Nondecoupling phenomena in QED in a magnetic field and noncommutative QED

0.15 MB·English

by  E. V. Gorbar

Nondecoupling phenomena in QED in a magnetic field and noncommutative QED
E.V. Gorbar,∗ Michio Hashimoto,† and V.A. Miransky‡
Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada
(Dated: February 7, 2008)
The dynamics in QED in a strong constant magnetic field and its connection with the noncom-
mutative QED are studied. It is shown that in the regime with the lowest Landau level (LLL)
dominance the U(1) gauge symmetry in the fermion determinant is transformed into the noncom-
mutative U(1) gauge symmetry. In this regime, the effective action is intimately connected with
nc
that in noncommutative QED and the original U(1) gauge Ward identities are broken (the LLL
anomaly). Ontheotherhand,itisshownthatalthoughacontributionofeachofaninfinitenumber
of higher Landau levels is suppressed in an infrared region, their cumulative contribution is not (a
nondecoupling phenomenon). This leads to a restoration of the original U(1) gauge symmetry in
the infrared dynamics. The physics underlying this phenomenon reflects the important role of a
5 boundarydynamics at spatial infinityin this problem.
0
0 PACSnumbers: 11.10.Nx,11.15.-q,12.20.-m
2

n Sincetheclassicalpapers[1,2],theproblemofQEDinaconstantmagneticfieldhasbeenthoroughlystudied(fora
a recentreview,seeRef. [3]). Inthisletter,weconsiderthisproblemforthecaseofastrongmagneticfield. Inparticular,
J
we study the connection of this dynamics with that in noncommutative QED (for reviews of noncommutative field
8
theories (NCFT), see Ref. [4]). The motivation for this study was the recent results obtained in Ref. [5], where
1
the connection between the dynamics in the Nambu-Jona-Lasinio (NJL) model in a strong magnetic field and that

in NCFT was established. The main conclusion of that paper was that although in the regime with the lowest
1
v Landau level (LLL) dominance the NJL model determines a NCFT, this NCFT is different from the conventional
5 ones considered in the literature. In particular, the UV/IR mixing, taking place in the conventional NCFT [6], is
3 absent in this case. The reason of that is an inner structure (i.e., dynamical form-factors) of neutral composites in
1 this model.
1
In this letter, some sophisticated features of the dynamics in QED in a strong magnetic field are revealed. It is
0
shownthat in the approximationwith the lowest Landaulevel (LLL) dominance, the initial U(1) gauge symmetry in
5
0 thefermiondeterminantistransformedintothenoncommutativeU(1)nc gaugesymmetry. Inthisregime,theeffective
/ action is intimately connected with that in noncommutative QED and the original U(1) gauge Ward identities are
h
broken (we call this phenomenon an LLL anomaly). In fact, this dynamics yields a modified noncommutative QED
t
- in which the UV/IR mixing is absent, similarly to the case of the NJL model in a strong magnetic field [5]. The
p
reason of that is an inner structure (i.e., a dynamical form-factor) of photons in a strong magnetic field. However,
e
h it is not the end of the story. We show that adding the contribution of all the higher Landau levels removes the
: LLL anomaly and restores the originalU(1) gauge symmetry. This restorationhappens in a quite sophisticated way:
v
althoughacontributionofeachofaninfinite numberofhigherLandaulevelsissuppressedinaninfraredregion,their
i
X cumulative contribution is not (a nondecoupling phenomenon). As will be discussed below, this phenomenon reflects
r the important role of a boundary dynamics at spatial infinity in this problem. We also indicate the kinematic region
a where the LLL approximation is reliable.
To put the dynamics in QED in a magnetic field under control, we will consider the case with a large number of
fermion flavors N, when the 1/N expansion is reliable. We also choose the current fermion mass m satisfying the
conditionm ≪m≪ |eB|,wherem isthedynamicalmassoffermionsgeneratedinthechiralsymmetricQED
dyn dyn
in a magnetic field [7]. 1 The condition m ≪ m guarantees that there are no light (pseudo) Nambu-Goldstone
dyn
p
bosons, and the only particles in the low energy effective theory in this model are photons. As to the condition
m≪ |eB|, it implies that the magnetic field is very strong.
Integrating out fermions, we obtain the effective action for photons in the leading order in 1/N:
p
1
Γ=Γ(0)+Γ(1), Γ(0) =− d4xf2 , Γ(1) =−iNTrLn[iγµ(∂ −ieA )−m], (1)
4 µν µ µ
Z
∗Electronicaddress: [email protected];Onleavefrom BogolyubovInstitute forTheoretical Physics,03143, Kiev,Ukraine
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected];Onleavefrom BogolyubovInstitute forTheoretical Physics,03143, Kiev,Ukraine
1 The dynamical mass is mdyn ≃ |eB|exp(−N) for a large running coupling α˜b ≡ Nαb related to the magnetic scale |eB|, and
mdyn∼ |eB|exp −α˜blnπ(N1/α˜b) pwhenthecouplingα˜b isweak[7]. p
p h i
2
where fµν =∂ A −∂ A and the vector field A =Acl+A˜ , where the classical part Acl is Acl =h0|A |0i. Since
µ ν ν µ µ µ µ µ µ µ
the constant magnetic field B is a solution of the exact equations in QED (see for example the discussion in Sec. 2
in the second paper in Ref. [7]), it is
Bx2 Bx1
Acl =(0, ,− ,0). (2)
µ 2 2
This field describes a constantmagnetic field directed in the +x3 direction and we use the so called symmetric gauge
for Acl.
µ
Forastrongmagneticfield|eB|≫m2,itisnaturallytoexpectthatintheinfraredregionwithmomentak ≪ |eB|,
the LLL approximation should be reliable. Indeed, let us consider the fermion propagator in a magnetic field [2]:
p
ie
S(x,y)=exp (x−y)µAext(x+y) S˜(x−y) , (3)
2 µ
(cid:20) (cid:21)
where the Fourier transform of the translationally invariant part S˜ can be decomposed over the Landau levels [8]:
k2 ∞ D (eB,k)
S˜(k)=iexp − ⊥ (−1)n n (4)
|eB| k2−m2−2|eB|n
(cid:18) (cid:19)n=0 k
X
with k ≡ (k1,k2) and k ≡ (k ,k ). The functions D (eB,k) are expressed through the generalized Laguerre
⊥ k 0 3 n
polynomials Lα:
m
k2 k2
D (eB,k) = (k γk+m) 1−iγ1γ2sign(eB) L 2 ⊥ − 1+iγ1γ2sign(eB) L 2 ⊥
n k n |eB| n−1 |eB|
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
(cid:0) k2 (cid:1) (cid:0) (cid:1)
+4(k1γ1+k2γ2)L1 2 ⊥ , (5)
n−1 |eB|
(cid:18) (cid:19)
where γk ≡ (γ0,γ3). Relation (4) seems to suggest that in the infrared region, with k ,k ≪ |eB|, all the higher
⊥ k
Landau levels with n ≥ 1 decouple and only the LLL with n = 0 is relevant. Although this argument is physically
p
convincing, there may be a potential flaw due to an infinite number of the Landau levels. As will be shown below,
this is indeed the case in this problem: the cumulative contribution of the higher Landau levels does not decouple.
But first we will consider the dynamics in the LLL approximation. In this case, the calculation of the effective
action (1) is reduced to calculating fermion loops with the LLL fermion propagators. Such a problem in the NJL
model in a magnetic field has been recently solved in Ref. [5]. The extension of that analysis to the case of QED is
straightforward. The effective action (1) in the LLL approximation is given by
iN|eB|
Γ =Γ(0)+Γ(1) , Γ(1) =− d2x Tr PLn[iγ||(∂ −ieA )−m] (6)
LLL LLL LLL 2π ⊥ || || || ∗
Z h i
(compare with Eq. (54) in [5]). Here ∗ is the symbol of the Moyal star product, which is a signature of a NCFT [4],
the projector P is P ≡[1−iγ1γ2sign(eB)]/2, and the longitudinal“smeared”fields A|| are defined as A|| =e4∇|e2⊥B|A||
[5], where ∇2 is the transverse Laplacian. Notice that P is the projector on the fermion (antifermion) states with
⊥
the spin polarized along (opposite to) the magnetic field and that the one-loop term Γ(1) in (6) includes only the
LLL
longitudinal field A = (A ,A ). This is because the LLL fermions couple only to the longitudinal components of
|| 0 3
the photon field [7].
Inaction(6),thetraceTr ,relatedtothelongitudinalsubspace,istakeninthefunctionalsenseandthestarproduct
||
relates to the space transverse coordinates. Therefore the LLL dynamics determines a NCFT with noncommutative
transverse coordinates xˆa, a=1,2:
⊥
1
[xˆa,xˆb ]=i ǫab ≡iθab. (7)
⊥ ⊥ eB
The structure of the logarithm of the fermion determinant in Γ(1) (6) implies that it is invariant not under the
LLL
initial U(1) gauge symmetry but under the noncommutative U(1) gauge one [4] (henceforth we omit the subscript
nc
|| in gauge fields):
i
A → U(x)∗A ∗U−1(x)+ U(x)∗∂ U−1(x), (µ=0,3) (8a)
µ µ µ
e
F → U(x)∗F ∗U−1(x), (µ,ν =0,3) (8b)
µν µν
3
where U(x)=(eiλ(x)) and the field strength F is
∗ µν
F =∂ A −∂ A −ie[A ,A ] (9)
µν µ ν ν µ µ ν MB
with the Moyal bracket
[A ,A ] ≡A ∗A −A ∗A . (10)
µ ν MB µ ν ν µ
Therefore the derivative expansion of Γ(1) should be expressed through terms with the star product of the field
LLL
F and its covariant derivatives:
µν
(1)
Γ =a S +a S +a S +a S +··· , (11)
LLL 0 F2 1 F3 2 (DF)2 3 D2F2
where
1
S ≡− d2x d2x F ∗Fµν, S ≡ie d2x d2x F ∗Fνλ∗F µ,
F2 4 ⊥ k µν F3 ⊥ k µν λ
Z Z
S ≡ d2x d2x D Fλµ∗DρF , S ≡ d2x d2x D F ∗DλFµν, (12)
(DF)2 ⊥ k λ ρµ D2F2 ⊥ k λ µν
Z Z
and the covariant derivative of F is D F = ∂ F −ie[A ,F ] . These are all independent operators which
µν λ µν λ µν λ µν MB
have the dimension four and six. In particular, by using the Jacobi identity,
[D ,[D ,D ] ] +[D ,[D ,D ] ] +[D ,[D ,D ] ] =0, (13)
µ ν λ MB MB ν λ µ MB MB λ µ ν MB MB
and the relation F = ie−1[D ,D ] , one can easily check that the operator d2x d2x D F ∗DµFνλ is not
µν µ ν MB ⊥ k λ µν
independent: it is equal to −1/2S .
D2F2 R
The coefficients a , (i=0,1,2,3,···) in Eq. (11) can be found from the n-point photon vertices
i
(ie)nN|eB|
T(n) =i d2x⊥d2x||···d2x|| tr S (x||−x||)Aˆ(x⊥,x||)...S (x||−x||)Aˆ(x⊥,x||) (14)
LLL 2πn 1 n || 1 2 2 || n 1 1 ∗
Z h i
by expandingthe verticesinpowersofexternalmomenta(here Sk(xk)= (d22πk)k2e−ikkxkkkγki−mP andAˆ≡γ||A||). In
particular, from the vertices TL(2L)L and TL(3L)L, we find the coefficients a0, aR1, a2, and a3 connected with the operators
of the dimension four and six in the derivative expansion (11) of Γ(1) :
LLL
α˜ |eB| 1 1
a = , a = a , a =− a , a =0, (15)
0 3π m2 1 60m2 0 2 10m2 0 3
where α˜ ≡Nα=Ne2/(4π)(since in the presence of a magnetic field the chargeconjugationsymmetry is broken,the
3-point vertex is nonzero).
The U(1) gauge Ward identities imply that the n-point photon amplitude Tµ1...µn(x1,...,xn) should be transverse,
i.e., ∂µiTµ1...µn(x1,...,xn)=0. It is easy to show that the 2-point vertex TLµL1µL2 yielding the polarization operator is
transverse indeed. Now let us turn to the 3-point vertex and show that it is not transverse, i.e., the Ward identities
connected with the initial gauge U(1) are broken in the LLL approximation. In the momentum space, the vertex is
Tµ1µ2µ3(k ,k ,k )=Ne3|eB|sin 1θ ka kb ∆µ1kµ2kµ3k(k ,k ,k ) (16)
LLL 1 2 3 2π 2 ab 1⊥ 2⊥ LLL 1k 2k 3k
(cid:18) (cid:19)
with 2
∆µ1kµ2kµ3k(k ,k ,k )≡ d2ℓk tr γkµ1[(/ℓ−k/1)k+m]γkµ2[(/ℓ+k/3)k+m]γkµ3(/ℓk+m) . (17)
LLL 1k 2k 3k i(2π)2 h (ℓ2−m2)[(ℓ−k )2−m2][(ℓ+k )2−m2] i
Z k 1 k 3 k
2 The explicit expression for ∆µ1kµ2kµ3k is given in terms of the two dimensional version of the Passarino-Veltman functions [9]. It is
LLL
quitecumbersomeandwillbewrittendownelsewhere.
4
The argument of the sine in Eq. (16) is the Moyal cross product with θ = ǫab/eB (see Eq. (7)). It is easy to find
ab
that the transverse part of the vertex (16) is not zero and equal to
2e 1
k Tµ1µ2µ3(k ,k ,k )=− sin θ ka kb Πµ2µ3(k )−Πµ2µ3(k ) , (18)
1µ1 LLL 1 2 3 i 2 ab 1⊥ 2⊥ k 2k k 3k
(cid:18) (cid:19)h i
with
4m2
1+ 1−
Πµν(k )=i2α˜|eB| gµν − kkµkkν Π(k2), Π(k2)≡1+ 2m2 ln s kk2 . (19)
k k π k kk2 ! k k kk2 1− 4km22 −1+ 1− 4m2
k k2
q s k
Herewedefinedg =diag(1,−1)[notethatΠµν coincideswiththepolarizationtensorinthe(1+1)-dimensionalQED
k k
iftheparameter2α˜|eB|hereisreplacedbythecouplinge2 inQED ]. Thus,theoriginalU(1)gaugeWardidentities
1 1+1
are broken in the LLL approximation. We will call this an LLL anomaly.
Note that all the vertices T(n) with n ≥ 3 are finite and a logarithmic divergence in the vertex T(2) , which is
LLL LLL
proportionalto the polarizationoperator,is absentif oneuses a gaugeinvariantregularization. Infact, itis sufficient
if the regularization is invariant under the longitudinal U(1) gauge group with phase parameters α(x||) depending
||
only on longitudinal coordinates. This U(1) is a subgroup of both the gauge U(1) and the noncommutative gauge
||
U(1) and it is the gauge symmetry of the whole action Γ (6). Indeed, while the free Maxwell term Γ(0) in (6)
nc LLL
is invariantunder the gaugeU(1), the one-looptermΓ(1) is invariantunder the U(1) . Another noticeable pointis
LLL nc
that the divergence (18) is not a polynomial function of momenta and have branch point singularities. Therefore the
fact that T(3) is not transverse is regularizationindependent.
LLL
The originofthe LLL anomalyis clear: theT(n) verticescome fromthe one-looppartΓ(1) ofthe actionwhichis
LLL LLL
invariantnot under the gaugeU(1) but under the U(1) . Therefore the Ward identities for the vertices T(n) reflect
nc LLL
not the U(1) gauge symmetry but the noncommutative symmetry U(1) .
nc
Notice that the action Γ (6) determines a conventional noncommutative QED only in the case of an induced
LLL
photon field, when the Maxwell term Γ(0) is absent. When this term is present, the action also determines a NCFT,
however,this NCFT is different from the conventionalones considered in the literature. In particular, expressing the
−∇2⊥
photon field Aµ through the smeared field Aµ as Aµ = e4|eB|Aµ, we find that the propagator of the smeared field
−p2⊥ −p2⊥
rapidly, as e2|eB|, decreases for large transverse momenta. The form-factor e2|eB| built in the smeared field reflects
an inner structure of photons in a magnetic field. This feature leads to removing the UV/IR mixing in this NCFT
(compare with the analysis of the UV/IR mixing in Sec. 4 of Ref. [5]).
It is clear howeverthat, since the initial QEDhas the usualU(1)gaugesymmetry, there shouldexist anadditional
contribution that restores the U(1) Ward identities broken in the LLL approximation. We will show that this
contribution comes from heavy (naively decoupled) higher Landau levels (HLL), and it is necessary to consider the
contribution of all of them in order to restore the Ward identities.
Let us consider the 3-point vertex replacing one of the LLL propagator by the full one in Eq. (4). We find that at
each n≥1 the contributions in the vertex of the first, second and third terms in Eq. (5) are respectively
∆µkνkλk(p,q,k) ∼ (−1)n fµνλ(p ,q ,k ), (20)
HLL n|eB| k k k k
∆µ⊥ν⊥λk(p,q,k) ∼ (−1)n gµνh (p ,q ,k )+ǫµνh (p ,q ,k )sign(eB) qλ − (q·k)k kλ Π(k2), (21)
HLL n|eB| ⊥ 1 ⊥ ⊥ ⊥ ⊥ 2 ⊥ ⊥ ⊥ k k2 k k
k !
h i
∆µ⊥νkλk(p,q,k) ∼ (−1)n hµ⊥(p ,q ,k ) gνλ− kkνkkλ Π(k2), (22)
HLL n|eB| 3 ⊥ ⊥ ⊥ k k2 k
k !
where g = diag(−1,−1), (p·q) = p0q0−p3q3, and fµνλ, h and hµ⊥ are some smooth functions of longitudinal
⊥ k k 1,2 3
and transverse momenta. As was expected, the contribution of each of the individual HLL with n ≥1 is suppressed
by powers of 1/|eB| in the infrared region. It is however quite remarkable that despite the suppression of individual
HLL contributions,their cumulativecontributionis notsuppressedinthe infraredregion(anondecouplingeffect). In
5
fact, using the relation [10]
∞
xz
(1−z)−(α+1)exp = Lα(x)zn (23)
z−1 n
(cid:18) (cid:19) n=0
X
and integratingit with respect to z, we can performexplicitly the summationoverthe HLL contributions and obtain
the 3-point vertex that satisfies the Ward identities for the U(1) gauge symmetry:
Tµ1µ2µ3(k ,k ,k )=Tµ1µ2µ3(k ,k ,k )+Tµ1µ2µ3(k ,k ,k )+Tµ1µ2µ3(k ,k ,k ), (24)
1 2 3 LLL 1 2 3 HLL1 1 2 3 HLL2 1 2 3
where
2e 1
Tµ1µ2µ3(k ,k ,k )= sin θ ka kb
HLL1 1 2 3 i 2 ab 1⊥ 2⊥
(cid:18) (cid:19)
−kµ1 kµ1kµ2 −kµ2kµ1 −(k ·k ) gµ1µ2
× 1⊥ Πµ2µ3(k )−Πµ2µ3(k ) + 1⊥ 2⊥ 1⊥ 2⊥ 1 2 ⊥ ⊥ k Πµ3ν(k )
k2 k 2k k 3k k2 k2 2kν k 3k
" 1⊥ 1⊥ 2⊥
(cid:16) (cid:17)
+ permutations of (k ,µ ), (k ,µ ), and (k ,µ ) , (25)
1 1 2 2 3 3
#
2e
Tµ1µ2µ3(k ,k ,k )=− sign(eB)
HLL2 1 2 3 i
(k ·k ) 1 kµ2ǫabka kb +(k ·k ) ǫµ2bkb
× exp − 1 2 ⊥ −cos θ ka kb 1⊥ ⊥ 1⊥ 2⊥ 1 2 ⊥ ⊥ 1⊥ Πµ3µ1(k )
2|eB| 2 ab 1⊥ 2⊥ k2 k2 k 3k
" (cid:26) (cid:18) (cid:19) (cid:18) (cid:19)(cid:27) 1⊥ 2⊥
kµ1ǫabka kb +(k ·k ) ǫµ1bkb gµ1µ2ǫabka kb +ǫµ1µ2(k ·k )
+ 2⊥ ⊥ 2⊥ 1⊥ 2 1 ⊥ ⊥ 2⊥ Πµ2µ3(k )+ ⊥ ⊥ 1⊥ 2⊥ ⊥ 1 2 ⊥ k Πµ3ν(k )
k2 k2 k 3k k2 k2 2kν k 3k
1⊥ 2⊥ 1⊥ 2⊥ !
+ permutations of (k ,µ ), (k ,µ ), and (k ,µ ) . (26)
1 1 2 2 3 3
#
Here we defined (p·q) = p1q1+p2q2. These contributions come from the HLL terms in Eqs. (21) and (22). The
⊥
HLL contribution coming from ∆µkνkλk in Eq. (20) is of a higher order in 1/|eB| and therefore is neglected here. 3
HLL
UsingEqs.(18), (25),and(26), onecaneasilycheckthatthe3-pointvertexTµ1µ2µ3 istransverse. Thecancellation
occurs between Tµ1µ2µ3 and Tµ1µ2µ3. As to the term Tµ1µ2µ3, it is transverse itself.
LLL HLL1 HLL2
Is there a kinematic region in which the LLL contribution is dominant? The answer to this question is “yes”. It is
the region with momenta k2 ≫|k2 |. In this case, the leading terms in the expansion of the LLL and HLL vertices
i⊥ ik
in powers of k are:
ik
2eα˜|eB| 1
Tµ1µ2µ3(k ,k ,k )=− sin θ ka kb (k −k )µ1gµ2µ3 +(k −k )µ2gµ3µ1+(k −k )µ3gµ1µ2 , (27)
LLL 1 2 3 3π m2 2 ab 1⊥ 2⊥ 2 3 k k 3 1 k k 1 2 k k
(cid:18) (cid:19)" #
2eα˜|eB| 1 −kµ1
Tµ1µ2µ3(k ,k ,k ) = − sin θ ka kb 1⊥ (k2 −k2 )gµ2µ3 −kµ2kµ3 +kµ2kµ3
HLL1 1 2 3 3π m2 2 ab 1⊥ 2⊥ k2 2k 3k k 2k 2k 3k 3k
(cid:18) (cid:19)" 1⊥ n o
kµ1kµ2 −kµ2kµ1 −(k ·k ) gµ1µ2
+ 1⊥ 2⊥ 1⊥ 2⊥ 1 2 ⊥ ⊥ k2 kµ3 −(k ·k ) kµ3
k2 k2 3k 2k 2 3 k 3k
1⊥ 2⊥
n o
+ permutations of (k ,µ ), (k ,µ ), and (k ,µ ) , (28)
1 1 2 2 3 3
#
3 Notethatthevertexfortheinitialnon-smearedfieldsAµ isgivenbye−k21⊥+4k|e22B⊥|+k23⊥Tµ1µ2µ3. Thus,forthesefields,anexponentially
dampingform-factoroccursnotinthepropagator butintheirvertices(comparewithadiscussionofthisfeatureinRef. [5]).
6
2eα˜eB (k ·k ) 1
Tµ1µ2µ3(k ,k ,k ) = exp − 1 2 ⊥ −cos θ ka kb
HLL2 1 2 3 3π m2 2|eB| 2 ab 1⊥ 2⊥
" (cid:26) (cid:18) (cid:19) (cid:18) (cid:19)(cid:27)
kµ2ǫabka kb +(k ·k ) ǫµ2bkb
× 1⊥ ⊥ 1⊥ 2⊥ 1 2 ⊥ ⊥ 1⊥ k2 gµ3µ1 −kµ3kµ1
k2 k2 3k k 3k 3k
1⊥ 2⊥
n o
kµ1ǫabka kb +(k ·k ) ǫµ1bkb
+ 2⊥ ⊥ 2⊥ 1⊥ 2 1 ⊥ ⊥ 2⊥ k2 gµ2µ3 −kµ2kµ3
k2 k2 3k k 3k 3k
1⊥ 2⊥
n o
gµ1µ2ǫabka kb +ǫµ1µ2(k ·k )
+ ⊥ ⊥ 1⊥ 2⊥ ⊥ 1 2 ⊥ k2 kµ3 −(k ·k ) kµ3
k2 k2 3k 2k 2 3 k 3k
1⊥ 2⊥ n o(cid:19)
+ permutations of (k ,µ ), (k ,µ ), and (k ,µ ) . (29)
1 1 2 2 3 3
#
It is clear from these expressions that in that region the LLL contribution dominates indeed. 4 This result is quite
noticeable. The point is that as was shown in Ref. [7], the region with momenta k2 ≫ |k2 | yields the dominant
i⊥ ik
contribution in the Schwinger-Dyson equation for the dynamical fermion mass in QED in a strong magnetic field.
Therefore the LLL approximationis reliable in that problem.
Nondecoupling of (heavy) HLL in the infrared region is quite unexpected phenomenon. What physics underlines
it? We believe that this phenomenon reflects the important role of a boundary dynamics at spatial infinity in this
problem. The pointis thatthe HLL arenotonly heavystates buttheir transversesizegrowswithoutlimit withtheir
gap m2+2|eB|nasn→∞. ThishappensbecausethetransversedynamicsintheLandauproblemisoscillator-like
one. Since in order to cancel the LLL anomaly one should consider the contribution of all higher Landau levels, it
p
implies that the role of the boundary dynamics at the transverse spatial infinity (corresponding to n→∞) is crucial
for the restoration of the gauge symmetry. In this respect this phenomenon is similar to that of edge states in the
quantum Hall effect: the edge states are created by the boundary dynamics and also restore the gauge invariance
[11]. Both these phenomena reflect the importance of a boundary dynamics in a strong magnetic field. It would be
interesting to examine whether similar nondecoupling phenomena take place in noncommutative theories arising in
string theories in magnetic backgrounds [4].
Acknowledgments
Discussions with A. Buchel, V. Gusynin, and I. Shovkovy are acknowledged. We are grateful for support from the
Natural Sciences and Engineering Research Council of Canada.
[1] W.Heisenbergand H.Euler,Z.f. Phys. 98, 714 (1936); V.F. Weisskopf, Kong.Dansk.Vid.Sels.Mat.-Fys. Medd.XIV,
No. 6 (1936).
[2] J. Schwinger, Phys. Rev. 82, 664 (1951).
[3] G. V.Dunne,hep-th/0406216.
[4] M. R.Douglas and N. A. Nekrasov,Rev.Mod. Phys. 73, 977 (2001); R.J. Szabo, Phys. Rep.378, 207 (2003).
[5] E. V.Gorbar and V. A.Miransky, Phys.Rev.D 70, 105007 (2004).
[6] S.Minwalla, M. VanRaamsdonk and N. Seiberg, JHEP 0002, 020 (2000).
[7] V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Phys. Rev. Lett. 83, 1291 (1999); Nucl. Phys. B563, 361 (1999);
Phys.Rev.D 67, 107703 (2003).
[8] A.Chodos,K.Everding,andD.A.Owen,Phys.Rev.D42,2881(1990);V.P.Gusynin,V.A.Miransky,andI.A.Shovkovy,
Nucl.Phys. B462, 249 (1996).
[9] G. Passarino and M. J. G. Veltman,Nucl. Phys. B160, 151 (1979).
4 Although the vertex Tµ1µ2µ3 is subdominant in this region, it is crucial for the restoration of the Ward identity. It is because while
vinerttheexWTµa1rµd2µid3enistimtyutlthHiepLrleiLe1daroentleyrmbysainsmthaellHloLnLgivtuerdtienxalTmHµ1oLµmL21eµn3tuwmhikch.are multiplied by a large transverse momentum k⊥, the LLL
LLL k
7
[10] I.S. Gradshtein and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press, New York,2000).
[11] X.G. Wen, Phys.Rev.Lett. 64, 2206 (1990).