{'fraction_non_alphanumeric': 0.054951522473128275, 'fraction_numerical': 0.02632964718272775, 'mean_word_length': 3.977489978415048, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 7, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 11, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
1-loop mass generation by a constant external magnetic field for an electron propagating in a thin medium

The 1-loop self-energy of a Dirac electron of mass m propagating in a thin medium simulating graphene in an external magnetic field B is investigated in Quantum Field Theory. Equivalence is shown with the so-called reduced QED_{3+1} on a 2-brane. Schwinger-like methods are used to calculate the self-mass \delta m_{LLL} of the electron when it lies in the lowest Landau level. Unlike in standard QED_{3+1}, it does not vanish at the limit m ->0 :\delta m_{LLL} ->(\alpha/2)\sqrt{pi/2}sqrt{\hbar|e|B/c^2}; all Landau levels of the virtual electron are taken into account and on mass-shell renormalization conditions are implemented. Restricting to the sole lowest Landau level of the virtual electron is explicitly shown to be inadequate. Resummations at higher orders lie beyond the scope of this work.


Introduction
Graphene is known as a quasi 2+1 dimensional medium with Dirac-like massless electrons (a gapless medium) -see for example [1]. Whether or not and in which circumstances a gap can open has important consequences, for example on electrical and optical properties [2]. In addition to purely theoretical motivations, this is one of the reasons why we study in this work the spontaneous emergence of a gap for a model which can naively mimic graphene in the presence of a constant external magnetic field B.
While it is generally admitted that the presence of B is likely to trigger chiral symmetry breaking (see for example [3]), the demonstrations usually rely on various approximations. In standard QED 3+1 , they are often based on the dimensional reduction that operates in the presence of a strong B [4] [5] and on resummations of a certain class of diagrams [6] [7] (which become suspicious after realizing that only double logs have been taken into account, leaving aside large single logs [8]). Also, various approximations to coupled Dyson-Schwinger equations are invoked, associated to the use of very special gauges to simplify the vertex (see [9]); this makes the demonstrations tedious, not very transparent and possibly controversial. In reduced QED 3+1 on a 2-brane, which is often considered to provide a fair description of graphene, other approximations are invoked, like the dominance of the lowest Landau level [10] while it was shown, for example in [7], that higher levels are important and trigger charge renormalization; moreover the language that is used is often confusing for people working in Quantum Field Theory.
The calculation of the 1-loop self-energy of an electron propagating in an external B that I present here uses the sole techniques of Quantum Field Theory. The external electron is chosen, for the sake of simplicity, to lie in the lowest Landau level (LLL), and, in this case, analytical (quasi-)exact formulae can be obtained by using the formalism of Schwinger [11] as it is carefully explained in [12].
I previously tackled the case of standard QED 3+1 in [8] by calculating the integral of Demeur [13] and Jancovici [14] beyond the leading ln |e|B m 2 2 approximation. I demonstrated that large logarithms had been overlooked and, then, neglected; they are tightly connected with the counterterms needed to implement suitable renormalization conditions. In this case, δm → 0 when m → 0.
These calculations are adapted here to a thin graphene-like medium. They are explained step by step such that they should appear fairly easy to reproduce, with no obscure gap to fill. They mostly go along the lines of [12], and differences are outlined. A massive Dirac electron is considered to propagate inside a thin film of thickness 2a, the Hamiltonian of which being deprived of its "p 3 γ 3 " term (see for example [2]). B, supposed to be static and uniform is considered to be directed along the z axis orthogonal to the medium strip. To make the calculation simpler and more transparent, no Fermi velocity different from the speed of light is introduced, such that I will be dealing with a special avatar of Quantum Electrodynamics, and extra degeneracies present in graphene [2] are eluded. The topic of symmetries will not be dwelt on either (see the review [3] on this subject).
As I will demonstrate by working in position space, this model yields for the electron self-energy the same expression as reduced QED 3+1 on a 2-brane [15] [16]: the effective photon propagator turns out, indeed, to be the one of standard QED 3+1 integrated over its k 3 momentum. For the internal electron propagator in presence of an external B I use Schwinger's [17] and Tsai's [18] expression, which accounts for all Landau levels, adapted to the particular situation and Hamiltonian under consideration. The calculations are (and should) be performed with a non-vanishing electron mass m before the limit m → 0 is taken. In the last part I only take into account the LLL of the internal electron, and show that neglecting higher levels is a bad approximation.
To avoid confusion, let me stress that all spinors and γ matrices that are considered in this work are 4-dimensional.
Any eventual connection with QED 2+1 , if any, can accordingly only be quite remote, and we shall not dwell on this any more.
So, though the result that I exhibit will certainly not be a surprise for many, I hope that the rigorous demonstration of a simple and exact formula that anyone can check with standard techniques will bring B-triggered mass generation from radiative corrections on a more solid ground. Like for QED 3+1 , renormalization conditions and the counterterms that must be introduced to fulfill them play important roles 5 .
A major challenge is also, there, to deal with a strongly coupled theory since a 1-loop result is certainly meaningless when the coupling constant gets of order 1. The necessary resummations look highly non-trivial since they do not only concern double and / or simple logs, but more complicated functions, and they have furthermore, of course, to be performed while satisfying at each order appropriate renormalization conditions. To my knowledge this last requirement has never been satisfied and tackling such formidable tasks lies largely beyond the scope of this work.
2 Propagation inside a thin, graphene-like medium; equivalence with reduced QED 3+1 on a 2-brane A general argumentation concerning reduced QED can be found, for example, in [15]. A more down-to-earth determination of the effective photon propagator is nevertheless instructive because it provides a simpler understanding of the mechanisms at work, and also because this approach can be applied to vacuum polarization [19], yielding less-trivial results.
Let us calculate in position space the electron propagator G(y, x) at 1-loop depicted in Fig. 1 (including external legs).
We call G 0 the tree-level electron propagator in the presence of B 6 (described by the double lines in Fig. 1) and ∆ µν the bare photon propagator.
in which the phase [18] Φ ensures the gauge invariance of the Green function (A µ is the vector potential).
To avoid confusion, the unit of electric charge we note |e| such that the electron charge is −|e|. In [11] and in [18], this unit of electric charge is instead noted e. In [12], e denotes instead the (negative) electron charge. We shall see that these precisions are important, in particular to get the appropriate propagator for the LLL of an electron.
Like in [11] and [18] we introduce q such that q|e| is the electron charge. Therefore q = −1 7 . This makes the covariant derivative D µ = ∂ µ − i|e|qA µ such that π µ = 1 i ∂ µ = p µ + |e|A µ . 5 In the work [19] I emphasize their role in the calculation of the photon vacuum polarization for the same graphene-like medium as the one considered here. 6 The results of this paragraph do not depend whether the external B is present or not. 7 This q should not be confused with the 4-momentum that appears in Fig. 1. The reader will easily make the difference.
For any 4-vector The 1-loop electron propagator depicted in Fig. 1 writes We now specialize to the medium under concern and consider "graphene-like" electrons propagating inside a thin film of thickness 2a. This situation has two consequences: * the vertices at which the electron and photon interact being located inside the strip, the integrals on their positions u 3 and v 3 along the z axis should be truncated to +a −a du 3 +a −a dv 3 . This gives The two integrations du 3 and dv 3 can be performed since +a −a dx e itx = 2 sin at t , which leads to iG(y, x) = −4e 2 e i|e|q x y dtµA µ (t) dp 3 2π In there the integration dq 3 can also be done explicitly since with σ = s 3 − r 3 , τ = p 3 − r 3 , which has the property to be independent of r 3 . We get now Going to the new variables h 3 = s 3 + p 3 , l 3 = s 3 − p 3 ⇒ dp 3 ds 3 = 1 2 dh 3 dl 3 yields The condition x 3 + y 3 ≤ 2a is verified because the electrons are constrained to propagate inside the strip, such that This yields The self-energy Σ is obtained from the 1-loop propagator above by chopping off the two external fermion iG 0 propagators, which leads to with the phase Φ given in (2) and to in which, to avoid conflicts between notations, we have made the change of variables p − q → k in the momenta, which amounts to labeling them like in [12].
No dependence on the thickness a of the medium occurs anymore (unlike for the vacuum polarization [19]). This is easily understood since we constrained the fermion to propagate inside the medium (while, for the vacuum polarization, the photon is allowed to also propagate in the "bulk").
3 The self-energy and self-mass of the electron In the whole paper, we use the metric (− + + +) like in [11], [18] and [12].
With these conventions, for an external magnetic field B along the z axis, the wave function of the lowest Landau level |LLL > is proportional to 8 The choice of a special gauge is of course not optimal but is justified by the property that the formalism of Schwinger is gauge invariant [17].

The self-energy in momentum space
We now proceed to calculating the self-energy expressed in (15), following the procedure given in [12]. To that purpose, we introduce 2 Schwinger parameters: s 2 for the photon and s 1 for the electron.
As far as the photon is concerned, instead of 1 (3.10) of [12]), that is used to represent the 4-dimensional photon propagator in the Feynman gauge, we shall use now, according to (14) and (15) However, it is important (see just above (18)) to use Tsai's [18] formulae and not the ones used in [12].
As for the electron, in general QED 3+1 , its propagator is given (see eq. (6) of [18]) by and, in position space by equations similar to (1) and (2). These expressions only need to be trivially adapted to the "truncated" momentap andk (see section 2).
As shown in appendix A, (17) leads to the adequate propagator for the LLL at the limit B → ∞. It is in particular proportional to the customary projector 1 − iγ 1 γ 2 , This is not the case of eq. (2.47b) of [12] (in there e < 0), which involves e iσ 3 z instead of e iqσ 3 z and leads to the wrong projector 1 + iγ 1 γ 2 and, later, to confusions and problems.
From (15) and using (16) and (17), one gets instead of (3.11) of [12] ("c.t." means "counter terms) Since the Hamiltonian of the Dirac electron is presently considered to be deprived of its One performs the same change of variable as (3.12) of [12] and one still introduces y = |e|Bsu.
The exponentials are then re-expressed in view of performing the d 3k integration. Following a procedure identical to that in [12] yields, instead of their (3.17) in which χ and ϕ are still given by (3.14), (3.15) of [12] The shifts in the integration variables are naturally k → k − up and k ⊥ → k ⊥ − − u tan y/y 1−u+u tan y/y p ⊥ .
One has to redo the k integrations (which only concerns the integral inside curly brackets in (20)) since it is now 4 for standard QED 3+1 . This is simple with the aid of the standard integral and leads to It is then simple matter to perform the Dirac algebra, which leads, instead of eq. (3.27) of [12], to Quite remarkably, in addition to the replacement σ 3 y → qσ 3 y in the exponentials, which originates from our taking the original Tsai's formula for G 0 instead of that of [12], and to a global factor 1/2, it only differs from (3.27) of [12] by du √ 1 − u instead of du and by the fact that, in the present situation, p 2 = −p 2 0 , p / = −γ 0 p 0 . We thus see that, after these lengthy but straightforward transformations have been done, the electron self-energy for QED 3+1 reduced on a 2-brane is formally very close to the one for QED 3+1 . The difference between the two integration measures for u is however at the origin of the completely different behaviors of the corresponding δm LLL at the limit m → 0, as we shall see in subsection 4.2.

Transforming the space representation of Σ
Unlike for the vacuum polarization in which the two opposite phases cancel, the phase Φ, given in (2), which occurs in the space representation (12) of the self-energy plays an important role. This makes the calculations all the more tedious as, like for QED 3+1 , the integrations on s and u for Σ(p) obtained in (24) cannot be done explicitly.
It is however possible, along the lines of p. 47-52 of [12] to get from the space representation Σ(x , x ) as written in (12) a useful expression for Σ(π) defined by Σ(π), which now depends on the covariant derivativeπ, has somewhat "swallowed" the phase Φ, and is the essential ingredient to get the self-mass δm of an electron on mass-shell (π / + m = 0).
The d 4 p in (12), which is at the root of the corresponding formal manipulations stays unchanged. One has to go through the steps of p. 34-36 and p. 47-50 of [12], which use in particular eq. (2.41) of [12] < tan q|e|Bs q|e|Bs (26) and its avatars, (2.45) and more specially (2.46) They entail, by simple changes of variables (ϕ is given in (21)) in which ∆(u, y) and the angle β have been introduced, which satisfy [12] sin After all terms inside (24) have been transformed via (28), one gets the result which differs from (3.38a) of [12] by the absence of γ 3 π 3 fromπ /, the same factor 1 2 that we already mentioned concerning (24), and the presence of q ≡ −1 in the exponentials (that was omitted in [12]).

Renormalization conditions and counterterms
The electron mass we define as the pole of its propagator, which is the only gauge invariant definition.
We briefly recall here the general procedure to fix the counterterms. It is then straightforwardly adapted to our concern by replacing everywhere p withp and π withπ (π µ = p µ + |e|A µ ).
At B = 0, the renormalized electron mass is defined by in which m 0 is the bare mass and Σ(p) the bare self-energy.
In the presence of and external field A µ , the propagator of a Dirac electron is and we define, in analogy with (31) the mass of the electron as the pole of its propagator by δm depends on the external B.
in which the superscript "ren" denotes the renormalized quantities.
They lead to the same counterterms as in [12] but for the simple modifications p →p, π →π, and one gets in which y, Θ, ∆ are given in (30) and (29).
The 2nd counterterm vanishes on mass-shell (since it must satisfy the 1st renormalization condition), and can therefore be forgotten in the calculation of δm.

The self-mass δm LLL for an electron in the lowest Landau level
The spectrum of a Dirac electron in a pure magnetic field directed along z is [22] 2 in which σ z = ±1 is 2 × the spin projection of the electron on the z axis. So, at n = 0, σ z = −1, p z = 0, n = m: this is the lowest Landau level.

We can consider
, N [21] = L y L z dimensions along y and z .
Gathering all information and simplifications leads finally to in which y = |e|Bsu as before, ∆(u, y) is the same as in (30), β the same as in (29), and Θ has shrunk down to 4 The "reduced" Demeur-Jancovici integralÎ(L)

General expression
We defineÎ(L) by such thatÎ By a successive change of variables, we cast it in a form similar to I(L) deduced by Jancovici in [14] from the formula obtained by Demeur in [13], and that was revisited in [8]. The calculations, which are detailed in appendix B, lead tô which is the expression which we shall focus on hereafter.
Calling (42) can be cast in the formÎ ThatÎ(L) would be divergent at z = 0 without the counterterm can be easily seen by expanding

Analytical evaluation ofÎ(L)
We split   The result is that, for L ≥ 20 and z 0 1 one can approximate at a precision better than 1/100 One has therefore On Fig. 2 we compare g(z) given in (46) (blue) with the one obtained in [8] for standard QED 3+1 g(z) = ln(z − 1 + z(z − 2))/ z(z − 2) ln z/z + π/2z 1.175 (yellow). We now proceed like M.I. Vysotsky in [23] and look for an interpolating function for g(z). One has g(1) ≈ 3.468, and an excellent fit for z ∈ [z 0 1, ∞] is It is plotted in yellow on Fig. 3, while the exact g is in blue. This approximation gives (using (40)) 10 The constant term comes from the contribution toÎ(L) of it is not sensitive to the precise value of z 0 = 1, but it is controlled by the leading behavior of g(z) ∼ 1/ √ z at z → ∞ 11 .
It is important to check that, at the limit of large L, the first integral z0=1 0 dz(. . .) is stable and can still be neglected with respect to the second integral. This is shown on Fig. 4-left, in which we plot the 1st integral as a function of L.
As already mentioned, the numerical cancellation of infinities requires that the lower bound of integration be set not to 0 but to smaller and smaller . The curve in blue corresponds to = 10 −3 , and the 3 other curves, green, yellow and red, corresponding to = 10 −6 , 10 −9 , 10 −12 are superposed;Î(L) as given by (50) is plotted on Fig. 4-right. We see that, even at very large values of L, the 1st integral can always be safely neglected insideÎ(L).
which shows that, in an external magnetic field, this model, equivalent to reduced QED 3+1 on a 2-brane, cannot stay massless at 1-loop. Notice that (52) fulfills the renormalization conditions (34), which are expressed at B = 0.
Since the role of the counterterms is slightly more subtle than for QED 3+1 (in which they yield the large logs (see [8])), it is useful to make some comments about them.
In (38), the (infinite) counterterm only depends on m through the exponential e −isu 2 m 2 inside the integrand.  (101), the counterterm has reacted on the bare contribution and the two become entangled (we introduced ±ζ to picture the fact that this dependence globally cancels but, in practice, one cannot "isolate" ζ ; also, strictly speaking, these terms are not defined before the infinities are regularized and canceled).
The "educated" splitting [14] of L. is due to h z0 1 (and the corresponding g(z) defined in (46) (47) (49)) which has an asymptotic expansion 1/ √ z at z → ∞. This makes the result insensitive to the precise value of z 0 . By contrast, as we have mentioned, in standard QED 3+1 , the asymptotic behavior of g(z) is ln z/z [8].
h z0 1 no longer represents the bare contribution for the reasons that we just evoked: * a change of variables introduced an extra dependence on L that mixes with the counterterm; * the splitting of the z interval of integration collects in the neglected (small) z0 0 dz(. . .), in particular, the two canceling infinite parts of the bare term and of the counterterm, establishing a second connection between the two. In this respect, both play crucial roles in the massless limit of δ LLL , that can hardly be disentangled.
Last, let us remark that it is necessary to make the z integration at m = 0 before taking the limit m → 0, otherwise, since L = |e|B/m 2 , one gets the undetermined expression 1 0 × 0. Had we started from a massless theory, we would have obtained such an undetermined result. This is why one can only state that the massless limit of the 1-loop δm LLL goes to a constant, or, equivalently, that the model under consideration cannot stay massless at 1-loop.
6 Restricting to the lowest Landau level of the virtual electron 6
Using the conventions and metric (− + ++) of Schwinger, the contribution of the LLL is in which we have introduced the Schwinger's parameter s 1 (see also appendix A).
To determine the contribution of the LLL of the virtual electron to the self-energy, we have to calculate (see (15)) One introduces as before (see (16)) the Schwinger parameter s 2 for the photon propagator and, instead of eq. (3.11) of [12], one gets We use again the change of variables (19) together with z = |e|Bs 1 , y = |e|Bus.
Like before, aiming at performing the integration d 3k , one rewrites the exponentials (watch the "i" which now occurs). Since s cannot be factorized everywhere, we have now included it into the definitions of χ 0 and ϕ 0 , unlike previously for χ and ϕ.
such that One then shifts the variables 1+i|e|Bs(1−u) . One has χ 0 = sr 2 + s(1 − u) + 1 i|e|B r 2 ⊥ . Then, (m − γ 0 (k 0 − p 0 )) = m − γ 0 (r 0 + (u − 1)p 0 ). χ 0 being even since it depends on r 2 0 , the odd term ∝ r 0 yields a vanishing contribution to the dk 0 . One can thus replace m − γ 0 (k 0 − p 0 ) by m − (u − 1)γ 0 p 0 . One gets With the help of (22) one gets and, since Next, one performs the Dirac algebra such that in which b 0 and b ⊥ are given in (58) and where we have now mentioned the counterterms (c.t.) that need eventually to be introduced to fulfill suitable renormalization conditions.
We are interested in δm 0 LLL concerning external electrons in the LLL. To get it we sandwich Σ(π) between two LLL eigenstates. Since these are annihilated by 1 + iγ 1 γ 2 , the only term that may play a role is the one proportional to m.

Accordingly, the quantity of interest to us is
+c.t. (65)

Transforming the space representation
One needs to determine Σ(π) satisfying (25). To that purpose, one must find the suitable change of variables to adapt (2.45) (2.46) of [12] to the present situation, that is to determine a 0 and a ⊥ in (27 (which is the same as (2.46) of [12]).
One must have Then One also has trivially This gives

Renormalization conditions and counterterms
Let us consider general on mass-shell external electrons. Since renormalization conditions have to be expressed at B = 0, let us also consider the limit B → 0 of Σ n=0 (π).
We then go through the successive changes of variables (u, s) → (u, y = |e|Bsu), t = iy, last z = ut, plus a Wick rotation (see subsection 6.4 below), to get If we now go on mass-shell, π / + m = 0, The 1st renormalization condition in (34) concerns the vanishing, on mass-shell, of Σ at the limit B → 0. We have therefore to introduce a 1st counterterm c.t. 1 (the terms ∝ (1 + iγ 1 γ 2 ) give vanishing contribution only to external LLL).
The second renormalization condition (see (34)) concerns the derivative of Σ. This leads to introducing a second set of counterterms. However, they have to vanish on mass-shell since they must satisfy the 1st renormalization condition.
Since, in order to calculate δm, we precisely work on mass-shell, we can forget about the second set of counterterms and proceed now with the calculation of δm 0 LLL .

Calculation of the 1-loop self-mass δm 0 LLL when both external and internal electrons are in the lowest Landau level
When acting on external LLL electrons, and on mass-shell, one has π 2 0 = m 2 , π 2 ⊥ = σ 3 eB = −eB = +|e|B. From (69) and (73) one then obtains After some calculations which are detailed in appendix C, one gets which diverges at z → ∞. Like before, one must eventually take the limit m → 0 after the integration has been performed.
The exponential e −2 tanh −1 z(1−u) u 2 +z(1−u) being bounded by 1 and going to 0 when z → ∞, we have to evaluate in which we have, like previously, factorized αm 4π , at the price of introducing an extra factor 2 in front of the integral. One accordingly defines now (compare with (42) (45)) Note that, unlike when taking all Landau levels of the internal electrons into account, the integralÎ 0 (L) is convergent at z = 0 without introducing any counterterm.
One has (to be compared with (46)) such that 6.4.1 Contribution of the counterterm to δm 0 LLL For external LLL, π 2 ⊥ → |e|B, σ 3 → −1, this counterterm contributes to δm 0 LLL by which is convergent. It yields in which g 0 is the same as that defined in (78). At the limit z → 0 such that which we shall truncate at the first term since the limit m → 0 should be taken afterwards. Accordingly, one finds a vanishing counterterm (which is in particular independent of the external B) Collecting (79) and (84) yields Notice that the bare δm 0 LLL (and, of course, the (vanishing) counterterm) are both finite, unlike when all Landau levels of the internal electron are accounted for.

The limit of δm 0
LLL when m → 0 In addition to the limit z → 0 given in (82) one has So, splitting the z interval of integration of (85) into 3 sub-intervals gives The bounds a and b are chosen such that, for z ∈ [0, a] the expansion (82) is valid, and for z ∈ [b, ∞] the expansion (86) is valid. Since one has To study the limit m → 0 we use which shows that it is the value at z = ∞ that controls δm 0 LLL . Finally 6.6 An approximate analytical expression for δm 0 LLL . Comparison with δm LLL It is easy to get a fair approximate analytical expression for δm 0 LLL given in (85) by using the following simple fit to g 0 (z) which has, in particular, the appropriate limits at z → 0 and z → ∞. On fig. 5 the exact g 0 is plotted in blue and the approximate one in yellow. This yields which has the limit (91) when m → 0. Notice also that the second contribution yields a finite δm 0 LLL → α 4π π 3/2 |e|B when m → ∞.
On Fig. 6, we plot m 2 |e|B ∞ 0 dz e −zm 2 /|e|B g 0 (z) in blue together with m 2 |e|B ∞ 0 dz e −zm 2 /|e|B g app 0 (z) in yellow, which corresponds to 4πδm 0 LLL /α |e|B. It shows that this rather crude approximation is good at better than 7% for m 2 |e|B ≥ .4, at ∼ 10% for lower values of m 2 |e|B and that it has, of course, the appropriate limit 2π 3/2 ≈ 11.14 at   While it is true that G n=0 (p, B) can indeed be obtained by formally taking the limit B → ∞ of G 0 (p, B) (see Appendix A), one should notice that: * this limit cannot be applied to the phase Φ; * the factor e −k 2 ⊥ /|e|B is not replaced by 1 inside G n=0 despite B → ∞; this is because, as the Larmor radius shrinks to 0 at this limit, k ⊥ can extend to ∞; * the (vanishing) counterterm is determined by taking first the limit B → 0, so as to fulfill renormalization conditions; then, eventually, the non-vanishing limit m → 0 is taken; therefore, naively taking the limit B → ∞ to "select" the LLL cannot be applied either to the counterterm.
Arguing that the limit m → 0 is equivalent to B → ∞ can accordingly only be wrong 13 .
The limits at m → ∞ (which should not be confused with those at B → 0) are also very different since δm LLL ∼ |e|B m e −m 2 /|e|B → 0 while δm 0 LLL ∼ cst × |e|B (see Fig. 7). Large cancellations therefore occur among multiple Landau levels of the virtual electron. However, they can only be estimated after going through the filter of renormalization, and infinities that need being tamed only arise when one accounts for all levels.

Conclusion and prospects
Unlike what happens for QED 3+1 , the massless limit of the 1-loop δm LLL in external B for QED 3+1 reduced on a 2-brane does not vanish. We have shown furthermore that it corresponds to an electron propagating inside a graphenelike medium. The latter cannot therefore stay "gapless" at 1-loop in the presence of a magnetic field. This result has been obtained with special attention paid to the renormalization conditions.
The result is very simple because we have restricted the external electron to lie in the lowest Landau level. For higher levels, the situation is much more intricate and analytical formulae certainly cannot be obtained.
We have also shown that restricting to the LLL of the internal electron largely overestimates the self-mass; in particular, its value when m → 0 triggers a multiplicative factor √ 2. Despite the case under concern has the peculiarity that taming infinities and renormalizing is only needed when accounting for all Landau levels, studies based on such an approximation appear rather suspicious. Note that, in the case of standard QED 3+1 , it was shown in [19] that accounting for the sole leading (ln) 2 terms largely increases the result, too.
I cannot pretend to have dealt with real graphene, in which, in particular, the smallness of the Fermi velocity with respect to the speed of light makes the theory strongly coupled. There, techniques have to be mastered which go beyond perturbative expansions, while respecting appropriate renormalization conditions.
It is also well known that the photon propagator gets modified in the presence of an external B (see for example [27]).
This modification has been included in calculations of the electron self-energy [26] [7] with the result that double logs are turned into single logs. However, the large single logs closely associated with counterterms (see [8]) were not taken into account. Furthermore, this modification of the photon propagator and the eventual screening of the Coulomb potential is obtained by resumming the infinite geometric series of 1-loop vacuum polarizations (see for example [27]); in contrast, Quantum Field Theory stipulates that renormalization conditions and the addition of the corresponding counterterms should be achieved consistently order by order in powers of the coupling constant or in the number of loops. In this framework, screening the Coulomb potential inside the electron self-energy at finite order raises many issues, both technical and conceptual.
Acknowledgments: It is a great pleasure to thank M.I. Vysotsky for his invaluable assistance and advice. 13 Eventually forcing the identity between the two limits at m = 0 of δm LLL and δm 0 LLL as a kind of renormalization condition must be rejected. then to notice that ∆(u, y) = (1 − u + u sin y y e +iqy )(1 − u + u sin y y e −iqy ) to simplify the previous expression intô The last operation to perform is a Wick rotation.
+i∞ 0 + 1/4 inf inite circle + 0 ∞ = 2iπ residues. Because of e −z m 2 |e|B the contribution on the infinite 1/4 circle is vanishing. That the residue at z = 0 vanishes is trivial as long as u is not strictly vanishing. The expansion of the terms between square brackets in (105) at z → 0 writes indeed u − 1 + (− 5 3 + 4 3u + u)z + − 7 3 − 1 u 2 + 7 3u + u z 2 + O(z 3 ), which seemingly displays poles at u = 0. However, without expanding, it also writes, then, 2 2z − 1 z = 0, which shows that the poles at u = 0 in the expansion at z → 0 are fake and that the residue at z = 0 always vanishes. Other poles (we now consider eq. (104)) can only occur when the denominator of the first term inside brackets vanishes. That the corresponding u pole = 2t 2t+e −2t −1 should be real constrains them to occur at t → inπ, n ∈ N > 0 and u → 1. In general, they satisfy 2t(1 − u) + u(1 − e −2t ) = 0 which, setting t = t 1 + it 2 , t 1 , t 2 ∈ R, yields the 2 equations e −2t1 cos 2t 2 = 1 + 2ηt 1 , e −2t1 sin 2t 2 = −2ηt 2 , η = 1−u u ≥ 0. Since t 1 → 0, one may expand the first relation at this limit, which yields cos 2t 2 − 1 = 2t 1 (η + cos 2t 2 ). As t 2 → nπ, cos 2t 2 > 0 and cos 2t 2 − 1 < 0, which, since η > 0, constrains t 1 to stay negative 14 . Therefore, the potentially troublesome poles lie in reality on the left of the imaginary t axis along which the integration is done and should not be accounted for when doing a Wick rotation. After changing u into v to work from now onwards with the same notation as in [14] and ease the comparison, one gets (42). 14 The 2nd relation then tells us that sin 2t 2 < 0, which means that the poles correspond to t 2 = nπ − , > 0.

C Demonstration of (75)
In (74), we go, like before (see (101)), to the variables u, y = |e|Bsu such that du ds = du dy |e|Bu and get Last, as before, we go to z = ut ⇒ du dt = du dz u .
As long as m = 0, the e −zm 2 /|e|B and the e − tanh −1 z(1−u) u 2 +z(1−u) ensure the convergence on the infinite 1/4 circle such that, supposing that no pole in the 1/4 quadrant causes problems, one may do a Wick rotation, which yields (75).