞ ðkÞ

It is also useful to consider the flavor basis π f , ρf , where 1 4N

that chiral symmetry largely dominates over flavor sym- 0.7

instead of approximating to flavor states π f . 0.4

the parameters of the model. Here we take the parameter eB [ GeV2 ]

is a small bump in the curve for the ρ̃ mass. This can be

ðkÞ ðkÞ ðkÞ ðkÞ

ð1Þ ð3Þ ð2Þ ð4Þ

Spin-isospin composition Spin-flavor composition

To conclude this subsection, in Fig. 5 we compare our

this can be achieved by setting α ¼ 0. In any case, as seen 0.0

mρSz = ±1(B) / mρ(0) 1.4

eB [ GeV2 ] eB [ GeV2 ] eB [ GeV2 ]

gv ðeBÞ. For definiteness we take the same dependence on B Ĵ0;reg

For the regularized imaginary part we get

APPENDIX B: INTEGRALS Imag

Here, we have used the definition αn ¼ 2 − δ0n , together with

where λ ¼ m2 =ð4Bf Þ. In the expression of I mag 2 mag

where we have denoted Be ¼ jeBj and sB ¼ signðBÞ. For mc N c sB Be 1

coefficients can be easily computed for a given value ð1Þ

[19] V. V. Braguta, P. V. Buividovich, M. N. Chernodub, A. Y. [51] E. V. Luschevskaya, O. E. Solovjeva, O. A. Kochetkov, and

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Z F fMM0 ðx0 ; xÞ ¼ N c trD ½S MF;f M MF;f M