Anatomy of the Magnetic Catalysis by Renormalization-Group Method

Physics Letters B 775 (2017) 283–289

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Anatomy of the magnetic catalysis by renormalization-group method ∗ Koichi Hattori a, Kazunori Itakura b,c, Sho Ozaki d,e, a Physics Department and Center for Particle Physics and Field Theory, Fudan University, Shanghai 200433, China b KEK Theory Center, Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization, 1-1 Oho, Ibaraki, 305-0801, Japan c Graduate University for Advanced Studies (SOKENDAI), 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan d Department of Physics, Keio University, Kanagawa 223-8522, Japan e Research and Education Center for Natural Science, Keio University, Kanagawa 223-8521, Japan a r t i c l e i n f o a b s t r a c t

Article history: We ﬁrst examine the scaling argument for a renormalization-group (RG) analysis applied to a system Received 21 June 2017 subject to the dimensional reduction in strong magnetic ﬁelds, and discuss the fact that a four-Fermi Received in revised form 3 November 2017 operator of the low-energy excitations is marginal irrespective of the strength of the coupling constant in Accepted 3 November 2017 underlying theories. We then construct a scale-dependent effective four-Fermi interaction as a result of Available online 9 November 2017 screened photon exchanges at weak coupling, and establish the RG method appropriately including the Editor: J.-P. Blaizot screening effect, in which the RG evolution from ultraviolet to infrared scales is separated into two stages by the screening-mass scale. Based on a precise agreement between the dynamical mass gaps obtained from the solutions of the RG and Schwinger–Dyson equations, we discuss an equivalence between these two approaches. Focusing on QED and Nambu–Jona-Lasinio model, we clarify how the properties of the interactions manifest themselves in the mass gap, and point out an importance of respecting the intrinsic energy-scale dependences in underlying theories for the determination of the mass gap. These studies are expected to be useful for a diagnosis of the magnetic catalysis in QCD. © 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

√ 1. Introduction π mdyn = eBexp − , (2) ρLLLGNJL Strong magnetic fields confine charged fermions in the low- where ρLLL and GNJL are the density of states in the LLL and a est Landau levels (LLLs), and they enjoy the properties of the dimensionful coupling constant of the four-Fermi interaction, re- + (1 1)-dimensional chiral fermions with the dispersion relation spectively. Their core observation is seen in the similarity between [ = ] eB (0, 0, eB), eB > 0 : the mass gap and the energy gap of superconductivity which is R/L given by ∼ ωD exp[−c /(ρFG )] with ωD and ρF being the De- =±pz . (1) LLL bye frequency and the density of states near the Fermi surface, Intuitively, this is a consequence of the formation of the small respectively. Also, G and c are a coupling constant and a posi- 1/2 cyclotron orbit with the radius ∼ 1/|eB| and the residual free tive number, respectively. In fact, this similarity is originated from motion along the field. It turned out that this dimensional reduc- the dimensional reduction in the low-energy domains of the both tion gives rise to rich physics phenomena. Especially, the magnetic theories, i.e., in the LLL and in the vicinity of the Fermi surface. catalysis of the chiral symmetry breaking and the chiral magnetic ef- We can clearly see the consequence of the dimensional reduc- fect have been addressed by many authors (see, e.g., Refs. [1,2] and tion by focusing on QED in the weak coupling regime. From the Refs. [3–6] for reviews). rainbow approximation of the Schwinger–Dyson (SD) equation, the The clear statement on the physical mechanism of the magnetic mass gap was obtained as catalysis was due to Gusynin, Miransky, and Shovkovy in terms of √ a simple four-Fermi interaction [7]. By solving the gap equation of π π mdyn eB exp − , (3) the NJL model, they found a mass gap 2 α with an unscreened photon propagator in the early studies [8–10], and also * Corresponding author. √ E-mail addresses: [email protected] (K. Hattori), 1/3 π mdyn 2eBα exp − , (4) [email protected] (K. Itakura), [email protected] (S. Ozaki). α log(Cπ/α) https://doi.org/10.1016/j.physletb.2017.11.004 0370-2693/© 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 284 K. Hattori et al. / Physics Letters B 775 (2017) 283–289 with a screened photon propagator [11,12]. Here, α = e2/4π and thus far, and we are not aware of the RG analysis in the literature C is a certain constant of order one. The constant was analyti- of which the result agrees with that from the SD equation (4). cally obtained as C = 1 when the momentum dependence of mdyn As we will discuss later in more detail, the screening effect is neglected. The authors of Refs. [11,12] observed that the gap should be appropriately incorporated in the derivation of the RG equation always has a nontrivial solution irrespective of the size of the equation, since the screening mass sets an intrinsic energy scale coupling constant, indicating that the strong magnetic fields cause of the underlying theory in between the ultraviolet and infrared the dynamical symmetry breaking without support of any other regimes. The essential technique was recently developed for the nonperturbative dynamics. This reminds us of the well-known fact analysis of the RG flow in “magnetically induced QCD Kondo that any weak attractive interaction causes superconductivity. effect” [29]. In the present Letter, we will show that the same tech- Our main assertion in this Letter is that all these aspects of the nique successfully works for the analysis of the magnetic catalysis magnetic catalysis can be understood with the Wilsonian renor- at weak coupling. malization group (RG) analysis. We will show that the emergence The structure of this Letter is the following. We first show the of the dynamical mass gap is informed from the RG flow for the connection between the magnetic catalysis and the dimensional effective four-Fermi operator that goes into the Landau pole. Bear- reduction which can be understood from a simple discussion of ing it in mind that four-Fermi operators are irrelevant in ordinary the scaling dimensions. Next, we construct an effective four-Fermi (3 + 1)-dimensional systems, we will clearly see from the RG point interaction from the underlying weak-coupling theory, i.e., QED, of view that the magnetic catalysis of the dynamical symmetry and appropriately include the energy-scale dependence of the tree- breaking is intimately related to the dimensional reduction. Our level interaction. Based on these discussions, we derive the RG approach shares the philosophy with the analysis of (color) super- equations and obtain the dynamical mass gap from their solutions. conductivity by the RG method [13–20]. We confirm that the energy-scale dependence of the interaction Our ultimate goal is to consistently understand the enhance- is necessary for obtaining the correct form of the gap, which was ment of the chiral symmetry breaking at zero or low temperature, however missing in the previous analyses. Finally, we discuss the and the inverse magnetic catalysis near the chiral phase transition correspondences between the RG and SD analyses, and the crucial temperature in QCD. There has been a discrepancy between the roles of the photon/gluon propagators in the magnetic catalysis. estimates of the chiral condensate from the lattice QCD simula- The derivation of the RG equation is briefly summarized in an ap- tion and typical model calculations [21]. For the magnetic catalysis pendix. and the inverse catalysis to be compatible with each other, it ap- peared to be important to explain a mechanism which makes the 2. Infrared scaling dimensions dynamical mass gap stay as small as the QCD scale QCD even in a We begin with looking into an analogy between the systems in strong magnetic field eB 2 [22] (see also Refs. [23–25]). The QCD the strong magnetic field and at high density. In the presence of a method of renormalization group is a potentially useful tool to ob- large Fermi sphere, the low-energy excitations near the Fermi sur- tain a clear insight on this issue on the basis of an argument of face show the dimensional reduction: The two-dimensional phase the hierarchy which we will elaborate in the present paper. space tangential to the large Fermi sphere is degenerated, and the However, to the best of our knowledge, even the correct form energy dispersion depends only on the momentum normal to the of the mass gap in weak-coupling gauge theories has not been ob- sphere. Then, the dimensional reduction enhances the infrared (IR) tained by the RG analyses in the presence of the screening effect. dynamics, leading to the instabilities near the Fermi surface. Based Therefore, before discussing the strong-coupling regime in QCD, on the analogy with this mechanism, Gusynin et al. clearly pointed one should understand how the screening effects are reflected in out that the chiral symmetry breaking occurs in the strong mag- the parametric form of the mass gap in a clear way. Moreover, it netic field no matter how weak the coupling is [7]. is a generic issue to establish a systematic way of including the One can see possible emergence of the IR instability from a screening effects in the RG analyses, which will be important in a simple argument of the scaling dimensions. The kinetic term for variety of systems. Note, for example, that there was an issue of the LLL reads the color magnetic screening in the RG analysis on the color superconductivity [17]. kin = ¯ 0 − 3 SLLL dt dpzψLLL(pz)(i∂t γ pzγ )ψLLL(pz), (5) We will show that all of the results in Eqs. (2), (3), and (4) from the SD equations are precisely obtained from the solutions of where we have suppressed the label specifying the location of the RG equations.√ Furthermore, we will clarify the origins of the the cyclotron center on the transverse plane. From this kinetic overall factor of eB and the exponents in the language of the RG term, one can find the IR scaling dimension of the LLL fermion method. We will find that the properties of the interactions in the field when the excitation energy goes down toward zero as −1 model/theory are directly reflected in the parametric dependences LLL → sLLL (t → s t) with s < 1. Since the LLL fermion has of the dynamical mass on the coupling constant and the magni- the (1 + 1)-dimensional dispersion relation (1), the longitudinal tude of eB. Ultimately, these studies will be useful for a diagnosis momentum pz also scales as pz → spz. On the other hand, the of the magnetic catalysis in QCD. We will come back to this point transverse momentum does not scale, because it serves just as with a brief comment on the perspective in the last section. the label of the degenerated states and does not appear in the More specifically, we will closely look into the screening effect dispersion relation (1). Therefore, when the kinetic term (5) is in- on the photon propagator. It would be instructive to mention a variant under the scale transformation, the LLL fermion field scales − successful application of the RG method to color superconductiv- as s 1/2 in the low-energy dynamics. ity in dense quark matter, where an appropriate treatment of the Bearing this in mind, we proceed to the effective four-Fermi dynamical screening effect on the magnetic gluons was important operator in the LLL: for obtaining the correct magnitude of the gap [17,20]. We should int = (i) (1) + (2) − (3) − (4) also mention that the RG analysis of the magnetic catalysis at weak SLLL dt dpz G δ(pz pz pz pz ) coupling was performed in Refs. [26,27]. Also, the magnetic catal- = i 1,2,3,4 ysis in QCD was investigated on the basis of both the SD and RG × ¯ (2) μ (4) ¯ (3) (1) equations in Ref. [28]. However, roles of the screening effect arising ψLLL(pz )γ ψLLL(pz ) ψLLL(pz )γ μψLLL(pz ) , from the quark loop in the magnetic field have not been identified (6) K. Hattori et al. / Physics Letters B 775 (2017) 283–289 285

μ 0 3 where G is an effective coupling constant and γ = (γ , 0, 0, γ ). The transverse momenta are again not written explicitly. Note that − the delta function has the scaling dimension s 1 due to the dimensional reduction in Eq. (1). Thus, we ﬁnd that the four-Fermi interaction term has the scaling dimension s0, meaning that the four-Fermi operator is marginal in the strong magnetic ﬁeld [26, 27]. This result suggests that the chiral symmetry could be broken even in weak-coupling theories like QED, which is just like the well-known fact of superconductivity that any weak attraction induces the BCS instability. Essentially, the magnetic catalysis occurs only for the dimensional reason. Below, we will explicitly see the indication of the magnetic catalysis by means of the RG approach. The logarithmic quantum correction will drive the effective cou- Fig. 1. Energy scales appearing in the RG evolution. The relevant region from the UV cutoff down to the Landau pole is divided into two regions by the scale of the pling constant G into the Landau pole. screening mass.

3. Effective interactions from underlying theories

To build a bridge between the underlying gauge theories and the scaling-dimension argument for the four-Fermi operator, we first construct an effective four-Fermi interaction based on the underlying theory. It is very crucial to correctly take into account the energy-scale dependence of the interactions in gauge theo- Fig. 2. Scattering diagrams contributing to the RG flow in the magnetic catalysis. ries which are mediated by gauge bosons. Elaborating this point, Yellow blobs capture the running effective coupling constants which are defined in we will obtain the correct form of the dynamical mass, and will terms of the exchanged gauge-boson propagators in the weak-coupling theory. show that the reliable effective theory gives a different RG evolution from that in the scale-independent four-Fermi interaction Importantly, we have an intrinsic energy scale given by the model which do not respect the scale dependences in the underly- screening mass sc ≡ mγ which cuts off the IR region of the ing theory. transverse momentum integral. Therefore, the result of the in- In QED as the underlying theory, we shall introduce the pho- tegral depends on if the scale of interest is larger than ton propagator in the strong magnetic field that has a screening sc [Region (I): > ] or is in the deeper IR region < [Re- mass [11,12,30–33]. The explicit form in a non-covariant gauge sc sc gion (II): < ]. As summarized in Fig. 1, one needs to examine reads [11,12] sc the RG evolution in these regions separately. On the other hand, μν g the upper√ boundary of the integral is, in the both cases, given by iDμν(q) = (7) UV ≡ 2eB which appears in the exponential factor of Eq. (8) q2 − m2 γ and also corresponds to the energy scale where the higher Landau μν μ ν + μ ν + μ ν levels start to contribute. Based on this scale-dependent four-Fermi g⊥ q⊥q⊥ q⊥q q q⊥ + − , interaction, we investigate the RG evolution in the next section. q2 (q2)2 μν μν where g = diag(1, 0, 0, −1), g⊥ = diag(0, −1, −1, 0), and 4. RG analysis at weak coupling μ = μν 2 = q ,⊥ g ,⊥qν . The screening mass mγ 2αeB/π comes from the one-loop photon self-energy composed of the LLL fermion lines in We derive the RG equation for the coupling constant of the the vanishing frequency limit.1 The other terms are coupled to nei- effective four-Fermi interaction in terms of the Wilsonian renor- ther the LLL fermion loop nor the scattering LLL fermions because malization group. More specifically, we compute the scattering am- μ = ¯ μ the LLL fermion current, jLLL ψLLLγ ψLLL, is longitudinal to the plitudes for the fermion and antifermion pair that forms the chiral magnetic field. Thus, those terms are completely irrelevant in the condensate (see Fig. 2), and integrate out the excited states. As present discussion. we have already learned that the four-Fermi operator is marginal Now, we note that the dispersion relation of the LLL fermion is in the LLL, we anticipate that the loop integral in the scattering given by the (1 + 1)-dimensional form (1), while the photons live amplitude generates a logarithm, and renormalizing the effective in the ordinary four dimensions. Therefore, we define the effective coupling constant with this logarithm will drive the system toward coupling constant G by integrating out the transverse momentum a strong coupling regime. in the photon propagator as [22,25,29] As shown in Fig. 2, the leading-order (tree-level) scattering am- 2 plitude is given by the one-photon exchange diagram: 2 − 2 q⊥ 2 d q⊥ ( ie) − G(q ) ≡ e 2eB , (8) (2π)2 2 − 2 − 2 2 q q⊥ mγ M0 = G(q ). (9) where the exponential factor comes from the transverse part of The relevant scattering channels for the formation of the chi- the fermion wave functions. This effective coupling allows us to ral condensate are those between the pairs carrying the oppo- identify the dimensionally reduced effective interaction discussed site chiralities, so that the two relevant spinor structures are in Eq. (6) which however possesses an appropriate energy depen- ¯ (3) (1) (4) μ ¯ (2) given by [ uR/L(p )γ μuR/L(p ) ] [ vL/R(p )γ vL/R(p ) ] with dence as we will see below. the spinors of the fermion u and the antifermion v in the LLL. The scattering amplitudes are the same for the both channels, and 1 While we discuss the one-flavor case, the extension to the multi-flavor cases below these trivial spinor structures will be suppressed for nota- just results in the overall summation of the charge in the screening mass. tional simplicity. 286 K. Hattori et al. / Physics Letters B 775 (2017) 283–289

The same result is obtained for the both scattering channels, so that we have again suppressed the trivial spinor structures which are common to the tree-level amplitude. The remaining one-dimensional integral has the anticipated logarithmic IR divergence. When the scale goes down from to −δ, the increment of the one-loop amplitude is found to be Fig. 3. Diagram for the scattering between the particle and antiparticle. G2() M ( − δ) − M () = log . (16) 1 1 − 2 π δ Since the coupling constant G(q ) has the scale dependence discussed in the previous section, the tree-level amplitude also The logarithm is absorbed by the renormalization of the effective contributes to the RG evolution. This situation is somewhat, though coupling constant G, which leads to the RG evolution. not exactly, similar to that in the color superconductivity in dense From the scattering amplitudes in Eqs. (12) and (16), the RG quark matter where the scale dependence of the dynamical screen- equation is obtained as ing mass in the tree-level diagram modifies the exponent in the d 1 2 critical temperature [17]. G() =−2α − G () Region I , (17a) d π In the present case, when the energy and momentum scales d 1 of the fermion are of the order of , the momentum transfer G() =− G2() Region II . (17b) 2 2 is − q 0, where we are only interested in the space- d π like region contributing to the fermion scatterings. Now, when The first term in Eq. (17a) comes from the tree-level amplitude ac- integrating out the fermionic degrees of freedom in the shell cording to the intrinsic scale dependence of the coupling constant ( − δ) ∼ , we obtain the increment of M0() as shown in Eq. (12). This term is absent in Eq. (17b), because the soft momentum transfer is cut off by the screening mass. The clear 2 2 M0( − δ) − M0() = G(−( − δ) ) − G(− ). (10) distinction between these two regions will result in an important consequence, as first discussed in the analysis of magnetically in- In Region II, the dependence on goes away, because the screen- duced QCD Kondo effect [29]. The other terms in these equations, ing scale dominates over . However, in Region I, the screen- sc which come from the logarithmic contribution in Eq. (16), have ing mass is negligible, so that there is the dependence on . Per- minus signs, so that these terms drive the RG flow toward the forming the transverse-momentum integral in Eq. (8), we have Landau pole when the interaction is attraction (G > 0) as explic- 2 itly seen below. 2 dq⊥ When the scale is reduced from 0 to√ ( >sc) where G(−( − δ)2) − G(−2) ∼ α . (11) 2 the initial scale is of the order of eB, we use the RG q⊥ 0 (−δ)2 equation (17a) in Region I. With the initial coupling G(0) = 2 α log(2eB/0) 1from Eq. (8), we find the solution as Therefore, the result at the tree level is summarized as √ α 2 M ( − δ) − M () G() 2απ tan − log . (18) 0 0 2π 2eB 2α log Region I −δ (12) Here, the 0 dependence appears in the higher-order terms of α, 0RegionII. and can be neglected. Then, we find the running coupling constant at the lower boundary of Region I as This intrinsic scale dependence of the effective coupling constant 3 partly drives the RG evolution, which should be taken into account 2eB α2 2eB in the RG equation below. G(sc) α log + log , (19) m2 6 m2 Next, let us proceed to the one-loop amplitude shown in Figs. 2 γ π γ and 3. It is written down as where we performed an expansion with respect to αlog(2eB/m2 )= γ d2k α log(π/α) 1. The leading term of order α corresponds to the − M = − 2 ¯ (3) μ ν (1) i 1 ( iG) u(p )γ S(k )γ u(p ) tree-level coupling constant (8), while the subsequent term ex- (2π)2 plains the growth of the coupling constant driven by the quantum (4) ¯ (2) × v(p )γ μ S(k − P )γ ν v(p ) , (13) effect in the scale region sc < <0. When the scale enters Region II, we solve the RG equation (1) (2) = where P = p + p . The LLL propagator in the Ritus basis is (17b) with the initial condition at 0 sc which was obtained given by from the RG evolution in Region I in Eq. (19). In this way, the evolutions in the two regions are smoothly connected. We find the ik/ solution in Region II as S(k) = P+ , (14) k2 + i G(sc) G() = . (20) + −1 with the spin projection operator P± = (1 ± iγ 1γ 2)/2in the di- 1 π G(sc) log(/sc) rection of the magnetic field. Clearly, this solution has a Landau pole at We are left with the two-dimensional loop integral in Eq. (13), −π/G(sc) which is a natural consequence of the dimensional reduction. After IR = sc e , (21) performing the elementary k0-integral as explained in Appendix A, which indicates the emergence of the dynamical IR scale. The pres- one finds the origin of the magnetic catalysis: ence of the strong four-Fermi interaction induces a minimum of dk 1 the effective potential at a nonzero value of the chiral condensate. M = 2 z 1 G . (15) The associated dynamical mass is of the order of the emergent 2π |kz| K. Hattori et al. / Physics Letters B 775 (2017) 283–289 287

scale IR, while the order of the chiral condensate is given in The RG equations (17a) and (17b) correspond to the resummation combination with the transverse degeneracy factor ∼ eBIR: Intu- of the ladder diagrams for the multiple photon exchanges. On the itively, the condensate is squeezed along √the magnetic field within other hand, the analysis by the SD equation is based on the rain- the size of the cyclotron motion ∼ 1/ eB, and has a number bow approximation, which holds only the planar diagrams of the of copies distributed with the density eB/(2π) in the transverse fermion self-energy. Cutting the intermediate fermion propagator plane [22] (see also a discussion in the last section). Therefore, the in the SD equation provides the same ladder diagrams which we scale of the dynamical mass gap is explicitly have included in the RG approach. Thus, solving the SD equation in the rainbow approximation is essentially equivalent to solving π π 1/6 the RG equation in the leading order. Note also that the numeri- mdyn mγ exp − + log log( / ) α π α α cal constant C in the exponent takes one in the both results. This √ π agreement originates from a correspondence between the regular- = 2eBα1/3exp − , (22) α log(π/α) izations involved in the analyses. The result from the SD equation is obtained assuming that the fermion self-energy is a constant where the first and second terms in the exponential correspond to without a momentum dependence, that is, mdyn, which regular- those in Eq. (19). izes an IR divergence of the SD equation in the (1 + 1)-dimensions. Notice that, no matter how small the coupling constant in the This regularization corresponds to the sharp cutoff scheme in our underlying theory is, the solution in Eq. (20) has the Landau pole RG analysis. The value of C could depend on such a regulariza- as long as the interaction is attractive. This is consistent with the tion as implied by a slightly different number C = 1.82 from the aforementioned observation made in the study of the SD equa- numerical result [11,12]. √ tions [11,12]. By the use of RG analyses, this fact is even more Now, we can identify the origins of the overall factors of eB, clearly seen in the RG flow informed by the beta function. In short, α1/3, and the exponent in Eq. (4) in the language of the RG when the magnitude of a magnetic field increases, the effect of method. We should first note that the dimensionful quantities en- the magnetic field shifts the UV fixed point that determines the ter in the RG evolutions (17a) and (17b) only through the energy critical coupling strength for the chiral symmetry breaking. Even- scales UV and sc that specify the hierarchy and the initial con- tually, the fixed point merges with the one at the origin, and the ditions. Therefore, the emergent IR scale has to appear in dimen- beta function is completely pushed out from the positive region, sionless combinations with those scales. leaving a vanishing critical coupling strength and the beta function We clearly see in Eqs. (20) and (21) that the initial√ energy scale entirely in the negative region (see Refs. [34,35] for pedagogical 0 = sc in Region II results in the factor of eB in the final discussions). This occurs only for the dimensional reason, as also result (22), and also that the initial condition G(sc) results in discussed in a little bit different way in Sec. 2 and implied by the exponent. Therefore, the dependence of the mass gap √on the the integral in Eq. (15), and thus is a generic consequence of the magnetic field comes from that of the screening mass ∼ αeB. dimensional reduction. We can immediately read off the beta func- Also, we could obtain the logarithmic exponent in Eq. (4) from the tion from Eq. (17a) and (17b) as initial condition (19) resulting from the evolution in Region I: The separation of the RG evolutions in the two regions naturally led 1 2 β () =−2α − G () , (23a) us to take the initial condition G√(sc) at the intermediate scale I π sc instead of at the UV scale ∼ eB. Note that this exponent is 1 2 independent of the magnetic field. This is because the only two βII() =− G () , (23b) π available scales are both proportional to eB. 1/3 for Region I and II, respectively. Clearly, the beta function is neg- The origin of the factor of α is more subtle. In the pass- ative for any value of the coupling strength G(), indicating that ing from Eq. (21) to Eq. (22), we notice that a part of this factor the broken phase is favored at zero temperature regardless of the comes from the initial energy scale sc in Region II and the other value of the coupling strength α. part comes from the initial condition G(sc) in the exponent. The We remark on the contributions of the higher Landau levels. latter contribution was correctly obtained as the result of the RG As is clear from Eq. (22), the location of the Landau√ pole is expo- evolution in Region I, as discussed below Eq. (19). In a word, with- nentially smaller than the Landau level spacing ∼ eB. Therefore, out the energy-scale dependence of the tree-level contribution in the higher Landau levels are decoupled from the low-energy dy- Eq. (17a), one cannot reproduce this factor. namics of the LLL and should not have any significant impact on The above observations clearly indicate the importance of re- the dynamical mass, consistently to the previous observations [36]. specting the interaction properties in the underlying theory. There- 2 fore, it is useful to compare with the results in the NJL model (2) Also, for sufficiently strong magnetic fields such that eB m f and the unscreened QED (3), and identify the origins of the differ- with m f being the intrinsic fermion mass, the dynamical mass, ences in the RG analyses. or IR, emerges much earlier than the m f when the energy scale QED with unscreened photons. Without the screening effect, the goes down. Therefore, the m f also should not have any significant relevant RG equation is only Eq. (17a) in the entire scale regions. impact. The solution is given in Eq. (18). As the scale decreases, this solution hits the Landau pole. Therefore, the dynamical IR scale is 5. Discussions and concluding remarks obtained from the following equation: Based on the RG analysis in the previous sections, we discuss 2 α IR π an equivalence between the analyses by the RG and SD equations, − log , (24) 2π 2eB 2 and digest our RG analysis to identify the origins of the parametric forms of the dynamical mass gap (22) in the screened QED as well which yields the mass gap as those in the NJL model and the unscreened QED [cf., Eqs. (2) √ π π and (3)]. mdyn 2eBexp − . (25) First, we would like to highlight the fact that our result (22) 2 2α has the same parametric form as that of the mass gap (4) ob- This result agrees with that from the SD equation (3) up to an tained from the SD equation. This is not an accidental coincidence. order-one constant factor. In unscreened QED, there is only one 288 K. Hattori et al. / Physics Letters B 775 (2017) 283–289

scale, that is, the initial scale √UV for the entire RG evolution. This Acknowledgements explains the overall factor of eB and the independence of the exponent from eB. S.O. thanks I. A. Shovkovy for useful discussions. K.I. and S.O. More in detail, as we have already discussed, the screening thank Y. Kuramoto for insightful discussion on related subjects. properties are reflected in the exponent. Without the screening K.H. thanks Toru Kojo for discussions. K.H. and S.O. are grateful effect, the exponential suppression is parametrically weaker√ than to KEK and Yukawa Institute for hospitalities and financial sup- that in the screened QED (22). The overall factor of eB comes ports during their visits and the YITP workshop “Strangeness and from the initial scale of the RG equation, of which the value charm in hadrons and dense matter,” where a part of this work is, however,√ different from that√ in screened QED: While it was was achieved. The research of K.H. is supported by the China sc ∼ αeB, it is now UV ∼ eB. This also means the absence Postdoctoral Science Foundation under grant Nos. 2016M590312 of the overall power factor of α in unscreened QED. and 2017T100266. The research of S.O. is supported by MEXT- NJL model. The coupling constant GNJL in the NJL model does Supported Program for the Strategic Foundation at Private Univer- not have any energy-scale dependence in its tree-level Lagrangian. sities, “Topological Science” under Grant No. S1511006. Therefore, the RG equation for the NJL model is formally the same as Eq.√(17b) for Region II, but with an initial scale at the UV region Appendix A. Logarithm from the one-loop scattering amplitude 0 ∼ eB. Since there is no photon propagator in Eq. (8), the effective coupling is, as the result of the Gaussian integral, given by Inserting the LLL propagator (14) into the one-loop scattering = ρLLLGNJL where ρLLL eB/(2π) is the density of states in the LLL. amplitude (13), we have Plugging the initial scale and initial condition into Eq. (21), one (3) μ ν (1) can precisely reproduce the mass gap (2). 2 i[u¯(p )γ k/ P+γ u(p )] 2 d k Although this result is similar to that in QED, the magnetic field M1 = i(−iG) 2 2 2 + dependence is different. As discussed just above, the exponent in ( π) k i QED is independent of the magnetic field, because of the absence (4) (2) i[v(p )γ μ(k/ − /P )P+γ ν v¯(p )] of the dimensionful quantities other than UV and sc which are × . (A.1) both proportional to eB. On the other hand, in NJL mode, the di- (k − P )2 + i mensionful coupling constant GNJL appears in combination with To proceed, we first perform the contour integral for the k0, which the factor of eB in the density of states ρLLL. Therefore, the mass gap in the NJL model increases much faster than that in QED with is given by the contributions from the residues of four poles at 0 =± − 0 0 =± + 0 ∓ − 0 an increasing eB due to the diminishing exponential suppression. k kz i sgn(k ) and k kz P P z i sgn(k ). After When discussing QCD, this behavior may not be regarded physi- inserting these poles into the numerator, one finds that the chiral- Q = ± 5 cal because of involved intrinsic energy-scale dependences in the ity projection operator ± (1 γ )/2 naturally arises through intermediate- to low-energy QCD, which implies one of limitations an identity of the NJL model. 0 ± 3 P = 0 ± 3 Q It will be interesting to investigate an extension to asymptotic- (γ γ ) + (γ γ ) ± . (A.2) free theories which offer an additional intrinsic IR scale, i.e., the To pick up the contributions to the scatterings between the parti- QCD scale QCD. Indeed, there has been an important issue that, cle and antiparticle pairs carrying opposite chiralities, one can use when eB QCD, typical effective models of QCD fail to explain useful formulas the linear dependence of the chiral condensate on eB, i.e., qq¯ ∼ eB which was observed in the lattice QCD simulations [21] (see μ α ν μα ν μν α αν μ γ γ γ = g γ − g γ + g γ , (A.3) also Ref. [37] for a brief summary). It was pointed out that the dynamical mass should stay at mdyn ∼ QCD with increasing B, be- and cause the dimensional reduction leads to a factorization which is ¯ 0 3 0 3 roughly qq ∼ eBmdyn [22] (see also Refs. [23–25]). Also, to ex- (γ ∓ γ )Q± = (γ ∓ γ )P− , (A.4) plain the inverse magnetic catalysis [38,21], this saturating behav- where the LLL spinors are orthogonal to P−, i.e., P−u = P− v = 0. ior of mdyn is thought to be important for the thermal excitations not to acquire too huge a mass gap to restore the chiral sym- After performing straightforward algebraic calculation, the scat- metry [22,25]. Thus, those issues at zero and finite temperatures tering amplitude is obtained as appear to reduce to a problem of explaining the saturation of the − − − 2 μ dkz θ(kz P z) θ( kz) mass gap with an increasing B. In Refs. [22,25], the IR-dominant M1 = G [γ ]R[γ μ]L 2π k − (P 0 + P )/2 interactions are shown to be important for reproducing the satura- z z − − + tion on the basis of the SD equation. In the√ above RG analyses, it is 2 μ dkz θ(kz) θ( kz P z) +G [ ] [ ] , (A.5) ∼ γ L γ μ R 0 clear that the mass gap has to be mdyn eB in any theory/model 2π kz + (P − P z)/2 containing only UV and sc: For example, NJL model may not [ μ] [ ] work as an effective model of QCD in the study of the magnetic where the shorthand notations γ R/L γ μ L/R denote catalysis for this reason. It is then interesting to see how the in- ¯ (3) (1) (4) μ ¯ (2) [ uR/L(p )γ μuR/L(p ) ][ vL/R(p )γ vL/R(p ) ], respectively. trinsic scale modifies the chart of the hierarchy in Fig. 1 and QCD The LLL spinors with the definite chiralities are uR/L = Q±u and manifests itself in the mass gap in the language of the RG and the v R/L = Q± v. Since the energies and momenta of the scattering functional RG [39,35,34,40–42,28,43]. 0 particles are less than , we have P , P z , and thus the loga- In summary, we have closely looked into the magnetic catalysis rithmic contribution arises in the form of the integral (15) for the phenomena by means of the RG method with a special care of both scattering channels up to a numerical factor. the scale separation in the RG evolution. Especially, we elaborated the treatment of the screening effect on the photon propagator, References and showed its crucial role in the determination of the dynamical mass gap. Our result on the mass gap agrees with that from the [1] I.A. Shovkovy, Magnetic catalysis: a review, Lect. Notes Phys. 871 (2013) 13, SD equation [11,12]. arXiv:1207.5081 [hep-ph]. K. Hattori et al. / Physics Letters B 775 (2017) 283–289 289

[2] V.A. Miransky, I.A. Shovkovy, Quantum field theory in a magnetic field: from [22] T. Kojo, N. Su, The quark mass gap in a magnetic field, Phys. Lett. B 720 (2013) quantum chromodynamics to graphene and Dirac semimetals, Phys. Rep. 576 192, arXiv:1211.7318 [hep-ph]. (2015) 1, arXiv:1503.00732 [hep-ph]. [23] P. Watson, H. Reinhardt, Quark gap equation in an external magnetic field, Phys. [3] D.E. Kharzeev, The chiral magnetic effect and anomaly-induced transport, Prog. Rev. D 89 (2014) 045008, arXiv:1310.6050 [hep-ph]. Part. Nucl. Phys. 75 (2014) 133, arXiv:1312.3348 [hep-ph]. [24] N. Mueller, J.A. Bonnet, C.S. Fischer, Dynamical quark mass generation in a [4] D.E. Kharzeev, J. Liao, S.A. Voloshin, G. Wang, Chiral magnetic and vortical ef- strong external magnetic field, Phys. Rev. D 89 (2014) 094023, arXiv:1401.1647 fects in high-energy nuclear collisions—a status report, Prog. Part. Nucl. Phys. [hep-ph]. 88 (2016) 1, arXiv:1511.04050 [hep-ph]. [25] K. Hattori, T. Kojo, N. Su, Mesons in strong magnetic fields: (I) general analyses, [5] X.-G. Huang, Electromagnetic fields and anomalous transports in heavy- Nucl. Phys. A 951 (2016) 1, arXiv:1512.07361 [hep-ph]. ion collisions—a pedagogical review, Rep. Prog. Phys. 79 (2016) 076302, [26] D.K. Hong, Y. Kim, S.-J. Sin, RG analysis of magnetic catalysis in dynamical sym- arXiv:1509.04073 [nucl-th]. metry breaking, Phys. Rev. D 54 (1996) 7879, arXiv:hep-th/9603157. [6] K. Hattori, X.-G. Huang, Novel quantum phenomena induced by strong [27] D.K. Hong, Magnetic catalysis in quantum electrodynamics, Phys. Rev. D 57 magnetic fields in heavy-ion collisions, Nucl. Sci. Tech. 28 (2017) 26, (1998) 3759, arXiv:hep-ph/9707432. arXiv:1609.00747 [nucl-th]. [28] N. Mueller, J.M. Pawlowski, Magnetic catalysis and inverse magnetic catalysis [7]. V.P Gusynin, V.A. Miransky, I.A. Shovkovy, Dimensional reduction and dynam- in QCD, Phys. Rev. D 91 (2015) 116010, arXiv:1502.08011 [hep-ph]. ical chiral symmetry breaking by a magnetic field in (3 + 1)-dimensions, Phys. [29] S. Ozaki, K. Itakura, Y. Kuramoto, Magnetically induced QCD Kondo effect, Phys. Lett. B 349 (1995) 477, arXiv:hep-ph/9412257. Rev. D 94 (2016) 074013, arXiv:1509.06966 [hep-ph]. [8]. V.P Gusynin, V.A. Miransky, I.A. Shovkovy, Dynamical chiral symmetry break- [30] K. Fukushima, Magnetic-field induced screening effect and collective excita- ing by a magnetic field in QED, Phys. Rev. D 52 (1995) 4747, arXiv:hep- tions, Phys. Rev. D 83 (2011) 111501, arXiv:1103.4430 [hep-ph]. ph/9501304. [31] K. Hattori, K. Itakura, Vacuum birefringence in strong magnetic fields: (I) pho- .[9] V.P Gusynin, V.A. Miransky, I.A. Shovkovy, Dimensional reduction and catalysis ton polarization tensor with all the Landau levels, Ann. Phys. 330 (2013) 23, of dynamical symmetry breaking by a magnetic field, Nucl. Phys. B 462 (1996) arXiv:1209.2663 [hep-ph]. 249, arXiv:hep-ph/9509320. [32] K. Fukushima, K. Hattori, H.-U. Yee, Y. Yin, Heavy quark diffusion in strong mag- [10] C.N. Leung, Y.J. Ng, A.W. Ackley, Schwinger–Dyson equation approach to chiral netic fields at weak coupling and implications for elliptic flow, Phys. Rev. D 93 symmetry breaking in an external magnetic field, Phys. Rev. D 54 (1996) 4181, (2016) 074028, arXiv:1512.03689 [hep-ph]. arXiv:hep-th/9512114. [33] K. Hattori, D. Satow, Gluon spectrum in quark–gluon plasma under strong mag- [11]. V.P Gusynin, V.A. Miransky, I.A. Shovkovy, Dynamical chiral symmetry breaking netic fields, arXiv:1704.03191 [hep-ph], 2017. in QED in a magnetic field: toward exact results, Phys. Rev. Lett. 83 (1999) [34] K. Fukushima, J.M. Pawlowski, Magnetic catalysis in hot and dense quark mat- 1291, arXiv:hep-th/9811079. ter and quantum fluctuations, Phys. Rev. D 86 (2012) 076013, arXiv:1203.4330 .[12] V.P Gusynin, V.A. Miransky, I.A. Shovkovy, Theory of the magnetic catalysis of [hep-ph]. chiral symmetry breaking in QED, Nucl. Phys. B 563 (1999) 361, arXiv:hep- [35] D.D. Scherer, H. Gies, Renormalization group study of magnetic catalysis in the ph/9908320. 3d Gross–Neveu model, Phys. Rev. B 85 (2012) 195417, arXiv:1201.3746 [cond- [13] J. Polchinski, in: Theoretical Advanced Study Institute (TASI 92): From Black mat.str-el]. Holes and Strings to Particles, Boulder, Colorado, June 3–28, 1992, 1992, [36] T. Kojo, N. Su, A renormalization group approach for QCD in a strong magnetic pp. 0235–276, arXiv:hep-th/9210046. field, Phys. Lett. B 726 (2013) 839, arXiv:1305.4510 [hep-ph]. [14] N.J. Evans, S.D.H. Hsu, M. Schwetz, An effective field theory approach to [37] G.S. Bali, et al., Thermodynamic properties of QCD in external magnetic fields, color superconductivity at high quark density, Nucl. Phys. B 551 (1999) 275, in: Proceedings, 10th Conference on Quark Confinement and the Hadron Spec- arXiv:hep-ph/9808444. trum, Confinement X, Munich, Germany, October 8–12, 2012, in: PoS, vol. Con- [15] N.J. Evans, S.D.H. Hsu, M. Schwetz, Nonperturbative couplings and color super- finementX, 2012, p. 197, arXiv:1301.5826 [hep-lat]. conductivity, Phys. Lett. B 449 (1999) 281, arXiv:hep-ph/9810514. [38] G.S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S.D. Katz, S. Krieg, A. Schafer, [16] T. Schafer, F. Wilczek, High density quark matter and the renormalization group K.K. Szabo, The QCD phase diagram for external magnetic fields, J. High Energy in QCD with two and three flavors, Phys. Lett. B 450 (1999) 325, arXiv:hep- Phys. 02 (2012) 044, arXiv:1111.4956 [hep-lat]. ph/9810509. [39] V. Skokov, Phase diagram in an external magnetic field beyond a mean-field .[17] D.T Son, Superconductivity by long range color magnetic interaction in high approximation, Phys. Rev. D 85 (2012) 034026, arXiv:1112.5137 [hep-ph]. density quark matter, Phys. Rev. D 59 (1999) 094019, arXiv:hep-ph/9812287. [40] J.O. Andersen, W.R. Naylor, A. Tranberg, Phase diagram of QCD in a magnetic [18] D.K. Hong, An effective field theory of QCD at high density, Phys. Lett. B 473 field: a review, Rev. Mod. Phys. 88 (2016) 025001, arXiv:1411.7176 [hep-ph]. (2000) 118, arXiv:hep-ph/9812510. [41] K. Kamikado, T. Kanazawa, Chiral dynamics in a magnetic field from the [19] D.K. Hong, Aspects of high density effective theory in QCD, Nucl. Phys. B 582 functional renormalization group, J. High Energy Phys. 03 (2014) 009, (2000) 451, arXiv:hep-ph/9905523. arXiv:1312.3124 [hep-ph]. [20] S.D.H. Hsu, M. Schwetz, Magnetic interactions, the renormalization group and [42] J. Braun, W.A. Mian, S. Rechenberger, Delayed magnetic catalysis, Phys. Lett. B color superconductivity in high density QCD, Nucl. Phys. B 572 (2000) 211, 755 (2016) 265, arXiv:1412.6025 [hep-ph]. arXiv:hep-ph/9908310. [43] K.-I. Aoki, H. Uoi, M. Yamada, Functional renormalization group study of the [21] G.S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S.D. Katz, A. Schafer, QCD Nambu–Jona-Lasinio model at finite temperature and density in an external quark condensate in external magnetic fields, Phys. Rev. D 86 (2012) 071502, magnetic field, Phys. Lett. B 753 (2016) 580, arXiv:1507.02527 [hep-ph]. arXiv:1206.4205 [hep-lat].