QCD phase diagram in a constant magnetic background. Inverse magnetic catalysis: where models meet the lattice
EEur. Phys. J. manuscript No. (will be inserted by the editor)
QCD phase diagram in a constant magnetic background
Inverse magnetic catalysis: where models meet the lattice
Jens O. Andersen a,1 Department of Physics, Norwegian University of Science and Technology, Høgskoleringen 5, N-7491 Trondheim, NorwayReceived: date / Revised version: date
Abstract
Magnetic catalysis is the enhancement of a con-densate due to the presence of an external magnetic field.Magnetic catalysis at T = T χ isincreasing as a function of the magnetic field B . This is indisagreement with lattice results, which show that T χ is adecreasing function of B , independent of the pion mass. Thebehavior can be understood in terms of the so-called valenceand sea contributions to the quark condensate and the com-petition between them. We critically examine these ideas aswell recent attempts to improve low-energy models usinglattice input. The phase diagram of QCD has received a lot of attentionsince the first ideas appeared in the 1970s. At that time, itwas thought that QCD has two phases, a hadronic phase atlow temperatures and a deconfined phase of quark and a glu-ons at high temperatures. In 1984, Bailin and Love [1] sug-gested that at high density, quark matter should be a colorsuperconductor. The ideas are analogous to those of ordi-nary superconductivity and BCS theory [2], namely the in-stability of the Fermi surface to form Cooper pairs underan attractive interaction. In QCD, an attractive interaction a e-mail: [email protected] is provided by one-gluon exchange in the triplet channel.Since then, there has been a huge effort to map out the phasediagram of QCD and study the properties of its differentphases [3–5]. The phase diagram has shown to be surpris-ingly rich at high baryon density and low temperatures. Itincludes a quarkyonic phase [6] as well as a number of su-perconducting phases, some of them being inhomogeneous.Most of these results have been obtained using low-energymodels of QCD, notably the quark-meson (QM) model andthe Nambu-Jona-Lasinio (NJL) model, with or without cou-pling to the Polyakov loop. The reason is that lattice simu-lations are notoriously difficult to perform at finite baryonchemical potential µ B due to the sign problem, so that onecannot use techniques involving importance sampling.The temperature T and baryon chemical potential µ B are not the only relevant parameters of QCD. For example,one can introduce a separate chemical potential µ f for eachquark flavor f . For two flavors, this leads to another inde-pendent chemical potential besides µ B , namely the isospinchemical potential µ I . For the three flavors, µ S = ( µ u + µ d − µ s ) is added. The addition of these chemical poten-tials gives rise to pion and kaon condensation. At T = µ I > m π , while kaon conden-sation takes place for | ± µ I + µ S | > m K (upper sign forcharged kaons and lower sign for neutral kaons). The for-mer is particularly interesting since finite µ I and vanishing µ B has no sign problem and is therefore amenable to latticesimulations.The final example of an external parameter, which is thetopic of this review, is a (constant) magnetic background.There are several areas of high-energy physics, where sucha background is relevant. One is non-central heavy-ion col-lisions, where large, time-dependent fields are generated.These fields are short-lived and have a maximum value ofapproximately | eB | = m π [7]. The basis mechanism is sim-ply that (in the center-of mass frame) the two nuclei repre- a r X i v : . [ h e p - ph ] F e b sent electric currents that according to Maxwell’s equationsgenerate a magnetic field. Another example where strongmagnetic fields appear, are magnetars [8]. This is a specialclass of neutron stars with relatively low rotation frequen-cies. It is believed that the magnetic field on the surface is10 − Gauss, while in the interior they can be as strongas 10 − Gauss.We consider QCD with an SU ( ) gauge group, a global SU ( N f ) vector symmetry and quark masses m f . The QCDLagrangian is L QCD = − F a µν F µν a + i ¯ ψ f γ µ D µ ψ f − m f ¯ ψ f ψ f + L gf + L ghost , (1)where the gluon field strength tensor is F a µν = ∂ µ A a ν − ∂ ν A a µ − g f abc A b µ A c ν , f abc are the structure constants the covariantderivative in the presence of an abelian background field A EM µ is D µ = ∂ µ + iq f A EM µ + igA µ . (2)Moreover, m f is the mass of a quark of flavor f and thereis a sum of flavors in Eq. (1). The nonabelian gauge fieldis A µ = t a A a µ , t a = λ a , and λ a are the Gell-Mann matrices.Finally L gf and L ghost are the gauge-fixing and ghost partof the Lagrangian, respectively.The partition function in QCD can be written as Z = (cid:90) D A µ D ¯ ψ f D ψ f e − S QCD = (cid:90) D A µ e − S g det (cid:0) / D ( B ) + m f (cid:1) , (3)where S QCD is the Euclidean action for QCD. In the sec-ond line, we have integrated over the fermions which canbe done exactly since L QCD is bilinear in the quark fields.Moreover, S g is the Euclidean action for the gluons and / D ( B ) = (cid:18) iXiX † (cid:19) , (4) iX = D + i σσσ · D . (5)This yieldsdet (cid:0) / D ( B ) + m f (cid:1) = det (cid:2) X † X + m f (cid:3) . (6)The last equation shows that the fermion determinant is man-ifestly positive. As in the case of finite isospin chemical po-tential, QCD in a magnetic field is also free of the sign prob-lem, and one can therefore carry out lattice simulations. In-terestingly, the combination of finite isospin and magneticfield is free of the sign problem only if the charges of the u and d -quark are the same. This is of course not real QCD,but it offers the possibility to compare lattice predictionswith those of low-energy effective theories and models.In this review, we will discuss (inverse) magnetic catal-ysis and the phase diagram og QCD in a strong magneticbackground, paying attention to recent developments. There are other reviews [9–12] focusing on different aspects of thefield. The paper is organized as follows. In the next section,we discuss the physics of magnetic catalysis at T =
0. InSection 3, we introduce the Polyakov loop and discuss mag-netic catalysis in model calculations at nonzero temperature.In Sec. 4, inverse magnetic catalysis on lattice focusing onthe competing sea and valence effects. In Appendix A, wediscuss renormalization of the quark-meson model in theon-shell scheme, while in Appendix B, we show how theparameters of the model are fixed.
Magnetic catalysis can be defined as1. The magnitude of a condensate is enhanced by the pres-ence of an external magnetic field B if the condensatealready is present at vanishing field.2. An external magnetic field induces symmetry breakingand a nonzero value of a condensate when the symmetryis intact for B = φ in low-energy models is an example of theformer, while ¯ ψψ is the chiral condensate in e.g. the NJLmodel or QCD is an example of the latter. On refers to thesecond case as dynamical symmetry breaking by a magneticfield. We will discuss both cases below. The first papers onmagnetic catalysis at T = L = − B + (cid:2) ( ∂ µ σ )( ∂ µ σ ) + ( ∂ µ π )( ∂ µ π ) (cid:3) + D ∗ µ π − D µ π + − m ( σ + π + π + π − ) − λ ( σ + π + π + π − ) + h σ + ¯ ψ (cid:104) i γ µ D µ − g ( σ + i γ τ · πππ ) (cid:105) ψ , (7)where D µ = ∂ µ + iqA µ is the covariant derivative σ , πππ =( π , π , π ) are the meson fields, π ± = √ ( π ± i π ) , τ a arethe Pauli matrices, ψ is a color N c -plet, a four-componentDirac spinor as well as a flavor doublet ψ = (cid:18) ud (cid:19) . (8) In the absence of an abelian gauge field in Eq. (7), the sym-metry is SU ( ) L × SU ( ) R for h =
0, otherwise it is SU ( ) V .In its presence, the Lagrangian Eq. (7) has a U ( ) L × U ( ) R symmetry for h =
0, otherwise it is U ( ) V . The reason is thatone cannot transform a u -quark into a d -quark due to theirdifferent electric charges. Defining ∆ ± = √ ( σ ± i γ π ) , thetwo sets of transformations are 1) u → e − i γ α u , d → e i γ α d , ∆ ± → ∆ ± e ± i γ α , and π ± → π ± u → e i α u , d → e − i α d ∆ ± → ∆ ± , and π ± → π ± e ± i α .After symmetry breaking, the sigma field has a nonzeroexpectation value φ . The classical potential is V = B + m φ + λ φ − h φ . The tree-level relations between the parameters of the La-grangian m , λ , g , and h and the physical masses m σ and m π , the pion decay constant f π , and the quark mass m q are m = − (cid:0) m σ − m π (cid:1) , λ = ( m σ − m π ) f π , (9) g = m q f π , h = m π f π . (10)Using the relations (9)–(10), we obtain V = B + m π f π ∆ m q − m σ f π ∆ m q + m σ f π ∆ m q − m π f π ∆ m q − m π f π ∆ m q , (11)where we have introduced ∆ = g φ . The minimum of theclassical potential is given by ∆ = g f π .The classical potential has by construction its minimumat ∆ = m q or φ = f π . In the large- N c limit, the mesons are included at tree level, while we include the Gaussianfluctuations of the fermions. Including the one-loop correc-tions from the fermions using a minimal subtraction scheme,leaves a renormalized one-loop effective potential that de-pends on the renormalization scale Λ . The minimum of theeffective potential therefore depends on Λ . In order to en-sure that the one-loop effective potential has its minimumat φ = f π for zero magnetic field B , several methods havebeen used in the literature. One method is simply to sub-tract the one-loop contribution to the effective potential for B =
0. Then the renormalization scale dependence drops outand the correction to Eq. (11) is a finite B -dependent termthat vanishes for B =
0. However this is inconsistent sinceone includes fermion fluctuations in the effective potential atfinite magnetic field, but not for B =
0. Moreover, it is alsoincorrect since Eqs. (9)–(10) are tree-level relations that re-ceive radiative corrections. One can also choose a specificvalue for Λ such that the one-loop correction to to the po-sition of the minimum of the effective potential vanishes.In this case, one has included quantum fluctuations also for B =
0, but again, the tree-level relations between the pa-rameters of the Lagrangian and physical quantities receiveloop corrections. In order to be consistent, the parameters ofthe Lagrangian must be determined to the same order in theloop expansion as one calculates the effective potential. Thesolution to the problem is to combine the minimal subtrac-tion scheme with the one-shell scheme [28–31]. In this wayone includes loop corrections to Eqs. (9)–(10), while at thesame time ensures that the effective potential has its mini-mum at g f π . Details of the renormalization of the one-loopeffective potential in the large- N c limit can be found in Ap-pendix A and the parameter fixing in Appendix B. It reads V + = m π f π (cid:40) − m q N c ( π ) f π m π F (cid:48) ( m π ) (cid:41) ∆ m q − m σ f π (cid:40) + m q N c ( π ) f π (cid:34)(cid:18) − m q m σ (cid:19) F ( m σ ) + m q m σ − F ( m π ) − m π F (cid:48) ( m π ) (cid:35)(cid:41) ∆ m q + m σ f π (cid:40) − m q N c ( π ) f π (cid:34) m q m σ (cid:16) log ∆ m q − (cid:17) − (cid:18) − m q m σ (cid:19) F ( m σ ) + F ( m π ) + m π F (cid:48) ( m π ) (cid:35)(cid:41) ∆ m q − m π f π (cid:34) − m q N c ( π ) f π m π F (cid:48) ( m π ) (cid:35) ∆ m q − m π f π (cid:34) − m q N c ( π ) f π m π F (cid:48) ( m π ) (cid:35) ∆ m q + B (cid:34) + N c ( π ) ∑ f q f log m q | q f B | (cid:35) − N c ( π ) ∑ f ( q f B ) (cid:20) ζ ( , ) ( − , x f ) + x f − x f log x f + x f log x f − (cid:21) , (12)where x f = ∆ | q f B | , ζ ( a , x ) is the Hurwitz zeta function, and F ( p ) and F (cid:48) ( p ) are defined in Eqs. (B.30)–(B.31). The first four lines of the one-loop effective potential areindependent of the magnetic field and this part was first cal-culated in Ref. [32]. The last line is the B -dependent correc-tion to V + . Note also that final result is independent of therenormalization scale Λ .In Fig. 1, we show the effective potential divided by f π at T =
0. The black line is the tree-level potential Eq. (11),while the green and blue lines are the one-loop effective po-tential Eq. (12) for | eB | = | eB | = m π , respectively.We have used m σ =
600 MeV, m π =
140 MeV, f π =
93 MeV,and m q =
300 MeV. The classical potential as well the one-loop effective potential with | eB | = ∆ = g f π by construction. Notice, however, that the lat-ter is significantly deeper. The blue line with | eB | = m π ,shows that the minimum of the effective moves to a largervalue, i. e. the system exhibits magnetic catalysis. One loop, B = =
10 m π - - - - Δ ( MeV ) V e ff f π Fig. 1
Effective potential normalized by f π at T =
0. The black line isthe tree-level result, the green and blue lines are the one-loop result forzero magnetic field and for | qB | = m π . See main text for details. While the above clearly demonstrates magnetic cataly-sis numerically, we would like to gain insight in the mech-anism behind the effect. Instead of analyzing Eq. (12), wewill discuss the gap equation in the NJL model. If G is thefour-fermion coupling, M is the mass gap, the gap equationat B = M G = M (cid:90) d p ( π ) p + M . (13)Conventionally, since the NJL model is non-renormalizable,one has used a three-dimensional or a four-dimensional mo-mentum cutoff Λ to regulate divergences. If Λ is a four-dimensional cutoff, the gap equation (13) reads for m (cid:28) Λ M (cid:20) π G − Λ + M log Λ M (cid:21) = . (14) M = G > G c = πΛ thereis also a nontrivial solution. Thus for G larger than the criti-cal value πΛ , quantum fluctuations induce symmetry break-ing in the model. At finite magnetic field, the gap equation becomes M G = M | q f B | π ∑ s = ± ∞ ∑ k = (cid:90) d p ( π ) p + p z + M B , (15)where M B = M + | q f B | ( k + − s ) and q f is the charge. Thedivergences in Eq. (15) can be isolated by adding and sub-tracting the right-hand side of Eq. (13). The right-hand sideof Eq. (15) minus the subtracted term is finite and is con-veniently evaluated using dimensional regularization in thesame way as done appendix A [34, 35]. Finally, we imposea four-dimensional cutoff on the added term as in Eq. (13).Factoring out the trivial solution M =
0, this yields the reg-ularized gap equation4 π G − Λ + M log Λ M − | q f B | (cid:104) ζ ( , ) ( , x f ) + x f − ( x f − ) log x f (cid:21) = . (16)This equation has only a nonzero M as solution. For G < G c ,the solution is [33, 36] M = | q f B | π exp (cid:20) − | q f B | (cid:18) π G − Λ (cid:19)(cid:21) . (17)In the limit | qB | →
0, this solution connects to the trivial so-lution M =
0. In the lowest Landau approximation, whereone excludes the k (cid:54) = M = Λ e − π / G | q f B | , which is reminiscent ofEq. (17) if we identify the cutoff Λ with (cid:112) | q f B | . We canthen think of magnetic catalysis as a 1+1 dimensional phe-nomenon, i.e. a dimensional reduction from 3+1 dimensionshas taken place. The functional form of the gap equation isthe same as for the gap equation in BCS theory of supercon-ductivity as well as the gap equation found in the large- N limit of O ( N ) -symmetric nonlinear sigma model in 1+1 di-mensions. The 1+1 dimensional nature of magnetic catalysisraises the question of whether this phenomenon is in conflictwith the Coleman theorem, which forbids spontaneous sym-metry breaking in less than two spatial dimensions at zerotemperature [37]. As pointed out in Ref. [36], the field ¯ ψψ is neutral with respect to the magnetic field. The neutral pionis the associated Goldstone boson that appears after break-ing the U ( ) symmetry. The charged pions are now massiveeven in the chiral limit.There are other ways of regularizing the gap equation (15)or the fermion contribution to the one-loop effective poten-tial (A.1), for example Schwinger’s proper time method [38].Let us illustrate this by computing the corresponding bosonicfunctional determinant, which shows up in chiral perturba-tion theory. It is based on the representation in Euclideanspace V = log det (cid:0) − D µ D µ + m (cid:1) = − (cid:90) ∞ dss Tr e − s ( − D µ D µ + m ) = − | qB | π ∞ ∑ k = (cid:90) p (cid:107) (cid:90) ∞ dss e − s ( p (cid:107) + | qB | ( k + )+ m ) , (18) where the sum over Landau levels k as well the momentumintegrals over p (cid:107) ( p (cid:107) = p + p z ) are convergent. The resultin d = − ε dimensions is V = − ( π ) (cid:90) ∞ dss − ε e − m s | qB | s sinh ( | qB | s ) . (19)The integral is divergent for small s , i.e. for large momen-tum. By adding and subtracting the divergent terms, we canisolate the divergences. One finds V = − ( π ) (cid:90) ∞ dss − ε e − m s + ( qB ) ( π ) (cid:90) ∞ dss − ε e − m s − ( π ) (cid:90) ∞ dss − ε e − m s (cid:20) | qB | s sinh ( | qB | s ) − + ( qBs ) (cid:21) . (20)The integrals in the first line are divergent for ε =
0, whilethe last integral is convergent. The divergences show up aspoles in ε . The first term in Eq. (20) is a vacuum energycounterterm while the second term corresponds to wave-function renormalization. The last integral in Eq. (20) can becalculating exactly and involves the Hurwitz zeta function.Using the proper time method with the momentum integralsevalauted in d = − ε dimensions yields the same results asthose obtained by combining dimensional regularization andzeta-function regularization, as done in Appendix A. Alter-natively, one can evaluate Eq. (20) with ε = / Λ as the lower limit of the s -integration.The regularization methods discussed so far separatesin clean way the B -independent divergences from the B -dependent terms whether they are finite or divergent. Thereare other regularization methods that do not separate thiscontributions, for example a sharp cutoff imposed directlyon the sum-integral in Eq. (20) or a form factor that is afunction of e.g. p z + k | qB | ( k + − s ) One has to be care-ful choosing such regulators since nonphysical oscillationsmay result [39, 40].After having discussed magnetic catalysis, we now turnto lattice gauge theory. The first lattice simulations werecarried out for an SU ( ) gauge group for magnetic fieldstrengths up to √ eB ∼ eB . The quark condensate itself was calculatedusing the Banks-Casher relation [41], which relates the den-sity of eigenvalues close to zero of the Dirac operator and thecondensate. Their calculations showed a monotonic increaseof the spectral density for typical gauge field configurations.This enhancement induced by the magnetic field can be con-sidered the basic mechanism behind magnetic catalysis. Be-low, we will discuss this mechanism further, here it suffices They are finite in the gap equation Eq. (15), but divergent in the effec-tive potential, cf. Eq. (20). to add that the enhancement of the spectral density as a func-tion of B for typical gauge configuration is also seen in fullQCD [42]. Even in the free case, there is a proliferation ofsmall eigenvalues due to the degeneracy of states, which ina constant magnetic field is proportional to | qB | [19]. In the previous section, we reviewed magnetic catalysis at T = eB .The effective potential of the quark-meson model in thelarge- N c approximation at finite temperature is V T + = V + − ∑ f N c | q f B | π T ∑ s = ± ∞ ∑ k = (cid:90) p z log (cid:104) + e − β E f (cid:105) , (21)where E f = (cid:112) p z + ∆ + | q f B | ( k + − s ) and V + is givenby Eq. (12). In Fig. 2, we show the results of a typical calcu-lation where the quark-meson model was used. The curvesshow the transition temperature for the chiral transition as afunction of | qB | / m π in the chiral limit (green points) and atthe physical point (red points). At B =
0, the gap between thetwo critical temperatures is approximately 10 MeV, whichdecreases as | qB | grows. In both cases, it is clear that thetransition temperature increases with the magnetic field. Herethe transition temperature was defined as the inflection pointof the curve φ ( T ) at the physical point and φ ( T ) = χ = ∂ (cid:104) ¯ ψ ¯ ψ (cid:105) ∂ T . (22)In QCD, two transitions take place as one increases thetemperature, namely the chiral transition and the deconfine-ment transition. Lattice calculations suggest that chiral sym-metry is “restored” at a temperature of approximately T χ c =155 MeV [50–54] though strictly speaking the transition isonly a crossover. The crossover temperature is defined by thepeak of the chiral susceptibility. This temperature is slightlyless than the crossover temperature for the deconfinement Using the peak of ∂Φ∂ T , where Φ is the Polyakov loop, yields a transi-tion temperature for deconfinement, which is in very good agreementwith the chiral transition temperature. π2 T c / M e V Chiral limit, h=0Physical point, h>0
Fig. 2 T pc ( B ) as a function of | qB | in units of m π in the quark-mesonmodel. The green points are in the chiral limit and the red points areat the physical point. See main text for details. Figure taken fromRef. [11]. transition, T dec c =
170 MeV. However this temperature differ-ence is observable dependent. In most cases, T dec c has beendetermined by the behavior of the Polyakov loop. Recently,it has been defined by the behavior of the quark entropy andin this case the two crossover temperatures agree within er-rors [54].We will next discuss the Polyakov loop and how it canbe incorporated in model calculations. The Wilson line isdefined as L ( x ) = P exp (cid:20) i (cid:90) β d τ A ( x , τ ) (cid:21) , (23)where P denotes path ordering, A = iA and A = t a A a .The Polyakov loop operator l is the trace of the Wilson line (23).Together with its Hermitian conjugate, it is defined as l = N c Tr L , l † = N c Tr L † , (24)where N c is the number of colors. The expectation values of l and l † are denoted by Φ and ¯ Φ . Under the center group Z N c of the gauge group SU ( N c ) , the Polyakov loop transforms as Φ → e π inNc Φ with n = , , ... N c −
1. In pure-glue QCD itis an order parameter for confinement, while for QCD withdynamical fermions it is only an approximate order parame-ter [55]. Note also that Φ = ¯ Φ at zero density, i.e. for µ f = N c = A = t A + t A . (25)Introducing the fields φ = β A and φ = √ β A , the ther-mal Wilson line reads for constant gauge fields L = e i ( φ + φ ) e i ( − φ + φ ) e − i φ . (26) Since the Polyakov loop is an approximate order parame-ter for deconfinement, the strategy put forward in Ref. [56]is to write down a phenomenological effective potential for Φ , ¯ Φ and the chiral condensate that describes the thermo-dynamics of the system. This potential consists of a gluonicpart U ( Φ , ¯ Φ ) as well as a matter part. The term U ( Φ , ¯ Φ ) is constructed such that it reproduces the pure-glue pressurecalculated on the lattice [57]. A number of different formsof U ( Φ , ¯ Φ ) have been proposed [58–61]. In Ref. [58], theyused a Polynomial expansion incorporating the Z centersymmetry, UT = − b ( T ) + b ( T ) (cid:2) Φ + ¯ Φ (cid:3) + b ( Φ ¯ Φ ) . (27)Here the coefficients are b ( T ) = . − . (cid:18) T T (cid:19) + . (cid:18) T T (cid:19) − . (cid:18) T T (cid:19) , (28) b = . , (29) b = . , (30)and T =
270 MeV, the transition temperature for pure-glueQCD [57]. A drawback of the proposed pure-glue potentialsis that they are independent of the number of flavors n f . Thetransition temperature for B = n f -dependent T . Once the coupling between the glu-onic sector and the matter sector has been implemented, thetwo transitions take place at approximately the same tem-perature: The chiral transition moves to larger temperatures,while the deconfinement transition moves to lower temper-atures. Finally, the Polyakov-loop potential is coupled tothe matter sector via replacing the partial derivatives in thefermionic part of the Lagrangian by covariant ones includingthe constant background gauge field. This is implemented bymaking the substitutionlog (cid:104) + e − β E f (cid:105) →
16 log (cid:104) + Φ e − β E f + Φ e − β E f + e − β E f (cid:105) +
16 log (cid:104) + Φ e − β E f + Φ e − β E f + e − β E f (cid:105) (31)in Eq. (21). In the same way, the Fermi-Dirac distributionfunction is generalized, n F ( β E f ) = + Φ e β E f + Φ e β E f + Φ e β E f + Φ e β E f + e β E f . (32)For small values of the Polyakov loop, Φ ≈
0, ¯ Φ ≈ E f , i.e. that of three quarks. For large temperatures, when Φ ≈
1, ¯ Φ ≈
1, the excitation energy is E f , which is the dis-tribution function of deconfined quarks. eB/m𝜋2 T p c ( B ) / T p c Fig. 3 T pc ( B ) / T pc as a function of eB / m π in the quark-meson model.Solid line is the mean-field result and the dashed line is the result fromthe functional renormalization group. See main text for details. Figuretaken from Ref. [47]. To the best of our knowledge, there are no systematicstudies of the critical temperature as a function of the mag-netic field B in various approximations. However, some in-teresting results using the quark-meson model exist.Fig. 3 shows the critical temperature from Ref. [47] intwo approximations, namely in the mean-field approxima-tion and using the functional renormalization group (FRG) [62].In this approach, one solves a flow equation for the effec-tive potential numerically by lowering a sliding scale froman initial UV cutoff k = Λ (where the effective potential isequal to the classical potential) down to k =
0. The bare pa-rameters at k = Λ are tuned such that one obtains the phys-ical values of the masses and the pion decay constant in thevacuum. In this way, all quantum and thermal fluctuationsare included. The black solid line is the mean-field result, i.e.the bosons are excluded from the flow equation, whereas thebrown line is the result using the functional renormalizationgroup. Clearly, the addition of bosonic fluctuations increasesthe critical temperature significantly.In Fig. 4, we show the transition temperature at the phys-ical point using the functional renormalization group [49].The green points are the results without the Polyakov loop,whereas the blue points are the results including it. Clearly,the Polyakov loop lowers the transition temperature for fixed B , but it is still increasing as we increase the magnetic field.The above FRG results are obtained in the so-called local-potential approximation. In Ref. [63], the authors added theeffects of wavefunction renormalization and the curve forthe critical temperature lies between mean-field and the local-potential approximation. Thus the coupling of the Polyakovloop to the chiral sector is not sufficient to reproduce (qual-itatively) the results seen on the lattice. |qB| / m π T φ ( B ) / T φ ( ) With Polyakov loopWithout Polyakov loop
Fig. 4 T pc ( B ) as a function of | qB | in units of m π in the quark-mesonmodel. The green points are without the Polyakov loop and the bluepoints are with the Polyakov loop. See main text for details. Figuretaken from Ref. [49]. After having discussed magnetic catalysis in low-energy mod-els and theories of QCD, we next consider QCD lattice sim-ulations. In the past decade, there have been a number of lat-tice calculations of QCD in a magnetic field [23–27, 42, 64–70], which have improved our understanding of QCD in amagnetic background.In order to discuss (inverse) magnetic catalysis as seenon the lattice, it is advantageous to take a look at the path-integral representation of a number of expectation values.The QCD Lagrangian is bilinear in the quark fields ψ f andso one can integrate over them, giving for the partition func-tion as a path integral over gauge configurations A µ Z ( B ) = (cid:90) dA µ e − S g det ( D / ( B ) + m ) , (33)where S g is the Euclidean gluon action and det ( D / ( B ) + m ) is the fermion functional determinant (suppressing flavors).The operator D / ( B ) contains the nonabelian gauge field, whichwe have suppressed, as well as the abelian background B thatwe have indicated. The quark condensate is given by (cid:104) ¯ ψψ (cid:105) = ∂∂ m log Z ( B )= Z ( B ) (cid:90) dA µ e − S g det ( D / ( B ) + m ) Tr ( D / ( B ) + m ) − . (34)We can think of P = Z ( B ) e − S g det ( D / ( B ) + m ) as a measurethat depends on the gauge-field configuration A µ , the mag-netic field, and the quark masses. Note that the B -dependenceis in the functional determinant as well as the the trace of thepropagator. In order to study the contributions to the quarkcondensate coming separately from change of the operator and the change of the measure, it is convenient to introducethe valence and sea contributions defined as (cid:104) ¯ ψψ (cid:105) val = Z ( ) (cid:90) dA µ e − S g det ( D / ( ) + m ) Tr ( D / ( B ) + m ) − , (35) (cid:104) ¯ ψψ (cid:105) sea = Z ( B ) (cid:90) dA µ e − S g det ( D / ( B ) + m ) Tr ( D / ( ) + m ) − . (36)This can be thought of as an expansion of the quark con-densate around B =
0. A priori, the sum of the two contri-butions needs not add up to the total quark condensate un-less we are at small fields. However, it turns out that writingthe condensate as a sum of the valence and sea contributionis remarkably good. This is clearly demonstrated in Fig. 5from Ref. [26], which shows the relative increment r of thevalence and sea contributions, their sum as well as the com-plete results for the quark condensate as a function of a di-mensionless quantity b . The relative increment is defined as r = (cid:104) ¯ ψψ (cid:105) B (cid:104) ¯ ψψ (cid:105) − , (37)where (cid:104) ¯ ψψ (cid:105) is the average of the u and d quark conden-sates. Within error, the additivity is confirmed for valuesof b up to 8, which corresponds to magnetic fields up to | eB | = ( ) [26]. It is also of interest to notice thatboth contributions work in the same direction, namely toincrease the quark condensate as B grows. This is unlikewhat happens at temperature around the critical tempera-ture T c , as we shall see below. As pointed out in Ref. [42],the sea can be thought of as the quark condensate of anelectrically neutral fermion flavor coupled to an electricallycharged fermion flavor, since the magnetic field only appearsin the functional determinant and not in the propagator. Onthe other hand, the valence contribution is reminiscent ofthe expression of the quark condensate in model calcula-tions, except in models one does not integrate over gauge-field configurations.Let us now turn to finite temperature. Inverse magneticcatalysis seems to have two somewhat different meaningsin the literature. The first meaning corresponds directly tothe concept magnetic catalysis discussed above: its simplymeans that a condensate, for example (cid:104) ¯ ψψ (cid:105) , decreases withthe magnetic field at a fixed temperature. The second mean-ing is that the transition temperature itself is a decreasingfunction of the magnetic field.The first finite-temperature lattice simulations of werecarried out in [23, 24] for SU ( ) gauge theory in the quenchedapproximation, focusing on the B -dependence of the chiralcondensate for temperatures below the transition. In two-flavor QCD, simulations at finite temperature were carriedout for pion masses in the range 200 −
480 MeV in Ref.[26]and it was concluded that the chiral and deconfinement tran-sition take place at the same temperature and that they in-crease slightly with the external magnetic field. The increase b full datavalence contributiondynamical contributionvalence + dynamical Fig. 5
Relative increment of the average of the u and d quark conden-sates as a function of b . Valence (red points) and dynamical (sea) (bluepoints) contributions, the sum of them (open circles), and the full quarkcondensate as a function of the dimensionless quantity b . See main textfor details. Figure taken from Ref. [26]. of the transition temperature with B is, at least qualitatively,in agreement with model calculations. Bali et al [27, 65]carried out lattice simulations at the physical point, i.e. forquark masses that correspond to m π =
140 MeV, and the re-sult were somewhat surprising; The transition temperatureturned out to be decreasing as B increases. The differentbehavior of T c is not a consequence of the different pionmasses, rather it results from lattice artefacts and that the re-sults of [26] were not continuum extrapolated. Today thereis consensus that the chiral transition temperature is a de-creasing function of the magnetic field. This behavior is il-lustrated in Fig. 6, which shows the results of a recent latticesimulation [67], namely the transition temperature in MeVas a function of the magnetic field eB in GeV for three dif-ferent pion masses. The pion mass is 343 MeV (red points),440 MeV (blue points), and 664 MeV (green points), whichis much larger than the physical pion mass of 140 MeV. Thetransition temperature increases as a function of the pionmass for fixed value of B , which is also known from B = B for fixed m π .In Ref. [42], the authors carried out a thorough analysisof the quark condensate around the critical temperature tounderstand the behavior of the transition temperature, focus-ing on disentangling the valence and sea effects. The valencecontribution Eq. (35) can also be written as (cid:104) ¯ ψψ (cid:105) = (cid:104) Tr ( D / ( B ) + m ) − (cid:105) , (38)where the subscript indicates that the quark determinant iswithout a magnetic field. The spectral density of the quarkoperator for different values of the magnetic field is shownin Fig. 7. From the figure, it is evident that there is an in-crease in the spectral density around zero with increasingmagnetic field. The corresponding ensemble was generated eB [GeV ] T c ( B ) m π =343 MeVm π =440 MeVm π =664 MeV Fig. 6
Transition temperature for the chiral transition as a function of eB in GeV for different values of the pion mass. See main text fordetails. Figure taken from Ref. [67]. at finite temperature, T =
142 MeV and for vanishing mag-netic background [42]. The Banks-Casher relation [41] thenimplies an increase of the valence contribution.
Fig. 7
Spectral density of the Dirac operator for three different val-ues of the magnetic field. See main text for details. Figure taken fromRef. [42].
Defining the quantity ∆ S f ( B ) = log det ( D / ( B ) + m ) − log det ( D / ( ) + m ) , (39)the full condensate can be written as (cid:104) ¯ ψψ (cid:105) = (cid:104) e − ∆ S f ( B ) Tr ( D / ( B ) + m ) − (cid:105) (cid:104) e − ∆ S f ( B ) (cid:105) . (40)Note that Eq. (40) reduces to the valence contribution Eq. (38)if one replaces ∆ S f ( B ) by unity. Fig. 8 from Ref. [42] showsa scatter plot of the condensate as a function of the change inthe action ∆ S f ( B ) due to the magnetic field. In this plot, the magnetic field strength is eB ≈ and T close to thetransition temperature. Each point represents a gauge con-figuration and they were generated at vanishing magneticfield. The plot suggests that larger values of the conden-sates corresponds to larger values of the weight e − ∆ S f ( B ) andtherefore suppresses the weight of the associated gauge con-figuration. As a result, this counteracts the valence effect,and leads to a decrease in the critical temperature. For pionmasses that are not too large, it also leads to a decrease ofthe condensate itself (see discussion below). Fig. 8
Scatter plot of the down-quark condensate as a function of ∆ S f ( B ) . See main text for details. Figure taken from Ref. [42]. In Refs. [67, 68], the effects of varying the pion mass onthe quark condensate as a function of the temperature havebeen studied in detail. Fig. 9 from [67] shows the differencebetween the quark condensates as a function of the temper-ature at B = eB = .
425 GeV (blue data points) and eB = .
85 GeV (red points) for three values of the pionmass. The authors find that the sea contribution is decreas-ing function of B around T c for the different values of thepion masses, while the valence contribution is on the otherhand an increasing function of the magnetic field for all tem-peratures and pion masses. In the upper panel it is clear thatthe sea contribution wins the competition around the transi-tion temperature implying inverse magnetic catalysis in thestrict sense of the word. This effect can barely been seen inthe middle panel and is completely absent in the lower panel.In other words, a decreasing function of the transition tem-perature does not imply that the chiral condensate decreasesas a function of temperature and it is therefore not clear thatthe latter is the driving mechanism of the former [67]. Thisnontrivial behavior was also demonstrated in [68], where theauthors fixed the magnetic field to eB = . and variedthe pion mass. In QCD, there is inverse magnetic catalysisfor pion masses up to 500 MeV, and magnetic catalysis forlarger values. -0.0100.010.02 m π = M e V eB=0.425 GeV eB=0.85 GeV m π = M e V eB=0.425 GeV eB=0.85 GeV
150 175 200 225 250
T [MeV] m π = M e V eB=0.20 GeV eB=0.425 GeV eB=0.85 GeV Fig. 9
Difference between the quark condensates at zero magneticfield and non-vanishing B for three different values of the pion mass.See main text for details. Figure taken from Ref. [67]. We finally comment on the nature of the chiral transitionand the temperature as a function of B . The simulations havebeen done with magnetic fields up to eB = . They allshow an analytic crossover and that the transition temper-ature is a decreasing function of B . However, it has beenconjectured that the transition would start increasing againfor sufficiently large temperatures, a phenomenon dubbeddelayed magnetic catalysis. In Ref. [66] the author went ashigh as eB = .
25 GeV in the simulations. The transition re-mains a crossover (albeit sharper), there is no sign of delayedmagnetic catalysis, and the chiral and deconfinement tran-sitions coincide. The sharper crossover suggests that theremay be a critical point for even larger values of the magneticfield. For asymptotically large fields, QCD can be mappedonto an anistropic pure-glue theory [71]. This theory wassimulated on the lattice and strong evidence for a first-ordertransition was found [66]. This implies the existence of acritical point and its position was estimated to be at eB (cid:39) . The failure of models to correctly describe the behavior ofQCD around the critical temperature, even after the intro-duction of the Polyakov loop, has lead to significant effortsto improve them, e.g. [72–82]. The temperature T that en-ters the Polyakov-loop potential depends on the number offlavors, it is therefore a reasonable assumption that it also de-pends on the magnetic field. In Ref. [72], the authors fittedthe strange-quark susceptibilities from their calculations inthe entangled PNJL model to the lattice results of Ref. [65].Their ansatz for the field dependence was a simple polyno-mial in ( eB ) up to quadratic order, giving two fitting pa-rameters, T ( eB ) = T ( ) + ζ ( eB ) + ξ ( eB ) . (41)An interesting feature here is that the model predicts a first-order transition for magnetic fields larger than approximately eB = .
25 GeV . As mentioned above, such a critical pointis expected in QCD, albeit at much larger magnetic fields [66].The crossover nature of the chiral transition was a guidefor the authors of Ref. [73], trying to incorporate the de-creasing behavior of the transition temperature in the (Polyakov-loop extended) two-flavor quark-meson model. In their mean-field analysis, they allowed the Yukawa coupling to varywith the magnetic field, g = g ( B ) using the boundary value g ( ) = .
3. This value is indicated by the vertical dotted lineand corresponds to a fixed quark mass in the vacuum. Thecritical temperature as a function of the Yukawa coupling forthree values of the magnetic field is shown in Fig. 10. Thegrey shaded region indicates the values of g for which thetransition is first order. Yukawa Coupling g T c [ M e V ] eB = 0eB = 5m π eB = 10m π st ordertransition Fig. 10 T c as a function of the Yukawa coupling g for various val-ues of the magnetic field. See main text for details. Figure taken fromRef. [73].1 However, to obtain a transition temperature which is de-creasing with the magnetic fields, any curve g ( B ) must startat g ( ) = . g ( B ) can simply not describe the correct B -dependence of the transition temperature, while at the sametime having a crossover transition.Similar approached have been used in the NJL allow-ing the coupling G to depend on both T and B . For exam-ple, in Ref. [74], the authors fix a set of parameters to geta reasonable fit to the lattice data for the sum of the lightquark condensates. The form of the B -dependent couplingwas motivated by the running of the coupling in QCD forstrong magnetic fields [83].One particular appealing idea was recently put forwardin Ref. [84] (see also Ref. [85]). Only T = T c or a cou-pling that depends both upon B and T . The authors firstperformed a determination of the baryon spectrum at thephysical point as a function of magnetic field using latticesimulations. The authors focused on strong magnetic fields,which are relevant for the phase diagram. Making the sim-ple assumption that the baryon masses can be written asthe sum of the masses of their constituents, they derived B -dependent constituent quark masses. This was used as in-put in the PNJL model at zero temperature: Using the gapequation with the B -dependent quark masses, a B -dependentfour-fermion coupling was obtained. The B -dependent con-stituent quark masses as well as G ( B ) are decreasing func-tions of the magnetic field. For B = M = .
097 MeV and G ( ) = . − and for the largest magnetic fieldused ( | eB | = . ), M = .
079 MeV and G ( B ) = . − . . . . . . . . . . . . . h ¯ ψψ i ( B = 0)) / = 229(3) MeV h ¯ ψψ i / h ¯ ψψ i ( B = ) eB [GeV ] lattice resultlattice-improved PNJLstandard PNJL χ PT Fig. 11
Quark condensate in the PNJL model as a function of eB . Seemain text for details. Figure taken from Ref. [84]. Fig. 11 shows the light quark condensate at T = χ PT and PNJL underestimate thequark condensate. This is in contrast with the lattice-improvedPNJL model, which is in quantitative very good agreementwith the simulations. It would be of interest to see the pre-dictions if one would include the effects of the s -quark in themodel calculations.We next consider the finite-temperature calculations ofRef. [84]. Fig. 12 shows the light quark condensate as afunction of T for different strengths of the magnetic field.The dashed lines are the predictions from the standard PNJLmodel without error bands, while the solid bands are thelattice-improved PNJL model predictions. We first note thatlight quark condensate at T = T compared to the PNJL model. This initself is not enough to conclude that we have inverse mag-netic catalysis, but the effect is so strong that the inflectionpoint moves to the left as a function of B so the transitiontemperature decreases. . . . . . . . . . .
02 0 .
14 0 .
16 0 .
18 0 . .
22 0 .
24 0 . h ¯ ψψ i [ G e V ] T [GeV]lattice-improved PNJLstandard PNJL eB [GeV ] = 0.0000.1030.2170.3320.4460.561 Fig. 12
Light quark condensate in the PNJL model as a function of T for different values of the magnetic field. See main text for details.Figure taken from Ref. [84].2 Fig. 13 shows the normalized transition temperature forthe chiral transition, as defined by the inflection point of thequark condensate, as a function of | eB | in units of GeV . Thepink band is from the lattice results of Refs. [65]. The widthof the band indicates the errors of the simulations. The light-blue points are the results of a calculation from the stan-dard PNJL model showing that the transition temperature in-creases as the magnetic field grows. This is in sharp contrastto the lattice-improved PNJL model, where the results areshown by the green points including uncertainties comingfrom the lattice determination of the baryon masses. Giventhe large uncertainties, the results for the transition temper-ature are in good agreement with the simulations. Similarly,the analytic crossover found also agrees with lattice results. . . . . . . . . .
25 0 0 . . . . . . T c / T c ( B = ) eB [GeV ] lattice resultlattice-improved PNJLstandard PNJL Fig. 13
Normalized transition temperature from the lattice and in thePNJL model as a function of | eB | . See main text for details. Figuretaken from Ref. [84]. The idea of magnetic catalysis at zero temperature has beenaround for three decades after its discovery in the NJL model.It is a robust phenomenon. For large values of the magneticfield, it can be understood in terms of dimensional reductionfrom 3 + + T - and B -dependent couplings and Polyakov-looppotentials. Fitting parameters can be considered an indirectway of incorporating the sea effect and requires input fromlattice simulations at finite temperature. In our opinion, acleaner approach is provided by the work [84], which useslattice input at T = B -dependent baryon masses measured on the lattice asinput in their PNJL-model calculations. It would be of inter-est to apply these ideas to other models such as the quark-meson model. Acknowledgements
The author would like to thank Prabal Adhikari, Patrick Kneschke,Willam Naylor, and Anders Tranberg for discussions andcollaboration on related topics. The author would like tothank Massimo D’elia, Gergely Endródi, Eduardo Fraga, andVladimir Skokov for permission to use their figures.
Appendix A: Renormalization of the one-loop effectivepotential in the quark-meson model
In this appendix, we will discuss renormalization of the one-loop effective potential in the quark-meson model using theon-shell scheme. The starting point is the one-loop contri-bution to the effective potential of a fermion of mass m f ina constant magnetic field, which is given by V = − ∑ (cid:90) B { P } log (cid:104) p (cid:107) + m f + | q f B | ( k + − s ) (cid:105) , (A.1)where p (cid:107) = p + p z and the sum-integral is short-hand no-tation for ∑ (cid:90) B { P } = | q f B | π ∑ s = ± ∞ ∑ k = (cid:18) e γ E Λ π (cid:19) ε (cid:90) p (cid:107) d d p (cid:107) ( π ) d , (A.2)where Λ is the renormalization scale associated with theMS-scheme. The sum is over spin s and Landau levels k . Theintegral over p (cid:107) can be evaluated using dimensional regular-ization in d = − ε dimensions. The result is V = | q f B | ( π ) Γ ( − + ε ) (cid:0) e γ E Λ (cid:1) ε ∑ s = ± ∞ ∑ k = M − ε B , (A.3)where M B = m f + | q f B | ( k + − s ) . The sum over spin s andLandau levels n can be expressed in terms of the Hurwitz ζ -function as ∑ s = ± ∞ ∑ k = M − ε B = ( | q f B | ) − ε ∞ ∑ k = (cid:34) k + m f | q f B | (cid:35) − ε − m − ε f = ( | q f B | ) − ε ζ ( − + ε , x f ) − m − ε f , (A.4)where x f = m f | q f B | is a dimensionless variable. The effectivepotential can then be written as V = ( q f B ) ( π ) (cid:18) Λ | q f B | (cid:19) ε Γ ( − + ε ) (cid:2) ζ ( − + ε , x f ) − x − ε f (cid:21) . (A.5) We next expand the result (A.5) in powers of ε to order ε .This yields V = ( π ) (cid:18) Λ | q f B | (cid:19) ε (cid:20)(cid:18) ( q f B ) + m f (cid:19) (cid:18) ε + (cid:19) − ( q f B ) ζ ( , ) ( − , x f ) − | q f B | m f log x f (cid:105) . (A.6)In the equation above, we have defined ζ ( , ) ( − , x f ) = ∂ζ ( − + ε , x f ) ∂ε | ε = . For renormalization purposes, it is conve-nient to isolate in Eq. (A.6) the terms in the functional deter-minant that equal the B = V , setting m f = ∆ , summing over quarkflavors and flavors yields V + = B + m g ∆ + λ g ∆ − hg ∆ + N c ∆ ( π ) (cid:20) ε + + log Λ ∆ (cid:21) + N c ( π ) ∑ f ( q f B ) (cid:20) ε + log Λ | q f B | (cid:21) − N c ( π ) ∑ f ( q f B ) (cid:20) ζ ( , ) ( − , x f ) + x f − x f log x f + x f log x f − (cid:21) . (A.7)Eq. (A.7) has simple poles in epsilon. The pole proportionalto ( q f B ) is eliminated by wavefunction renormalization ofthe electromagnetic field, while the pole propertional to ∆ is eliminated by renormalization of the parameters in the La-grangian. The bare parameters are replaced by the parame-ters in MS-scheme, e.g. m → m MS + δ m MS and the runningparameters given by Eqs. (B.22)–(B.25) are substituted intothe renormalized expression for the effective potential. Thecouplings at the reference scale Λ are determined usingEqs. (B.17)–(B.20) and expressed in terms of physical quan-tities. The result is Eq. (12). Appendix B: Parameter fixing
The mass of a particle is given by the pole of the propagator.In the on-shell scheme, the sum of the self-energy evaluatedon-shell and the counterterms vanishes, Σ σ , π ( p = m σ , π ) + counterterms = . (B.1)In addition, the residue of the propagator evaluated on shellis unity. This implies Σ (cid:48) ( p = m σ , π ) + counterterms = . (B.2) The self-energies are Σ σ ( p ) = − ig N c (cid:2) A ( m q ) − (cid:0) p − m q (cid:1) B ( p ) (cid:3) + i λ φ N c m q m σ A ( m q ) , (B.3) Σ π ( p ) = − ig N c (cid:2) A ( m q ) − p B ( p ) (cid:3) + i λ φ N c m q m σ A ( m q ) , (B.4)where the integrals A ( m ) and B ( p ) are defined in Eqs. (B.28)–(B.29). The second line in Eqs. (B.3)–(B.4) corresponds tothe tadpole which is one-particle reducible. It is cancelled bya counterterm, when we impose the condition that φ = f π .The counterterms are given by the expressions δ m σ , π = − Σ σ ( p ) (cid:12)(cid:12)(cid:12) p = m σ , π , δ Z σ , π = Σ (cid:48) σ , π ( p ) (cid:12)(cid:12)(cid:12) p = m σ , π . . (B.5)This yields δ m σ = ig N c (cid:2) A ( m q ) − ( m σ − m q ) B ( m σ ) (cid:3) , (B.6) δ m π = ig N c (cid:2) A ( m q ) − m π B ( m π ) (cid:3) , (B.7) δ Z σ = ig N c (cid:2) B ( m σ ) + ( m σ − m q ) B (cid:48) ( m σ ) (cid:3) , (B.8) δ Z π = ig N c (cid:2) B ( m π ) + m π B (cid:48) ( m π ) (cid:3) , (B.9) δ t = − ig N c f π A ( m q ) , (B.10)where we have added the counterterm δ t for the one-pointfunction. We next need to relate the above counterterms tothe counterterms of the parameters of the Lagrangian. Theserelations follow immediately from Eqs. (9)–(10), δ m OS = − ( δ m σ − δ m π ) , (B.11) δ λ OS = ( δ m σ − δ m π ) f π − λ δ f π f π , (B.12) δ g OS = δ m q f π − g δ f π f π . (B.13)In the large- N c limit, δ m q = δ g OS = − g δ f π f π .There is also no loop correction to the quark-pion vertex.This implies that the associated counterterms must cancel aswell, leading to δ g OS = − g δ Z OS π . We can therefore write δ λ OS = ( δ m σ − δ m π ) f π − λ δ Z OS π . (B.14)The counterterm δ h OS is found from the one-point function.At tree level, we have t = h − m π f π =
0, which yields δ t = δ h OS − δ m π f π − m π δ f π = δ h OS − δ m π f π − m π f π δ Z OS π . Fi-nally, we need the counterterm for the electromagnetic field, δ Z OS A = − N c ( π ) ∑ f ( q f B ) B ( ) . (B.15)Since the bare parameters are independent of the renormal-ization scheme, we can immediately write down the rela-tions between the renormalized parameters in the on-shell and MS schemes. For example g MS + δ g MS = g OS + δ g OS . FromEq. (B.9), we find δ g = g N c ( π ) (cid:34) ε + log Λ m q + F ( m π ) + m π F (cid:48) ( m π ) (cid:35) , (B.16) where F ( p ) and F (cid:48) ( p ) are defined in Eqs. (B.30)–(B.31).The counterterm in the MS-scheme is simply the pole part, δ g MS = N c g ( π ) ε . From this, one finds the running coupling g MS using g MS = g OS + δ g OS − δ g MS and given by Eq. (B.19). Therunning parameters are m MS = − (cid:0) m σ − m π (cid:1) − m q N c ( π ) f π (cid:34)(cid:0) m σ − m π (cid:1) log Λ m q + m q + (cid:0) m σ − m q (cid:1) F ( m σ ) − m π F ( m π ) (cid:35) , (B.17) λ MS = (cid:0) m σ − m π (cid:1) f π + g N c ( π ) f π (cid:20) (cid:0) m σ − m π − m q (cid:1) log Λ m q + (cid:0) m σ − m q (cid:1) F ( m σ )+ (cid:0) m σ − m π (cid:1) F ( m π ) + (cid:0) m σ − m π (cid:1) m π F (cid:48) ( m π ) (cid:21) , (B.18) g MS = m q f π (cid:40) + m q N c ( π ) f π (cid:34) log Λ m q + F ( m π ) + m π F (cid:48) ( m π ) (cid:35)(cid:41) , (B.19) h MS = m π f π (cid:40) + m q N c ( π ) f π (cid:34) log Λ m q + F ( m π ) − m π F (cid:48) ( m π ) (cid:35)(cid:41) , (B.20) A µ MS = A µ (cid:34) − N c ( π ) ∑ f q f Λ m q (cid:35) . (B.21)The running parameters satisfy renormalization group equa-tions that follow from Eqs. (B.17)–(B.20) upon differentia-tion with respect to Λ . The solutions are m MS ( Λ ) = m − g N c ( π ) log Λ m q , (B.22) g MS ( Λ ) = g − g N c ( π ) log Λ m q , (B.23) λ MS ( Λ ) = λ − g N c ( π ) log Λ m q (cid:16) − g N c ( π ) log Λ m q (cid:17) , (B.24) h MS ( Λ ) = h − g N c ( π ) log Λ m q , (B.25)where m , g , λ , and h are the values of the running massand couplings at the scale Λ determined by (cid:34) log Λ m q + F ( m π ) + m π F (cid:48) ( m π ) (cid:35) = . (B.26)This equation in conjunction with Eqs. (B.17)–(B.20) canbe used to determine the values of the couplings at the scale Λ expressed in terms of physical quantities. For example, itfollows that g = g MS ( Λ ) = m q f π . We need a few divergent integrals space in four dimen-sions. Going to Euclidean space via Wick rotation, we canuse dimensional regularization in d = − ε dimensions.The integrals needed are (cid:90) k log (cid:2) k + m (cid:3) = − m ( π ) (cid:18) Λ m (cid:19) ε (cid:20) ε + + O ( ε ) (cid:21) , (B.27) A ( m ) = (cid:90) k k − m im ( π ) (cid:18) Λ m (cid:19) ε (cid:20) ε + + O ( ε ) (cid:21) , (B.28) B ( p ) = (cid:90) k ( k − m q )[( k + p ) − m q )= i ( π ) (cid:32) Λ m q (cid:33) ε (cid:20) ε + F ( p ) + O ( ε ) (cid:21) , (B.29)where Λ is the renormalization scale associated with the MSscheme and the functions are F ( p ) = − r arctan (cid:18) r (cid:19) , (B.30) F (cid:48) ( p ) = m q rp ( m q − p ) arctan (cid:18) r (cid:19) − p , (B.31)where r = (cid:114) m q p − References
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