The quark mass gap in a magnetic field

0.78 MB·English

by  Toru Kojo

The quark mass gap in a magnetic field
Toru Kojo, Nan Su
Faculty of Physics, University of Bielefeld, D-33615 Bielefeld, Germany
(BI-TP 2012/50)
Abstract
3
1
A magnetic field and the resulting Landau degeneracy enhance the infrared contributions to the
0
2
quark mass gap. The gap does not grow arbitrarily, however, for models of asymptotic free

n
interactions. For B → ∞, the magnetic field decouples from the dimensionally reduced self-
a
J (cid:112)
consistent equations, so that the gap behaves as ∼ Λ (or less), instead of ∼ |eB|. On the
QCD
6
2 other hand, the number of participants to the chiral condensate keeps increasing as ∼ |eB| so that

|(cid:104)ψ¯ψ(cid:105)| ∼ |eB|Λ . After the mass gap stops developing, nothing tempers the growth of screening
] QCD
h
p effects as B → ∞. These features are utilized to interpret the reduction of critical temperatures
-
p for the chiral and deconfinement phase transitions at finite B, recently found on the lattice. The
e
structures of mesons are analyzed and light mesons are identified. Applications for cold, dense
h
[
quark matter are also briefly discussed.

2
v
8
1. Introduction
1
3
7 Inpastdecades, systemsinamagneticfield(B)havebeenusefullaboratoriestotesttheoretical
.
1
ideas. A famous example is a system of cold atoms, in which a magnetic field controls the
1
2 strength of the interactions. In QCD, similar utilities are also expected for the lattice Monte
1
: Carlo simulation at finite B [1–4]. In particular, we can study the nonperturbative gluon dynamics
v
i through polarization effects, controlling quark dynamics by a magnetic field. Such information
X
r may help the studies of cold, dense quark matter [5].
a
It seems that lattice studies already confirmed some of the theoretical ideas. At T = 0, a
magnetic field enhances the size of the chiral condensate due to magnetic catalysis [6, 7]. A key
feature of this phenomenon is the effective dimensional reduction. For B (cid:54)= 0, the phase space for
(cid:82)
the low energy particles and anti-particles is ∼ |eB| dp , increasing the number of participants
(cid:107)
to the formation of the chiral condensate. This should be contrast to the B = 0 case, where phase
Email addresses: [email protected] (Toru Kojo), [email protected] (Nan
Su)
Preprint submitted to Elsevier January 29, 2013
(cid:82)
space quickly decreases as ∼ |p|2d|p| in the infrared region. In this case, due to the small number
of participants, the system needs sufficiently strong attractive forces to form chiral condensates.
On the other hand, some surprises have been provided as well [4]. While B increases the
chiral condensate below the (pseudo-)critical temperatures for the chiral restoration (T ) and
χ
deconfinement (T ), those temperatures themselves decrease. This might contradict with our
D
intuitions, if we think that a larger chiral condensate should generate a greater quark mass gap.
Such thinking would suggest that (i) a larger quark mass gap should suppress thermal quark
fluctuations, leading to increasing T , and (ii) a larger quark mass gap suppresses quark loops, so
χ
that the results should approach to the pure gauge results, leading to increasing T .
D
To resolve this apparent contradiction, we shall argue that the quark mass gap at T = 0 can
stay around ∼ Λ (or less) for large B. Then we can imagine that the gap at T < T also
QCD χ,D
stays around ∼ Λ , without strongly suppressing thermal quark fluctuations and quark loops.
QCD
If this is the case, the decreasing of critical temperatures would not be so unnatural.
In addition, the aforementioned behavior of the quark mass gap does not contradict with
the growing behavior of the chiral condensate, but instead naturally explains its B-dependence
at T = 0. In fact, the lattice results in [4] showed the behavior (cid:104)ψ¯ψ(cid:105)B ∼ |eB|Λ , for
T=0 QCD
|eB| ≥ 0.3GeV2 (cid:29) Λ2 ((cid:39) 0.04GeV2). Noting that the relation under the Landau quantization,
QCD
¯ ¯
(cid:104)ψψ(cid:105) ∼ |eB|×(cid:104)ψψ(cid:105) , (1)
4D 2D
¯
we can see that (cid:104)ψψ(cid:105) or the quark mass gap must be nearly B-independent and O(Λ ).
2D QCD
In this work we will carry out all the calculations in the large N limit1. The use of the
c
large N is motivated by at least three reasons: (i) At large N , gluons are not screened, so the
c c
nonperturbative forces (i.e. the forces in the infrared) are stronger than the N = 3 case. Such
c
forces can be used to set the upper bound of the quark mass gap. (ii) The large N limit has
c
captured many qualitative aspects of the confined phase at B = 0. Therefore it is worth thinking
and testing this approximation in the confined phase at finite B, since its validity and invalidity
are not evident apriori. (iii) It is easy to imagine how the 1/N corrections qualitatively modify the
c
large N results, and such corrections just provide welcomed effects for our scenario (see below).
c
WewillusethelargeN limittojustclaimthatthequarkmassgapdoesnotgrowmuchbeyond
c
Λ . To explain the reduction of the critical temperatures, in addition we have to argue the
QCD
1/N corrections. The quark loops as the 1/N corrections screen the nonperturbative forces. As
c c
B increases, the screening effects become larger because more low energy particles can participate
to the gluon polarization, due to the enhanced Landau degeneracy ∼ |eB| in the lowest Landau
1The large N limit in a magnetic field was also studied in Ref. [9] from a different perspective from ours.
c
2
level (LLL) [8]. If the quark mass gap stops growing as suggested in our scenario, there is nothing
to suppress the growth of the screening effects as B increases2. Therefore the nonperturbative
forces are reduced at large B, and such reduction should lower the critical temperatures for given
B. In addition, hadronic fluctuations as the 1/N corrections also grow as B increases, helping
c
the chiral symmetry to restore [10]3.
We will argue that the demanded (nearly) B-independent gap of O(Λ ) can be derived,
QCD
provided that it is dominantly created by the nonperturbative force mediated by the IR gluons.
In particular, both the IR enhancement (that is more drastic than the perturbative 1/p2 case)
and the UV suppression of the gluon exchanges are crucial for our discussions. Since we will deal
with the LLL which is essentially soft physics in the present paper, IR enhanced gluon is a key
feature in this study (see [12] for a review). If we include only the perturbative 1/p2 force, the gap
is much smaller than Λ and depends on B at most logarithmically. Similar arguments have
QCD
been used in studies of the quark mass function at finite quark density [13, 14].
To illustrate our points, we first consider the NJL model which does not have the abovemen-
tioned properties. For |eB| → ∞, the gap equation within the LLL approximation is
(cid:90)
|eB| dq M (B)
z NJL
M (B) = GtrS(x,x) −→ G f(q ,B;Λ), (2)
NJL (cid:112) z
2π 2π q2 +M2 (B)
z NJL
where f(q ,B;Λ) is some UV regulator function. The contact interaction couples all states in the
z
LLL so that the Landau degeneracy factor |eB| for the LLL appears. The intrinsic property of
the model is that the chiral condensate has the same B-dependence as the mass gap:
1
(cid:104)ψ¯ψ(cid:105)B (cid:39) − M (B). (3)
NJL G NJL
Depending on the regularization schemes, M (B) can be ∼ |eB|1/2 (proper time regularization
NJL
[15]), or∼ Λ(fourmomentumcutoff[10]), orelse. Eachschemehasitsownproblems. Forschemes
predicting the growing behavior of the chiral condensate [16], the quark mass gap also develops
as B increases. Then at finite temperature, thermal quark contributions are largely reduced so
that the increasing chiral restoration temperature is naturally expected. On the other hand, if the
mass gap approaches to constant, the chiral condensate also does, contradicting with the lattice
results. Therefore, as far as the relation like (3) is retained, it seems that we have to abandon
either the increasing chiral condensate or the reduction of critical temperatures.
This dilemma can be bypassed if we use the gluon exchange type interactions with the IR
enhancement and UV suppression. To emphasize the point, we consider a simple model for the
2This suggests that even at T =0, the nonperturbative gluons will be screened out at some critical value of B
such that the screening mass, m ∼g |eB|1/2 ∼N−1/2|eB|1/2, becomes comparable to Λ (see also Sec.4).
D s c QCD
3 For hadronic fluctuations at small |eB|, see Ref. [11], where chiral perturbation theory should be at work.
3
Figure 1: (a) The Schwinger-Dyson equation at large N for the model in Eq.(4). (b) The distribution of states
c
in the lowest Landau level for fixed p . A state with momentum p can strongly couple to states within a domain
z ⊥
of |q −p |≤Λ ∼Λ .
⊥ ⊥ f QCD
gluon exchange with these features (for the moment we ignore spinor structures),
D(q) = Gθ(Λ2 −(cid:126)q2), (Λ ∼ Λ ) (4)
f f QCD
which was proposed in [14]. In this model, the quark mass function appears to be momentum
dependent. For B → ∞, the Schwinger-Dyson equation at large N (Fig.1a) is
c
(cid:90) d4q
M(p;B) (cid:39) G trSLLL(q )θ(Λ2 −|p(cid:126)−(cid:126)q|2), (5)
(2π)4 2D z f
where B is used to reduce the quark propagator to the (1+1)-dimensional one and to separate
higher Landau levels from the LLL. In contrast to the previous case, the factor |eB| does not
appear in front of the integral. This is because the interaction (4) does not couple all the states
in the LLL, but couples the states having similar momenta (Fig.1b). This feature makes the gap
B-independent. In fact, carrying out the integral over the transverse momenta, we get
(cid:90)
G dq M(q;B)
M(p;B) (cid:39) z θ(Λ2 −|p −q |2)F(p −q ), (6)
2π 2π (cid:112)q2 +M2(q;B) f z z z z
z
(cid:113)
where F(k ) = Λ2 −|k |2. Note that the equation does not have any explicit B dependence, so
z f z
the mass gap is solely determined by the scale Λ , i.e. M(p;B) = M (p).
f Λ
f
Another important feature is the damping behavior of the mass function at large p. To see
this, in (6) we take p → ∞, for which q must go to ∞. Then the integrand in the RHS goes to
z z
zero. The phase space integral cannot compensate the damping behavior of the integrand, because
the integral is limited within the finite domain Λ around p . So M(p ) → 0 as p → ∞. Thus
f z z z
there must be some damping region of M(p ) which we will denote Λ as a function of Λ . The
z χ f
4
emergence of the damping scale Λ makes the chiral condensate UV finite4:
χ
(cid:90) Λχ dq M(q )
¯ z z
(cid:104)ψψ(cid:105) ∼ −|eB| ∼ −|eB|F(Λ ), (7)
(cid:112) f
2π q2 +M2(q )
−Λχ z z
where F(Λ ) is a function of Λ . The chiral condensate depends on B linearly, as advertised.
f f
Note that all of the discussions presented so far did not ask whether the interaction is confining
ornot. AllweneededwastheIRenhancement. Nevertheless, itisinterestingtofurtherinvestigate
a model which not only contains properties of the IR enhancement, but also captures certain
aspects of the confinement. For this purpose, we will use the model of Gribov-Zwanziger [17]
which contains the linear rising potential between colored charges. Confinement is expressed as
the absence of the quark continuum in the meson spectra. We will analyze the self-consistent
equations for the quark self-energy and meson states at large N and large B, by reducing them
c
into those of the ’t Hooft model. Then we will rederive the aforementioned conclusions.
2. Dimensional reduction
We consider the Euclidean action (convention: g = δ , γ = γ†):
µν µν µ µ
(cid:90)
S = d4x ψ¯[∂/ +iQA/ +m]ψ +S , (8)
E int
where A is a U(1) gauge field, Q is a flavor matrix for electric charges, and m is the current
em
quark mass matrix. The color gauge interaction is treated as S .
int
We apply an external, uniform magnetic field in spatial 3-direction5, which can be given by a
vector potential (A ,··· ,A ) = (0,Bx ,0,0). For later convenience, it is useful to introduce the
1 4 1
projection matrices. We decompose fermion fields,
1±sgn(QB)σ
PQ = 3 , σ = iγ γ , ψ = PQψ, σ ψ = ±sgn(QB)ψ . (9)
± 2 3 1 2 ± ± 3 ± ±
We expand the fermion fields in bases which diagonalize the unperturbed Lagrangian,
(cid:88)(cid:90) d3p˜
ψ (x) = ψ (l,p˜)e−ip˜x˜H (x ;p ), (10)
± (2π)3 ± l 1 2
l=0
where p˜≡ (0,p ,p ,p ). H (x ;p ) is the harmonic oscillator base with mω = |QB|, whose center
2 3 4 l 1 2
is located at x = p /QB. If we want to write the Lagrangian as a sum of the Landau levels, we
1 2
4In vacuum, Λ effectively plays the role of the UV cutoff in the NJL model.
χ
5We try to explain only the lattice results for uniform B in the quenched QED. If we had some charged
condensates(seerecentdiscussions,[18–20]),thedynamicalQEDwouldgeneratevorticesinelectricsuperconductor.
5
should relabel ψ (n,x˜) → ψ (x˜), ψ (n−1,x˜) → ψ (x˜), ψ (0,x˜) → PQχ(x˜). Now the index
+ n+ − n− + +
n characterizes the Landau level. The unperturbed action becomes
(cid:32) (cid:33)
(cid:90) (cid:90)
(cid:88)
d4xL (x) = d3x˜ L (x˜)+ L (x˜) , (11)
unpert. 0 n
n=1
after carrying out the integral over x . The Lagrangian density is (γ = (γ ,γ ) and γ = (γ ,γ ))
1 L 3 4 ⊥ 1 2
L = χ¯(x˜)(∂/ +m)PQχ(x˜), L = ψ¯ (x˜)(cid:16)∂/ +isgn(QB)(cid:112)2n|QB|γ +m(cid:17)ψ (x˜). (12)
0 L + n n L 2 n
The propagators for Landau levels are6
(cid:90) d3p˜ i
(cid:10)χ(x˜)χ¯(y˜)(cid:11) = PQ e−ip˜(x˜−y˜),
(2π)3 + p/ +im
L
(cid:90) d3p˜ i
(cid:10)ψ (x˜)ψ¯ (y˜)(cid:11) = e−ip˜(x˜−y˜). (13)
n n (2π)3 p/ −sgn(QB)(cid:112)2n|QB|γ +im
L 2
We also expand the color gauge interactions in the Ritus bases instead of usual Fourier bases [21],
(cid:90) (cid:88) (cid:90) d3p˜d3q˜
d4x ψ¯γ t ψA (x) = ψ¯(l,p˜)γ t ψ(l(cid:48),q˜)All(cid:48)(p˜−q˜;p ,q ), (14)
µ a µa (2π)6 µ a µa 2 2
l,l(cid:48)=0
where the gauge field is convoluted with the harmonic oscillator bases,
(cid:90) (cid:90)
dk
All(cid:48)(p˜−q˜;p ,q ) = 1 A (p˜−q˜,k ) dx H (x ;p )H (x ;q )e−ik1x1. (15)
µa 2 2 2π µa 1 1 l 1 2 l(cid:48) 1 2
For later convenience, it is useful to prepare the propagator for this convoluted form,
(cid:68) (cid:69)
All(cid:48)(p˜−q˜;p ,q )Aj(cid:48)j(q˜−p˜;q ,p )
µa 2 2 νb 2 2
(cid:90) (cid:90) (cid:90)
dk
= 1 dx dy H (x ;p )H (x ;q )Dab(p˜−q˜,k )H (y ;q )H (y ;p )e−ik1(x1−y1), (16)
2π 1 1 l 1 2 l(cid:48) 1 2 µν 1 j(cid:48) 1 2 j 1 2
where we used (cid:104)A (p)A (q)(cid:105) = (2π)4δ4(p + q)Dab(p). Let us note that the H (x ;p ) depends
µa νb µν l 1 2
on coordinates in the combination, x −p /QB. We can simultaneously shift x −p /QB → x ,
1 2 1 2 1
and y −p /QB → y , and get
1 2 1
H (x ;p )H (x ;q )H (y ;q )H (y ;p ) → H (x )H (x ;q −p )H (y ;q −p )H (y ), (17)
l 1 2 l(cid:48) 1 2 j(cid:48) 1 2 j 1 2 l 1 l(cid:48) 1 2 2 j(cid:48) 1 2 2 j 1
6It is convenient to attach the Ritus wavefunctions to the vertices instead of propagators, because the different
Landau orbits couple only after interactions are turned on.
6
keeping the other part in Eq.(16) invariant. Thus the integral depends on p and q only through
2 2
p −q . This is natural consequence of the gauge invariance, A = Bx → B(x −c), that affects
2 2 2 1 1
the origin of p and q but should not affect the final result.
2 2
This complicated expression can be drastically simplified if (i) the interaction shows an IR
enhancement around ∼ Λ , while it damps quickly in the UV, and (ii) the magnetic field is
QCD
sufficiently strong, |QB| (cid:29) Λ . The gluon propagator rapidly damps for p˜− q˜ and k much
QCD 1
larger than Λ , cutting off the domain of integral. Note also that the function H(x ) contains
QCD 1
(cid:112)
the Gaussian function, exp[−|QB|x2/2], cutting off the domain of x by ∼ 1/ |QB|, as far as we
1 1
consider not very high Landau orbits.
Assembling all these properties, we can make replacements at very large B,
(cid:16) (cid:112) (cid:17)
e−ik1(x1−y1) → 1, |k (x −y )| ∼ Λ / |QB| (cid:28) 1
1 1 1 QCD
(cid:16) (cid:112) (cid:17)
H (x ;p −q )H (x ) → H (x )H (x ), |p −q | ∼ Λ (cid:28) |QB| (18)
l 1 2 2 l(cid:48) 1 l 1 l(cid:48) 1 2 2 QCD
which simplify the Eq.(16) as
(cid:90)
(cid:68) (cid:69) dk
All(cid:48)(p˜−q˜;p ,q )Aj(cid:48)j(q˜−p˜;q ,p ) −→ 1 δ δ Dab(p˜−q˜,k ). (19)
µa 2 2 νb 2 2 2π ll(cid:48) j(cid:48)j µν 1
(Note that indices are for the orbital quanta. Whether they coincide with the Landau level indices
depends on γ-matrices to which gluons couple, because γ ’s flip spins.)
⊥
The final expression itself coincides with a naive expectation and is not surprising. Ultimately,
as B → ∞ a hopping from one Landau orbit to others (transverse dynamics) is completely sup-
pressed. A nontrivial consequence of the asymptotic free theories, however, is that the separation
of the Landau levels is achieved at a relatively small magnetic field, compared to forces like 1/p2.
For the 1/p2 force, the damping of the UV force is much milder so that the LLL and higher
levels do not decouple quickly. The validity of separation is quantified by examining how the
replacements in Eq.(18) work.
3. The self-consistent equations at large N within a confining model
c
Now we consider the self-consistent equations, the Schwinger-Dyson and Bethe-Salpeter equa-
tions. Below we set m = 0. We use the gluon propagator of the Gribov-Zwanziger type [17],
δ 8πσ
Dab(k) = ab D (k), D (k) = −δ , D (k) = D (k) = 0. (20)
µν CF µν 44 44 ((cid:126)k2)2 4j ij
which is motivated by Coloumb gauge studies. D gives a linear rising potential for the color
44
charges, and we dropped off terms without the strong IR enhancement. σ is a string tension of
O(Λ2 ), C = (N2 −1)/2N is Casimir for the adjoint representation.
QCD F c c
7
We will assume that the self-energies are diagonal for each Landau level, and are translational
invariant. Then the Schwinger-Dyson equation, after applying our approximations (19), becomes
(cid:90) d3q˜ (cid:20) (cid:21)(cid:90) dk
(cid:0) (cid:1) 1
Σ/ +Σ/ +Σ (n,p˜) = iγ S(n,q˜;Σ)iγ D (p˜−q˜,k ), (21)
L ⊥ m (2π)3 4 4 2π 44 1
that is diagonal for the Landau orbits.
The computations are particularly simple for the LLL. Below we drop the subscript 0 for the
LLL. The matrix γ drops off because of the projection operator PQ, so that RHS of the equation
⊥ +
does not have γ , meaning that Σ (p˜) = 0 in LHS. Then the equation looks like
⊥ ⊥
(cid:90) d3q˜ i(cid:2)/q −Σ/ (q˜)(cid:3)+Σ (q˜) (cid:90) dk
(cid:0)Σ/ +Σ (cid:1)(p˜) = iγ L L m iγ 1 D (p˜−q˜,k ). (22)
L m (2π)3 4 (cid:2)q −Σ (q˜)(cid:3)2 +Σ2 (q˜) 4 2π 44 1
L L m
Recall that the quark transverse momentum is gauge dependent in the sense that A = Bx →
2 1
B(x −c)affectstheoriginofthetransversemomentaasp → p +QBcandq → q +QBc. Doing
1 2 2 2 2
this shift, the gluon propagator is unaffected while the modification appears only through the self-
energy as Σ(p ,p ) → Σ(p ,p +QBc), and Σ(p ,q ) → Σ(p ,q +QBc). This replacement does
L 2 L 2 L 2 L 2
not change the structure of the self-consistent equation at all, so that we obtain the same solutions
for the two cases, Σ(p ,p ) = Σ(p ,p +QBc), which means that Σ(p) is p independent.
L 2 L 2 2
As a consequence, we can factorize the equation:
(cid:90) d2q i(cid:2)/q −Σ/ (q )(cid:3)+Σ (q ) (cid:90) dk dq
(cid:0)Σ/ +Σ (cid:1)(p ) = L iγ L L L m L iγ 1 2 D (p˜−q˜,k ). (23)
L m L (2π)2 4 (cid:2)q −Σ (q )(cid:3)2 +Σ2 (q ) 4 (2π)2 44 1
L L L m L
The gluon propagator is smeared by integrating out the transverse momentum,
(cid:90) dk dq (cid:90) d2(cid:126)k −8πσ 2σ
D2D(p −q ) = 1 2 D (p˜−q˜,k ) = ⊥ = − , (24)
44 z z (2π)2 44 1 4π2 (cid:0)(pz −qz)2 +(cid:126)k⊥2(cid:1)2 (pz −qz)2
which is a confining propagator in (1+1) dimensions. The form of the Schwinger-Dyson equation
is exactly same as that of ’t Hooft model [22] in axial gauge, A = 0, whose solution is known.
z
The string tension is related to the two dimensional gauge coupling as N g2 = 4σ.
c 2D
The magnetic field disappears from the equation. The only role of the magnetic field is to
make the unperturbed quark propagator (1+1)-dimensional and to separate the LLL from the
other Landau orbits. The only scale in the equation is σ ∼ Λ2 . Using the dressed (1+1)-
QCD
dimensional quark propagator for Eq.(7), we can get the chiral condensate at finite B.
Since the quark self-energies are not always well-defined, we will also consider the meson states
which are free from any ambiguities. One can estimate an effective or constituent quark mass by
examining the meson spectra. To do this, we study the Bethe-Salpeter equation.
8
We shall move to Minkowski space, and treat the homogenous Bethe-Salpeter equation assum-
ing that the total momentum of a quark and an anti-quark is sufficiently close to the pole. The
equation for a color singlet channel is
(cid:90) d4k
Ψ(P;q)ff(cid:48) = − (cid:2)S(k −P)γ Ψ(P;k)γ S(k +P)(cid:3)ff(cid:48)D (k −q), (25)
αβ (2π)4 0 0 αβ 00
where Ψ(P;q) is a meson wavefunction with total momentum P and relative momentum q for a
quark and an anti-quark. The indices α,β(f,f(cid:48)) are for spinor (flavor) indices. We can decompose
the wavefunction into different spinor combinations and flavor structures,
Ψff(cid:48) = (Ψ +Ψ γ +Ψ γ +···) 1 +··· . (26)
αβ S 5 5 L L αβ ff(cid:48)
We must apply further decompositions or take appropriate linear combinations, because a mag-
netic field breaks isospin and rotational symmetries explicitly. Below we restrict ourselves to the
meson states with electric charges and spinor combinations for which we can close the equa-
tion only by the LLL. One can select out such mesons using the projection operator PQ =
+
(1 + isgn(QB)γ γ )/2. For the two flavor case, the following structures satisfy the condition,
1 2
¯ ¯
(uu¯,dd)⊗(1,γ ,γ ,γ γ ,σ ,σ ), (ud,du¯)⊗(γ ,γ γ ,σ ). (27)
5 L L 5 LL(cid:48) ⊥⊥(cid:48) ⊥ ⊥ 5 L⊥
(cid:112)
For instance, neutral pions remain light, while charged ones acquire the masses ∼ |eB|. Other
examples are vector mesons. The longitudinal (transverse) component of neutral (charged) vector
mesons can remain light, while others not7. This observation seems to be consistent with recent
lattice results [19] and model calculations [23].
Below we consider the neutral scalar component, Ψ , as an illustration, taking only the LLL
S
into account. As we saw, the dressed quark propagator is independent of the transverse momen-
tum. Then we can conclude that Ψ (P;q) is independent of P , because P -dependence appears
S ⊥ ⊥
only through Ψ so that different P gives the same equation and thereby the same solution. We
S ⊥
can also conclude that Ψ (P;q) is independent of q , because simultaneous shift of momenta,
S ⊥
q → q +c and k → k +c, does not affect the equation.
⊥ ⊥ ⊥ ⊥
Therefore we can again factorize the equation,
(cid:90) d2k (cid:90) d2k
L (cid:2) (cid:3) ⊥
Ψ (P ;q ) = − S(k −P )γ Ψ (P ;k )γ S(k +P ) D (k−q), (28)
S L L (2π)2 L L 0 S L L 0 L L (2π)2 00
7More careful considerations are necessary beyond the large N limit where the annihilation diagrams may
c
contribute. In this respect studies of instantons in a magnetic field are important [24].
9
Figure 2: A structure of mesons listed in Eq.(27). With a magnetic field, a quark and an anti-quark aligns along
direction of the magnetic field. The meson can move in the transverse direction without costing much energy.
and obtain the Bethe-Salpeter equation in the ’t Hooft model. There is no B-dependence in the
equation, so masses of mesons are characterized by current quark masses and Λ . There are an
QCD
infinite tower of meson spectra, due to the string oscillations along z-direction.
The total momentum of a quark and an anti-quark is independent of transverse momentum,
meaning that the meson can move freely in the transverse direction without costing energy8. The
inclusion of the higher Landau level can generate the dependence on the transverse momentum,
which is suppressed by a factor of Λ /|eB|1/2. This leads to a spatially anisotropic meson
QCD
Lagrangian [8] for which mesonic fluctuations are strong due to enhanced phase space in the
infrared. It may significantly affect dynamics at finite temperature [25], like a system close to the
quasi-long range order9 [27].
On the other hand, the independence of the relative momentum means that the internal meson
wavefunction behaves as
(cid:90)
Ψrel.((cid:126)r) = dq d2(cid:126)q Ψrel.(q )eiq(cid:126)·(cid:126)r ∼ δ2((cid:126)r )ψ (r ), (29)
S z ⊥ S z ⊥ S z
that is, a quark and an anti-quark align along the z-direction, and the inter-particle distance
is ∼ 1/Λ (Fig.3). Couplings with the higher Landau levels will introduce the width for the
QCD
(cid:112)
transverse wavefunction, and the mean square radius should be (cid:104)(cid:126)r2(cid:105) ∼ Λ /|QB|.
⊥ QCD
4. Conclusion
We have discussed that the quark mass gap in the presence of a magnetic field stays around
∼ Λ or less, provided that it is mainly generated by the nonperturbative part of the gluon
QCD
exchange. Accordingly the chiral condensate grows at most linearly as a function of B. This
8The meson dynamics is effectively (1+1)-dimensional, in the sense that the transverse momenta do not affect
much the energy dispersion. This means that mesons can easily propagate in the transverse directions.
9 In the strict quasi-long range order, (cid:104)ψ¯ψ(cid:105) ∼ ρ(cid:104)eiθ(cid:105) → 0 due to the IR divergence of the phase fluctuations.
But the quark mass gap can remain finite as far as ρ(cid:54)=0 [26].
10