Nambu-Goldstone mesons in strong magnetic field
aa r X i v : . [ h e p - ph ] J un Nambu–Goldstone mesons in strong magneticfield
V.D.Orlovsky and Yu.A.Simonov,Institute of Theoretical and Experimental Physics117218, Moscow, B.Cheremushkinskaya 25, Russia
Abstract
We study the q ¯ q structure embedded in chiral mesons in responseto external magnetic fields (m.f.), using the chiral Lagrangian with q ¯ q degrees of freedom derived earlier. We show that GMOR relations holdtrue for neutral chiral mesons, while they are violated for the chargedones for eB > σ = 0 . . The standard chiral perturbation theoryalso fails in this region. Masses of π + and π mesons are calculatedand compared to lattice data. Chiral Lagrangians introduced to clarify the dynamics of Nambu–Goldstonemesons have created a new selfconsistent formalism [1] prior to the emergenceof QCD.One of the basic conceptual relations in QCD is the relation of the purelychiral particles – the Nambu–Goldstone (NG) mesons – to all other QCDstates, which are mostly nonchiral. It other words one can define this asa connection of Nambu–Goldstone to ordinary states, which can be calledpure confinement or the flux-tube states. The models treating most statesin the phenomenological chiral-like Lagrangians are now numerous, but un-fortunately they do not clarify this chiral – confinement connection.In [2, 3, 4, 5] one of the authors suggested a way of derivation the chiralNambu–Goldstone spectrum from the QCD Lagrangian, where the basic chi-ral relations: chiral condensate, GMOR relation and expressions for m π , f π m i , string tension σ and α s , and the FCM provides agood description of the QCD spectrum in all channels and for all masses m i ,except for NG mesons: π, K, η .The connection of NG and flux-tube mesons [2, 3, 4, 5] described above,which may be called the chiral-confinement relations (CCR), allows to expressNG meson masses, wave functions and quark decay constants in terms of thesame basic input and in this way completes the theory. One should note,that in the CCR one calculates not only ground states, but also excited NGstates [4, 5] and, moreover, one can study how chiral properties fade awaywith growing quark current masses m i [7].Recently a wide interest has occurred in the literature in the effects, whichcan be produced in hadron dynamics due to strong external magnetic fields(m.f.) [8]. In particular, strong m.f. are expected in neutron stars [9], earlyuniverse [10], heavy ion collisions [11] and possibly m.f. can produce strongreconstruction of the vacuum [12].From the theoretical point of view, strong m.f. play the role of crucialtest of the dynamics used in the model. For the QCD as a strong interactiontheory one must use the relativistic dynamical formalism, incorporating con-finement and perturbative gluon exchanges, producing all effects of strongdecays. This is naturally imbedded in the FCM formalism, based on theQCD path integral, where one derives the relativistic Hamiltonian (RH) forthe q ¯ q, q etc. states.The inclusion of m.f. is done automatically in the RH, and the firstresults for the masses were already obtained in [13], while the important roleof color Coulomb interaction in strong m.f. was studied in [14], and magneticmoments of mesons in [15].Of special interest is the influence of m.f. on chiral dynamics, and in thisway one can check that the CCR sustain their reliability in the presence ofm.f. [16]. On the lattice several analysis were done [17, 18, 19] on chiral dy-namics in m.f., e.g. the dependence of π + mass and chiral condensate on m.f.was done in unquenched QCD with physical pion mass [20]. These resultswere compared with the CCR prediction for the h ¯ uu i and h ¯ dd i dependenceon m.f. and a good agreement was found in [16].On thee other hand, this dependence found on the lattice was comparedin [20] with the what one expects from the chiral theory, and a strong dis-2greement was found for eB > . . This implied that the standardchiral theory [4], which lacks quark degrees of freedom, is unable to be agood working tool for distances less than 0.5 fm.In this paper we study the m.f. dependence of the NG spectrum, whichfollows from CCR. The latter expresses the NG masses through nonchiralPS isovector states, and we approximate several lowest states, which givethe dominant contribution to CCR, in their m.f. dependence and obtain NGmasses.One of the most important result of this paper is the violation of GMORrelations and of the standard chiral formalism in the m.f. There appearadditional terms in the NG Lagrangian, proportional to m.f., which disclosethe internal quark-antiquark structure of NG mesons, not accounted for inthe standard chiral formalism of [1]. As a result the dependence of NGmeson mass on m.f. contains new terms, which do not vanish in the chirallimit m q = 0. This behavior is supported by lattice data [20] for the behaviorof the π + mass in m.f.In what follows we introduce first in section 2 RH for neutral and charged q ¯ q states in m.f., in section 3 we write down the basic equations of CCR andgeneralize them to the case of nonzero m.f. Section 4 is devoted to thecalculation of NG states and in section 5 results are compared with chiralperturbation theory and lattice data. The RH for the q ¯ q system in m.f. was derived recently in [13, 21] from thepath integral in QCD and we follow these notations and definitions. H = H + H σ + W, (1)where H = X i =1 (cid:16) p ( i ) − e i ( B × z ( i ) ) (cid:17) + m i + ω i ω i , (2) H σ = − e σ B ω − e σ B ω , (3) W = V conf + V Coul + V SD + ∆ M ss . (4)3ne defines ω i → ω (0) i from the extremum values of eigenvalues of the oper-ator ¯ H H + H σ + V conf = ¯ H ; ¯ H Ψ = M (0) n Ψ , (5) ∂M (0) n ( ω ω ) ∂ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω i = ω (0) i = 0 , i = 1 , . (6)We now treat H and try to separate c.m. and relative motion R = ω z (1) + ω z (2) ω + ω , η = z (1) − z (2) , (7)For two-body systems q ¯ q the c.m. and relative motion can be separatedin two cases:a) e + e = 0 , neutral case:b) e = e , m = m , ω = ω .We shall consider both cases below.In case a) one introduces the so-called “phase factor”,Ψ( η , R ) = exp( i Γ) ϕ ( η , R ) , (8)Γ = PR − ¯ e B × η ) R , ¯ e = e − e , (9)and defines a new operator H ′ from the relation H Ψ = exp( i Γ) H ′ ϕ, (10) H ′ = 12˜ ω − ∂ ∂ η + e B × η ) ! + X i =1 m i + ω i ω i , (11)At this point it is convenient to replace linear confinement by the quadraticone, with adjustable coefficient γ , which yields a deviation < ∼
5% in resultingmasses, V conf = ση → ˜ V conf = σ η γ + γ ! (12)and now the average mass M n is the eigenvalue of the operator ¯ H ′ ¯ H ′ = H ′ + H σ + ˜ V conf ; ¯ H ′ ϕ = ¯ M n ϕ, (13)4efines the extremal values of ( ω , ω , γ ) ∂ ¯ M n ( ω , ω , γ ) ∂ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω = ω (0)1 = ∂ ¯ M n ∂ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( ω = ω (0)2 = ∂ ¯ M n ∂γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ = γ = 0 . (14)The resulting form of ¯ M (0) n = M n ( ω (0)1 , ω (0)2 , γ ) defines the total mass ofthe meson, M n = ¯ M (0) n + ∆ M coul + ∆ M SE + ∆ M ss . (15)The form of ¯ M n (prior to stationary point insertions) is¯ M n = ε n ⊥ ,n z + m + ω − e B σ ω + m + ω + e B σ ω , (16)where ε n ⊥ ,n z = 12˜ ω "s e B + 4 σ ˜ ωγ (2 n ⊥ + 1) + s σ ˜ ωγ (cid:18) n z + 12 (cid:19) + γσ . (17)The retaining three terms in (15) are defined as follows:1) ∆ M Coul = h V Coul i , where averaging is done with wave functions ϕ n defined in (13), and V Coul is the OGE interaction V Coul ( q ) = − πα s q , and atlarge m.f. q is augmented by the ( q ¯ q ) loop contribution, see details in [14].2), 3) ∆ M SE and ∆ M ss are given in [13] and we rewrite those in theAppendix.We now turn to the case b), and consider two-body system with equalcharges and masses. It is clear, that relativistic charged pions and kaonscontain charges e = e, e = e , in contrast to the case b), however themain new feature in the case b) is the contribution of the c.m. motion inm.f. to the total mass and this is pertinent also to the realistic case, thedifference between the case b) and the realistic case can possibly be treatedin a perturbative manner. As it is clear from (8), (9), the phase factor Γ isnot necessary in the case b), and one obtains the Hamiltonian as in Eq.(34),we also put below ω = ω = ω, e = e = e.H = P ω − e ( P ( B × R ))2 ω + e ω ( B × R ) + π ω + e ω ( B × η ) ++ 2 m + 2 ω − e ( σ + σ ) B ω + σ η γ + γ ! + V coul + ∆ W. (18)5olution of (18), treating V Coul and ∆ W as a perturbation, immediately yields M = m + ω ω + h V coul i + h V ss i + h ∆ M SE i ++ eB ω (2 N ⊥ +1)+ vuut(cid:18) eB ω (cid:19) + 2 σγ ω (2 n ⊥ +1)+( n k + 12 ) s σγ ω − e ( σ + σ ) B ω + γ σ , (19)where γ = β ( B, ω ) (cid:18) σω (cid:19) − / β / ( B, ω ) = 12 + 1 q β ( eB ) (4 σω ) / (20)Finally, ∂M ( ω ) ∂ω (cid:12)(cid:12)(cid:12) ω = ω = 0 , ω ( B ) = a ( B ) √ σ .For the lowest states and eB ≫ σM ee = ω + eB + q ( eB ) + ¯ c σ − ( σ + σ ) eB ω + const ≥ . (21)To be compared with the neutral case (Eq. (16) of our work) M e, − e = ω + 1 ω √ e B + ¯ c σ − e B ( σ − σ )2 ω + ... ≥ σ z + σ z = 0 and M ( eB → ∞ ) ≈ √ eB , while for the neutral case, Eq. (22) for σ z = − σ z = − M ( eB → ∞ ) → const.However, for σ z = σ z the stationary values of ω and ω can be differentfor large eB , and having our results in (20), (21) as a first approximation,we now turn to the case e = e, e = e, e >
0, and introducing the “phasefactor” as in (9), with ¯ e = e − e = e , one obtains the Hamiltonian H ′ = P ω + ω ) + ( ω + ω )Ω R R ⊥ π ω + ˜ ω Ω η η ⊥ X LP BL P ++ X L η BL η + X P ( B × η ) + X ( B × R ) · ( B × η )+6 X π ( B × R ) + m + ω ω + m + ω ω . (23)Here Ω R , Ω η are Ω R = B ( e + e ) ω ω (24)Ω η = B ω ( ω + ω ) " ( e ω + ¯ eω ) ω + ( e ω − ¯ eω ) ω . (25)All coefficients X i ( i = 1 , ,
3) are given in the Appendix 1 of [15].One can see in (23) that the c.m. and relative coordinates can be sepa-rated, provided the terms X , X , X vanish, or else one can treat them as aperturbative correction∆ M X = h X P ( B × η ) + X ( B × R )( B × η ) + X π ( B × R ) i . (26)Then one can write the total eigenvalue M (0) n of the Hamiltonian ¯ H ′ in (13)as M (0) n = M (0) ( P ) + M (0) ( π ) + ∆ M X + H σ (27)where M (0) ( P ) = P z ω + ω ) + Ω R (2 n R ⊥ + 1) + X LP L P B , (28) M (0) ( π ) is the eigenvalue of the operator H π , h π = π ω + ˜ ω Ω Ω η η ⊥ X L η BL η + V conf + V OGE . (29)Note, that we take in the zeroth approximation the q ¯ q state with L p = L η = 0, in which case ∆ M X vanishes in the first approximation, and one hasthe following result for the lowest mass, as in (15), (16), but with additionalc.m. contribution Ω R . M n = ¯ M (0) n + ∆ M Coul + ∆ M SE + ∆ M ss (30)¯ M n = Ω R + ε (+) n ⊥ ,n z + m + ω − e B σ ω + m + ω − e B σ ω (31)where ε (+) n ⊥ ,n z = s Ω η + σγ ˜ ω (2 n ⊥ + 1) + s σγ ˜ ω ( n z + 12 ) + γσ eB ≫ σ, ¯ M n has the form (for n ⊥ = n z = 0)¯ M n ( eB → ∞ ) = Ω R + Ω η + ω + ω − e B σ ω − e B σ ω (33)provided ∆ M SE and ∆ M ss grow slower than eB . We follow here [3, 4] to write first the effective lagrangian of the light quarkin the confining field of the antiquark in m.f., starting with the standardQCD partition function in Euclidean space-time Z = Z DADψDψ + exp[ L + L + L int ] (34)where L = − Z d x ( F aµν ) , (35) L = − i Z f ψ + ( x )( ˆ D + m f ) f ψ ( x ) d x (36) L int = Z f ψ + ( x ) g ˆ A ( x ) f ψ ( x ) d x (37)and ˆ D = γ µ ( ∂ µ − ie f A ( e ) µ ( x )) , A ( e ) µ ( x ) = 12 [ x × B ] . (38)Note, that in m.f. ˆ D can be considered as a diagonal 2 × e f = e u or e d .Averaging Z over vacuum gluonic field and keeping only lowest (bilocal)correlators of color fields D µν,λσ ( x, y ) = N c tr h F µν ( x )Φ( x, y ) F λσ ( y )Φ( y, x ) i ,one finds as in [2, 3, 4, 5], h Z i A = Z DψDψ + exp( L + L (4) eff ) , (39)where L (4) eff = 12 N c Z d xd y f ψ + aα ( x ) f ψ bβ ( x ) g ψ bγ ( y ) g ψ aδ ( y ) J αβ,γδ ( x, y ) (40)8nd J αβ,γδ ( x, y ) = ( γ µ ) αβ ( γ ν ) γδ J µν ( x, y ) (41) J µν ( x, y ) = g Z xC du α Z yC dv β D αµ,βν ( u, v ) (42)Here indices f, g refer to flavor, a, b to color and α, β, µ, ν to Lorentzindices. Eq. (42) implies that some contour gauge is used for simplicity,but the final result is gauge invariant and the most important property of J µν ( x, y ) is that it is proportional to the distance of the average point (cid:16) x + y (cid:17) to the contour C (linear confinement), and the effective distance between x and y (nonlocality) is of the order of the vacuum correlation length λ ≈ . N c limit the four-quark expression in L (4) eff can be replacedby the quadratic one, using the limit f ψ bβ ( x ) g ψ bγ ( y ) → δ fg N c f S βγ ( x, y ) , (43)where f S βγ ( x, y ) is the quark propagator.As a result one obtains the form L (4) eff → − i Z d xd y f ψ + aα ( x ) ( fg ) M αβ ( x, y ) g ψ aβ ( y ) , (44) fg M αδ ( x, y ) = − iJ µν ( x, y )( γ µ fg S ( x, y ) γ ν ) αδ , (45)and the quark propagator satisfies the equation( − i ˆ D − im f ) f S ( x, y ) − i Z ( fg ) M ( x, z ) g S ( z, y ) = δ (4) ( x, y ) . (46)It is convenient to use the following parametrization of f M ( x, y ) in terms ofscalar functions, flavor singlet M s ( x, y ) and flavor triplet φ a ( x, y ) , a = 1 , , M ( fg ) αβ x, y ) = M s ( x, y ) exp( iγ t a φ a ( x, y )) ( fg ) αβ ≡ M s ( x, y ) ˆ U ( fg ) αβ ( x, y ) (47)As a result the effective Lagrangian assumes the form L φ = Z d xd y n f ψ + aα ( x )[( i ˆ D + im f ) αβ δ (4) ( x − y ) δ fg + iM s ˆ U ( fg ) αβ ( x, y )] g ψ aβ ( y ) o , (48)and the partition function can be written as h Z i A = Z DψDψ + DM s Dφ a exp L φ . (49)9ntegrating over DψDψ + one obtains the effective chiral Lagrangian (ECL) L ECL , h Z i A = Z DM s Dφ a exp L ECL , (50)where L ECL = N c tr log [( i ˆ D + im f )ˆ1 + iM s ˆ U ] . (51)Finally,the ECL at the stationary point in the integral (50) is defined byconditions δL ECL δM S = δL ECL δφ a = 0 , which yields iM (0) s ( x, y ) = ( γ µ S (0) γ ν ) J µν ( x, y ) , φ (0) a = 0 , (52)and S (0) = S φ ( φ a = 0) , S φ = − [( i ˆ D + im f )ˆ1 + iM (0) s ˆ U ] − . Insertion of (52) in (51) yields the effective action for pseudoscalar fields φ a L ECL → − W ( φ ) = N c tr log[( i ˆ D + im f )ˆ1 + iM (0) s ˆ U ] . (53)Our final step here is the local limit of J µν ( x, y ) and M (0) s ( x, y ) proved in[2, 3, 4, 5], which yields φ a ( x, y ) → φ a ( x ) , M (0) s ( x, y ) → M (0 s ( x ) δ (4) ( x − y ) (54)Expanding W ( φ ) in powers of φ a and keeping quadratic terms, one has W (2) ( φ ) = 12 Z d kd k ′ (2 π ) (2 π ) φ + a ( k ) N ( k, k ′ ) φ a ( k ′ ) , (55)where ˆ N ( k, k ′ ) = N c Z dxe i ( k + k ′ ) x tr ( t a Λ M s t a ) xx + N c Z d ( x − y ) d (cid:18) x + y (cid:19) exp " i ( k + k ′ )2 ( x + y ) + i k − k ′ )( y − x ) ×× tr[Λ( x, y ) t a M s ( y ) ¯Λ( y, x ) t a M s ( x )] , (56)with the definitions M s ≡ M (0) s , Λ = ( ˆ D + m + M s ) − , ¯Λ = ( ˆ D − m − M s ) − . (57)10t is important at this point to make explicit the dependence of Λ and¯Λ on charges. We shall consider below the cases of neutral and chargedNG mesons and their treatment will be different, since charged NG mesonscontain additional c.m. term in m.f. We start with the neutral case anddefine in (55), (56) a = 3 and Λ + = ( ˆ D + + m + M s ) − , ¯Λ + = ( ˆ D + − m − M s ) − , ˆ D + / − = ˆ ∂ ∓ ie q ˆ A ( e ) , and the same for Λ − , ¯Λ − .Using translation invariance of traces in (55) one can rewrite it for neutralNG mesons as W (2) ( φ ) = N c Z φ ( k ) φ ( − k ) ¯ N ( k ) d k (2 π ) , (58)where¯ N ( k ) = 12 tr { ( Λ + + Λ − M s ) + 12 Z d ze ikz Λ + (0 , z ) M s (0) ¯Λ − ( z, M s (0) +(59)+ 12 Z d ze ikz Λ − (0 , z ) M s (0) ¯Λ + ( z, M s (0) } . At this point it is important to make clear, how the GMOR relationsoccur from the effective Lagrangian (55) from the expression for ¯ N (0) in thecase, when m.f. is absent, and how they are violated by m.f.In the case of no m.f. one can write¯ N (0) = 12 tr n Λ M s + Λ M s ¯Λ M s o = 12 tr n Λ M s ¯Λ( ˆ ∂ − m ) o = (60)= m − ¯Λ) = 12 m trΛ + O ( m ) , where we have used identity ¯Λ( ˆ ∂ − m − M s ) = 1 in the first step, vanishingof the vector part of the expression in the second step, and another identity M s = Λ − − ¯Λ − − m in the last step.Since ¯ N (0) = m π f π N c , one obtains in (60) the GMOR relation, as shown in[3, 4].Another situation occurs in the case of m.f. Indeed, Eq. (60) in this caseacquires the form¯ N (0) = 14 tr { (Λ + + Λ − ) M s + Λ + M s ¯Λ − M s + Λ − M s ¯Λ + M s } , (61)11nd one can rewrite this expression as¯ N (0) = 12 tr (cid:26) − m + M s ¯Λ − + Λ − M s ¯Λ + ) + ∆ ¯ N (0) (cid:27) (62)where the new term is∆ ¯ N (0) = M s ( m + M s )2 tr[ G − ˆ D − ˆ D + G + − G − ˆ D − G + ˆ D − + (63)+ ˆ D − G − G + ˆ D + − G − ˆ D + G + ˆ D + ] , and we have introduced quadratic Green’s functions G + ≡ m + M s ) − ˆ D , G − = 1( m + M s ) − ˆ D − (64)In (63) ˆ D − , ˆ D + are acting at the vertices as follows e.g. for the secondterm inside the tr sign, Z ˆ D − ( x ) G + ( x, y ) ˆ D − ( y ) G − ( y, x ) d ( x − y ) ⇒⇒ Z ˆ D − ( x ) h x | e − ˆ H + − T | y i ˆ D − ( y ) d ( x − y ) = (65) Z h x | [ m − i (ˆ p − + 2 e ˆ A ( e ) ( x ))] e − M + T ( m − i ˆ p − ) | y i d ( x − y ) . However A e ( x = 0) = 0, and therefore magnetic field B cannot act oncharges at the vertices x, y but only can act via the magnetic moment terms,which are the same in the denominators of all four terms in (63), but theseterms also appear in the products ˆ D − ˆ D + and ˆ D + ˆ D − in the first and thirdterm under the tr sign, namelyˆ D = ( D + µ ) + e σ B , ˆ D + ˆ D − = D + µ D − µ − e σ B , (66)ˆ D − = ( D − µ ) − e σ B , ˆ D − ˆ D + = D − µ D + µ + e σ B . Therefore in the sum of these terms, ˆ D − ˆ D + G + G − + ˆ D + ˆ D − G − G + onecan take into account, that G + and G − commute in the constant m.f. andtherefore the sum due to (66) vanishes. Thus we come to the conclusion,that ∆ ¯ N (0) for neutral mesons vanishes, and the GMOR relation surviveswith ¯ N (0) = m π f π N c and ¯ N (0) is given in (62) with ∆ ¯ N (0) = 0.The calculation of the quark condensate in this case was done in [16].12 Masses of NG mesons in magnetic field
We start with the mass of the neutral NG meson and as shown in the previoussection, one can use the GMOR relation with m.f. induced quark condensateand f π .The GMOR relation with additional O ( m ) correction, found in [7], is m π f π = ¯ mM (0) M (0) + ¯ m |h ¯ uu i + h ¯ dd i| , ¯ m = m u + m d , (67)and the quark condensate in m.f. was defined in [16], |h ¯ qq i i | = N c ( M (0) + m i ) ∞ X n =0 | ψ (+ − ) n,i (0) | m (+ − ) n,i + | ψ ( − +) n,i (0) | m ( − +) n,i , (68)where i = u, d, s and the superscripts(+ − ) and ( − +) refer to the quarkantiquark spin projections on m.f. B . In a similar way one can use thederivation of f π , given in [3, 4] to generalize it to (+ − ) and ( − +) projectionsof the Green’s function, namely f π = N c M (0) ∞ X n =0 | ψ (+ − ) n,i (0) | ( m (+ − ) n,i ) + | ψ ( − +) n,i (0) | ( m ( − +) n,i ) . (69)Actually in (68), (69) the summation is over n ≡ ( n , n ⊥ ) and whilemasses m n ,n ⊥ strongly grow with n ⊥ in m.f., the sum over n cut off due tofactors exp( − m n λ ) in (68) and exp( − m n λ )(1 + m n λ ) in (69), see [3, 4] fordetails. As a result only few first terms contribute in (68), (69), and as wasargued in [16] one can replace | ψ n,i (0) | by | ψ (+ − ) n ⊥ =0 ,n (0) | ∼ = √ σ q e q B + σ (2 π ) / , (70) | ψ ( − +) n ⊥ =0 ,n (0) | ∼ = ( σ c − + ) / vuut (cid:18) e q Bσ (cid:19) c − + , (71)and c − + ( B ) = (cid:16) e q Bσ (cid:17) / As it is seen in (16), (17), (22), the mass of the (+ − ) state does not growwith | e q B | , m (+ − ) n ⊥ =0 ,n = O ( √ σ ), while the mass of the ( − +) state grows as13 .0 0.2 0.4 0.6 0.8 1.00.60.70.80.91.0 m (eB)/m (0) eB, GeV Figure 1: The normalized mass of π meson as a function of magnetic fieldstrength (solid line) in comparison with prediction of ChPT (85) (brokenline).2 q | e q B | + σ , therefore we can neglect the sum over ( − +) states in (69),and write for eB ≫ σf π ( B ) = N c M (0)( ¯ m (+ − ) ) X | ψ (+ − ) n,i (0) | ( m (+ − ) n,i ) . (72)On the other hand, |h ¯ qq i i | was estimated, using (68), in [16] as |h ¯ qq i i ( B ) | = |h ¯ qq i i (0) | s (cid:18) e q Bσ (cid:19) + vuut (cid:18) e q Bσ (cid:19) c − + (73)and finally the mass of π at large m.f. | eB | ≫ σ can be written as m π = ¯ mM (0) ( ¯ m (+ − ) ) { A } , A = (cid:16) e q Bσ (cid:17) c − + (cid:16) e q Bσ (cid:17) / (74)where ¯ m (+ − ) is close to the lowest (+ − ) mass with n ⊥ = n = 0.14e can find the π mass numerically, keeping for |h ¯ qq i| and f π the firstfew terms in sums over n (68) and (69). The masses m (+ − ) n,i and m ( − +) n,i aretaken as eigenvalues (16), (17) with appropriate spin directions, while for thevalues of wave function we have the following expression: | ψ n ,n ,n (0) | = n ! n ! n !2 n ( n )! n ( n )! n ( n )! π / r ⊥ r , (75)if all n , n , n are even, for odd n , n or n | ψ n ,n ,n (0) | = 0. The transverseand longitudinal scale parameters r ⊥ = q eB (cid:16) σ ˜ ωγe B (cid:17) − / , r = (cid:16) γσ ˜ ω (cid:17) / .The cut-off parameter λ is taken to be about 1 GeV − . The resulting nor-malized mass m π ( eB ) m π (0) is given in Fig. 1 in comparison with prediction ofchiral perturbation theory (ChPT) (85). This behavior is in agreement withlattice data for π in [22].We now turn to the case of charged mesons, e.g. π + , and one can expect,that, neglecting the internal structure of π + the energy in m.f. will be E π ( eB ) = q | eB | + ¯ m , (76)where ¯ m can depend on m.f. more slowly than | eB | .This kind of behavior was found on the lattice [20], and we shall findbelow whether it appears in our formalism and what ¯ m is.Actually, the behavior m π + ( eB ) in (76), found on the lattice, shows that¯ m ∼ = m π (0) and π + at eB > m π (0) can be considered to some extent asan elementary pseudoscalar meson seemingly without internal q ¯ q structure.However, the derivation of the GMOR relation for π + meson similarly tothe π case does not work for two reasons. First of all,the cancellation inthe ∆ ¯ N (0) term which we observed for π , in the case of π + is absent.Secondly, the total charge motion of π + in m.f. creates its own quantumenergy ∆ E ∼ eB which adds to m π , as it is seen e.g. in (76). Hence, theGMOR relations do not apply to π + at eB > ∼ m π and π + mass m π + ( eB ) doesnot vanish in the limit m q , m ¯ q →
0. We shall show below, however, that thebehavior m π + ( eB ) at eB > ∼ σ can display the q ¯ q structure and, moreover,the lattice data [20] possibly show the beginning of the new pattern.We start with the expressions (30), (31) for ρ + ( S z = 0) and π + states,which can be expressed as combinations √ ( | + − > ±| − + > ) of ( u ¯ d ) spinprojected states. These two states can be considered first in the approxima-15ion eB ≫ σ , when u and ¯ d quarks are independent, then M + − ( B ) = (cid:18)q m u + p z + q m d + p z + 2 | e d | B (cid:19) P z =0 ≈ s eB (77) M − + ( B ) = (cid:18)q m u + p z + 2 e u B + q m d + p z (cid:19) P z =0 ≈ s eB. (78)These two curves M + − ( B ) and M − + ( B ) are below and above the “ele-mentary” behavior q m π + eB, Eq. (76), see Fig. 2.However, we have not taken into account the hf interaction, which mixesthese two states, and therefore one should diagonalize the spin-dependentpart of interaction M ∼ = eB ω + eB ω + ω + ω V SD (79)(see a similar treatment of the neutral meson in [14]) V SD = a σ σ + eBσ z ω − eBσ z ω (80)As it is seen from (79),(80) the stationary values of ω , ω denoted as ω (0)1 , ω (0)2 depend on the state, and at eB ≫ σω (0)1 (+ − ) ∼ = ¯ m, ω (0)2 (+ − ) ∼ = s eB, ω (0)1 ( − +) ∼ = s eB, ω (0)2 ( − +) ∼ = ¯ m, ¯ m ≈ √ σ. (81)From [23] a = cπ / q λ + r ( λ + r ⊥ ) , c = 8 πα s ω ω (82)and λ ∼ − , while r ≈ O (1 / √ σ ) , r ⊥ ∼ eB . Hence the magnetic moment part of V SD (the last two terms on the r.h.s.of (80)) is always dominating for eB > ∼ σ and one expects in this region thatthe asymptotic result for ρ + and π + are m as ( ρ + ( S z = 0)) = M − + ( B ) ≈ s eB (83)16
20 40 60 80 1000123456789101112 m as ( + ) m as ( + ) eB, GeV sqrt(m (0) +2/3(eB)) sqrt(m (0) +4/3(eB)) sqrt(m (0) eB) eB, GeV Bali sqrt(m (0) +2/3(eB)) sqrt(m (0) +(eB)) Figure 2: The masses of charged π + and ρ + mesons (in GeV) as a function ofmagnetic field strength at asymptotically large fields (left) and in the region eB < in comparison with lattice data of [24] (right). m as ( π + ) = M + − ( B ) ≈ s eB. (84)At smaller m.f., when eB < ¯ m ≈ σ , one can diagonalize V SD , and thisprocedure is given in Appendix.The results of numerical calculations of asymptotic behavior for the π + and ρ + masses with the account of Coulomb and self-energy corrections aregiven in Fig. 2 (left graph). The contribution of spin-spin interaction canbe neglected in this region. We extrapolate these asymptotic to small fieldsand compare them with the lattice data [24] (right graph). One can see,that at large eB > . the lattice data for π + possibly prefer the lowerasymptotic (84), rather than the “elementary π + pion curve” of Eq. (76). As was discussed above, our two examples, π and π + ( π − ) mesons behavequite differently at strong m.f. and while the first obeys GMOR relations,the charged meson looses all chiral properties at eB > . . These factsare in agreement with the results of chiral perturbation theory [25]-[30]. In17articular, it was shown in [27, 28] that GMOR relations hold for the π meson, while they are violated for the π + , and π retains its NG propertiesin chiral perturbation theory.However, as shown in [16] and above, both h ¯ qq i and f π are not any moreobjects of ChPT in strong m.f. and at eB > m π the q ¯ q degrees of freedomdefine the values of h ¯ qq i and f π .This in particular is present in the m.f. dependence of m π , which ac-cording to ChPT is [27, 28] m π ( eB ) m π (0) = 1 − eB π ln 2 , (85)and e is the meson charge in ChPT, while in the ( q ¯ q ) system two components(¯ uu ) and ( ¯ dd ) enter in an admixture, with e q = e or e . We compare thedependence (85) with our result (74) in Fig. 1.For π + meson the ChPT is not applicable for eB > m π , while the q ¯ q structure is clearly seen for eB > σ , as it is clear from Fig.2, where the curve m π + ( eB ) deflects from m ρ + ( eB ), as discussed in the previous section.As it is, one can distinguish three regions: 1) eB < ∼ m π (0), 2) m π (0) ≪ eB < ∼ σ , 3) eB ≫ σ , where different dominant mechanisms of meson massformation are present. In the region 1) the ChPT is active for NG mesons,while in the region 2) the q ¯ q structure is evident and both m.f. effects andstrong q ¯ q interaction (confinement and gluon exchange) are important. Fi-nally in the region 3) one can consider q and ¯ q in π + as independent in thestrong m.f. with asymptotic calculated in section 4, while for π the situationis more complicated and the mass is defined by GMOR relations with h q ¯ q i and f π computed in the nonchiral theory.The authors are grateful to N. O. Agasian, M. A. Andreichikov andB. O. Kerbikov for useful discussions.18 ppendix m ( ρ + ( S z = 0) , B ) = 12 ( M + M ) + s(cid:18) M − M (cid:19) + 4 a a , (A. 1) m ( π + , B ) = 12 ( M + M ) − s(cid:18) M − M (cid:19) + 4 a a , (A. 2)where M = ¯ M − eB ω (+ − ) + eB ω (+ − ) − a (+ − ) (A. 3) M = ¯ M + eB ω ( − +) − eB ω ( − +) − a ( − +) (A. 4)and a ik is given in (82), with c ik defined as c = 8 πα s ω (+ − ) ω (+ − ) , c = 8 πα s ω ( − +) ω ( − +) , (A. 5)and c c = (cid:18) πα s (cid:19) ω (+ − ) ω ( − +) ω (+ − ) ω ( − +) . (A. 6)The values of ¯ M can be calculated from (30) or (32), or else for eB ≫ σ they can be estimated as ¯ M ≈ ω + ω = √ e B + √ e B . References [1] S. L. Glashow and S.Weinberg, Phys. Rev. Lett. , 224 (1968);S. Weinberg, Physica, , A 327 (1979);M. Gell-Mann and M. L´evy, Nuovo Cimento , 53 (1960);M. Gell-Mann, R.L Oakes, and B. Renner, Phys. Rev. , 2195 (1968);J. Gasser and H. Leutwyler, Phys. Rept. C , 77 (1982); Ann. Phys.(N.Y.) , 142 (1984); Nucl. Phys. 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