The Anomalous Magnetic Moment of Quarks

The Anomalous Magnetic Moment of Quarks Pedro J. de A. Bicudo, J. Emilio F. T. Ribeiro and Rui Fernandes Deartamento de Física and Centro de Física das Interacções Fundamentais, Edifício Ciência, Instituto Suerior Técnico, Av. Rovisco Pais, 96 Lisboa, Portugal In the case of massless current quarks we find that the breaking of chiral symmetry usually triggers the generation of an anomalous magnetic moment for the quarks. We show that the kernel of the Ward identity for the vector vertex yields an imortant contribution. We comute the anomalous magnetic moment in several quark models. The results show that it is hard to escae a measurable anomalous magnetic moment for the quarks in the case of sontaneous chiral symmetry breaking. Theoretically, the various hadronic electromagnetic form factors are usually described in terms of ole dominance together with contributions arising from virtual mesonic exchanges. A third contribution to the electromagnetic form factors should come from the quark microscoic interaction itself, in close analogy with QED. It is clear that these three scenarios should not be indeendent but just three different asects of the same model. This desideratum can be achieved, at least qualitatively, in terms of a quark field theory dislaying sontaneous breaking of chiral symmetry, (SχSB. In such a descrition any hadron, when seen from the trivial vacuum Fock sace, aears as a collection of an infinite number of quark antiquark airs together with the aroriate valence quarks. It haens that the contributions of this quark sea can be summarized in terms of a new set of valence quasiquarks which now carry the information on the details of the hysical vacuum trough a modified roagator. In this fashion we recover the simlicity of the constituent quark icture. It is the role of the Ward identities to ensure charge conservation throughout this rocess. And this they do at the exenses of the quark magnetic moment which, in general, becomes non-zero. As will be shown in this aer, to maintain throughout the rocess of SχSB a zero anomalous magnetic moment for the quarks, constitutes the excetion rather than the rule and is just the consequence of articular choices for the Lagrangian. However, the B.C.S. diagonalization of the Hamiltonian, (mass ga equation does not reclude quark air creation or anihilation rocesses to occur. In fact it sets the strength of mesonic contributions for such hysical rocesses as decay widths and meson-nucleon interactions 3, among others. The counterarts of these rocesses, when seen from the oint of vue of the hoton couling, are recisely ole dominance and mesonic cloud contributions for the electromagnetic form factors. The objective of this aer is to set u the general formalism for the evaluation of electromagnetic form factors in the resence of SχSB and to use it to evaluate the u, d anomalous magnetic moments for various models. In the Pauli notation, the electromagnetic current u to first order in the hoton momentum q ν is, j µ = eūγ µ u = eū γ µ + i σµν M q νa u µ = µ ( + a, µ = e h ( Mc where a stands for the anomalous art of the magnetic moment µ and M the article mass. The magnetic moment of ground state hadrons is measured exerimentally. For instance we have for the roton and neutron a =.79 and a n =.9. In the constituent quark model for light hadrons we have, µ = 3 (4µ u µ d =.85µ µ n = 3 (4µ d µ u =.97µ ( and the quark magnetic moments are nearly roortional to the charges e u = 3 e, e d = 3e, which suggests that the gyromagnetic factor (+a is nearly flavor indeendent. The quantity which can be measured is M/( + a. For quark flavors u and d we have, M u ( + a u 338 MeV, M d ( + a d 3 MeV (3 The constituent quark model can be alied to fit the hadron sectrum, with a confining interaction, an hyerfine interaction, and a zero oint energy, 4 The required arameters are of the order of α s =.974,M u M d = 4 MeV which would suggest a sizeable a of the order of.5 to.3. It is also clear that we will need a d a u.5 in order to recover the isosin symmetry. The Ward Identity, iq µ S( + q/γ µ S( q/ = S( + q/ S( q/ q µ Γ µ = is ( + q/ is ( q/ (4 is obeyed both by the bare vertex Γ µ and by the Bethe- Saleter vertex Γ µ 5. We will show that in the limit of

small momentum q, this identity has the following solution for the vertex, Γ µ (, q =i S (+q ν T νµ (+o(q (5 µ where q ν T νµ ( is defined as the kernel which is not determined by the Ward identity, q µ q ν T νµ ( =. (6 The Ward identity ensures that charge conservation survives renormalization. However it does not constrain the kernel, which is a signature of the renormalization. In articular the kernel contributes to the anomalous magnetic moment of fermions. This can clearly be seen in QED where the infrared and ultraviolet divergences can be removed from the hoton roagator, i ( i ( λ i ( Λ (7 The vertex is given by, Γ µ =Γ µ o i µ Σ+q ν T νµ o (8 and, u to first order in α, the contributions from the self energy and the kernel to the anomalous magnetic moment are resectively, ( ln λ ( α M π, +ln λ α M π. (9 In the case of actual QED, where λ, Λ,they are both infrared divergent but their sum is finite: α/π. As for QCD, there has been a considerable effort on how to derive quark models by integrating out, under various aroximations, the gluonic degrees of freedom. An interesting and romising aroach is rovided by the cummulant exansion of the interaction term of the QCD Lagrangian 6. A non-local Nambu Jona-Lasinio tye Lagrangian (NJL is obtained when we retain only bilocal correlators. Therefore we hold the view that such quark models are aroriate to study electromagnetic roerties of hadrons, even for light quarks, rovided we have small enough hoton momenta and the hysics of chiral symmetry breaking is treated correctly. Therefore, at this stage, rather than focusing on a secific examle of NJL we will study the static electromagnetic roerties of a wide class of quark effective quartic interactions. In quark models with dynamical SχSB, the vector vertex Γ µ is a solution of the Bethe-Saleter equation, q q q q =Γ µ + + = + q, = q, = + q, = q ( where the strong interaction, which is described by a dotted line in the diagrams is iterated to all orders in the Bethe Saleter equation. This equation can be written, d Γ µ (, q =Γ µ 4 i (π 4 V (, +, qω a S( Γ µ (, q S( Ω a V (q, +, + Ω a tr {S( Γµ (,qs( Ω a} ( where the factor from the fermion loo was included in the tadole term. The momentum deendence of the otential is only assumed to conserve the total momentum, and in this case it deends on 3 momenta. The Dirac, flavor and color structure of the interaction is determined by the Ω a matrices. In order to have dynamical SχSB, we require this structure to be chiral invariant. Substituting the Ward Identity in the ladder Bethe Saleter equation for the vertex we get, is ( is ( =is ( is ( (π 4 V (, +, qω a S( S( Ω a V (q, +, + Ω a tr {S( S( Ω a}. ( For articular cases of the otential V (, +, we recover the BCS mass ga equation, S ( =S ( (3 rovided that either the rainbow diagram vanishes or, V (, +, q =V(, +,, (4 and that either the tadole diagram vanishes or, V (q, +, + =V(, +, +. (5 Equation (3 can be written, is ( =is ( (π 4 V (, +, Ω a S( Ω a V (, +, + Ω a tr {S( Ω a }. (6 Now we insert the exression (5 for Γ µ in the Bethe Saleter equation (, in order to find the kernel qt and exand it u to first order in q. The equation for the tensor T, which is antisymmetric, is then, T νµ = T νµ i T νµ = (π 4 V Ω a( tr {ST νµ SΩ a } (π 4 V Ω a( tr {J νµ Ω a } J νµ = ν (S S µ (S µ (S S ν (S (7

This is a self consistent forced linear integral equation. Let us consider a general quark roagator, solution of the mass ga equation, of the form, S( µ = if ( M( (8 where = µ µ. The integrand J νµ is then, J νµ F = i ( M {, γν,γ µ }+Mγ ν,γ µ Ṁ (ν, γ µ µ, γ ν (9 where the dot suerscrit denotes d/d. In general we find, T νµ = t ({, γ ν,γ µ }+t (Mγ ν,γ µ +t 3 (( ν, γ µ µ, γ ν ( U to o(q the electromagnetic current of the quark is then, j µ = e F ū γ µ µ M ( M µ F F + Fqν T νµ u e( Ṁ = ū γ µ + a (i σµν F M q ν u, F(t + t a=ṁ+4m Ṁ ( where the mass shell condition = M was used together with the Gordon identities. The anomalous magnetic moment a turns out to be indeendent of t 3 and F. However the deendence on M is crucial in models where t and t are finite. In those models a can be thought as a measure of SχSB. The quark condensate qq is also a functional of the dynamically generated mass, qq = = n c tr S( ( (π 4 where the trace sums colors with n c = 3, but the flavor is ket fixed. Thus, at the onset of the sontaneous χsb, we will obtain an imlicit relation between a, qq and the constituent quark mass, which were simultaneously vanishing before the occurrence of this hase transition and now become non-zero. We will now comute F, M, a and qq in articular models which are aradigmatic cases of chiral symmetry breaking and comly with the constraints of the Ward identity. Model I is the first original NJL model 7. The Lagrangian of model I is, L I = qi q + G ( qq ( qγ 5 q (3 where L I is secific to the case of flavor, but its results are similar to the ones of flavor symmetric U A (n f extended NJL models. The equations will be solved in the momentum reresentation. As usual the integrals are done in Euclidean sace. A momentum cutoff Λ is included in order that the integral in the loo momentum is finite. Since the cutoff cannot be adscribed to the otential which has to be constant in momentum sace, is must be included in the roagator, S( = i F( M +iɛ, F( Θ Euclidian(Λ. (4 With a constant otential and this momentum cutoff, the loos turn out to be constant, indeendent of the external momentum. It is convenient to evaluate the integrals, if I = (π 4 ( M = 6π Λ M ln (+ Λ M, F I = i = 6π (π 4 ( M Λ Λ + M +ln (+ Λ, (5 M where the solid angle π is included. The mass ga equation is, M = G if (π 4 ( M ( + M γ 5 ( + Mγ 5 tr{ + M} (6 With the solutions M =or=8n c GI (M,Λ. The arameters Λ and G are determined once the quark dynamical mass and the quark condensate are fixed. We now study the kernel in model I. Because the integrals are constant, the antisymmetric tensor T is indeendent of. ThusT νµ has to be of the t tye, roortional to γ ν,γ µ. Including the structure factors Ω a we find that the tadole-like term vanishes since σ νµ and σ νµ γ 5 have a null trace. In this case of model I the rainbow diagram also cancels since the structure γ 5 γ 5 rojects on the terms with an odd number of Dirac γ matrices, of tye t but γ ν,γ µ is even. Thus model I roduces no kernel for the vector vertex and no anomalous magnetic moment for the quark 8. Model II is the second original NJL model 9. The Lagrangian is, L II = qi q + G ( qq ( qγ 5 τq (7 where L II is used for flavors u and d. It only has an SU( A symmetry and breaks U( A from the onset. Its 3

results are similar to those of flavor symmetric SU A (n f extended NJL models. The anzats for the roagator is that of Eq.(8, and model II only differs from model I in the algebra. The mass ga equation is changed since τ τ = 3 in the fermion line. We get, M = G if (π 4 ( M ( + M n f γ 5 ( + Mγ 5 tr{ + M} (8 n f ( M = or = n f +4n f n c GI (M,Λ. n f were n f and n c stand resectively for the number of flavors and colors. The rainbow diagram contributes in this case. The tadole diagram contribution also changes. The referred values for the arameters Λ and G are Λ =.65GeV and G =.3GeV which yield M =.33GeV, qq = (.5GeV 3 and f π =.9GeV. As in model I, the tadole will not contribute to the antisymmetric tensor which will be again of the t tye, T νµ = t γ ν,γ µ. The first order term is a function of, (π 4 J νµ = I M γ ν,γ µ. (9 In order to evaluate the higher order terms, we calculate, (π 4 SMγν,γ µ S = im I q ν M γ ν,γ µ (3 In this case we have two flavors with two different charges e u = 3, e d = 3, and two anomalous magnetic moments a f, ( eu ( ed e u t u = GI M e u t u +( M e d t d (u d (3 The natural arameter is GM I (Λ,M=.4. inverting this equation we find the solution, a u (GM I e d =.4, M u = 339 MeV e u a d (GM I e u =.6, M d = 37 MeV (3 e d Although this effect is small, it has the right sign to correct the M u and M d inversion. If the tadole term was removed from the mass ga equation then the a d a u would be bigger. This is ossible for instance when the otential has a λ. λ deendence, being λ the Gell-Mann matrices. Model III is the simlest QCD insired model. The Lagrangian is, L III = qi q+ q(xγα λ q(x λ d 4 yv (x y q(yγ α q(y In the case of model III, the Dirac structure γ µ γ µ is U A (n f chiral invariant. For V( we will choose a color confining square well otential because of its calculational simlicity. V ( = GΘ Euclidian (Λ, (33 The mass ga equation is, M F = 4i 3 (π 4 V ( F +4M M which includes the color factor of 4 3. We now calculate the kernel. The first order term for the kernel is a functional of, γ α J νµ γ α = if {, γν,γ µ } ( M (34 The terms with two gamma matrices, of the form σ µν are now cancelled by the γ α γ α of the interaction, and only the t tye term remains. Therefore this model differs from the revious ones insofar it covers the form factor t. The self consistent equation for the antisymmetric tensor T will also close, γ α S {, γ ν,γ µ }Sγ α =F +M We get, T νµ = t{, γ ν,γ µ } t(=t o 8 d 4 3 i + M ( M t(, ( M {, γν,γ µ } (π 4 V ( F ( t o ( = d 4 3 i (π 4 V ( F ( ( M. (35 For the euclidean integration it is convenient to evaluate the angular integrals, + ( I 3 (,=G dwθ Λ + w = G( + I 6 θ( I 6 θ( + I 6 +Gθ(I 6 > + ( I 4 (,=G dwwθ Λ + w = G I 6 θ( I 6 θ( + I 6 > I 5 = Λ (36 and the nonlinear integral mass ga equation for F and M, the integral for t o, the linear integral equation for T, and the integral for qq canbesolvedsimultaneously, 4

F ( = + M(=F( t o (= t( =t o qq = d I 5(, 6π 4 +M ( F( d I 4(, 3π 3 F( +M ( M( d I 5(, 4π d I 5(, 6π 4 F ( + M ( d 3 3 F (M( π + M ( 4 F( M ( +M ( t( (37 The mass term has a trivial solution M = and another solution which breaks sontaneously chiral symmetry. A dimensional simlification occurs if we work in units of Λ =. In this case the only arameter is G which is now adimensional. We find a critical value G c = 3 above which chiral symmetry occurs. In Fig. we deict the values of M, qq and a. We solve the integral equations numerically for F,Mand t with the Gauss iterative method and using the Gauss integration. We find that at = these functions decrease by a factor of just.9.7. Since we cannot continue analytically the numerical solution we use the aroximation of nearly constant F, M and t and comute the mass and the anomalous magnetic moment for =. The literature refers a < qq >= (.5 GeV 3. A dynamical quark mass M =.33 GeV wouldcorresondto G= 45Λ,Λ=.74 GeV and a =.5. If we now consider a M =(+a.33 GeV then the lowest ossible condensate is < qq >= (.8 GeV 3 which corresonds to G = 3Λ, Λ =.69 GeV, M =.4 GeV and a =.8, see Fig.. NJL models I and II are the simlest models with chiral symmetry breaking. In the NJL model I the anomalous magnetic moment a vanishes. In the model II the U( breaking interaction yields a too small a, which nevertheless rovides an examle of an isosin deendence for a and, therefore, contributes to the u d mass inversion. The reason for the smallness of the anomalous magnetic moment stems from the resence of tadole contributions and were not for this contribution and we would have obtained a much larger a. Thisisrecisely the case of model III where a larger a is derived, comatible with the nonrelativistic constituent quark models. We also find that M, a, < qq >, are functions of (G G c with critical exonents which are resectively, and. The resent work constitutes a first ste on a more elaborate model unifying hadronic sectroscoy (including decay widths and the electromagnetic form factors which have been shown to be consistent with the simle quark constituent icture recisely because of SχSB. H. Ito,Phys. Rev C 5, R75, (95; H. Forkel, M. Nielsen, X Jin, T. Cohen, Phys Rev C 5, 38, (994. A. Le Yaouanc, L.Oliver, O. Pene and J-C. Raynal, Phys Rev D 9, 33 (984; A. Le Yaouanc, L.Oliver, S. Ono, O. Pene and J-C. Raynal, ibid 3, 37 (985;P.Bicudo and J. Ribeiro, Phys Rev D 4, 6 (99. 3 P.Bicudo and J.Ribeiro Phys Rev D 4, 635 (99; P.Bicudo and J.Ribeiro Phys Rev C 55, 834 (997. 4 N. Isgur, G. Karl, Phys. Rev. D8, 478 5 S. Adler and A. C. Davis, Nucl. Phys. B 4, 469 (984. 6 H. G. Dosh, Phys Lett. B9 77 (987; H. G. Dosh and U. Marquard, Nuc Phys. A56 333 (993; N. Brambilla and A. Vairo Phys. Lett B47 67 (997; Yu. A. Simonov he-h/9748. 7 Y. Nambu, G. Jona-Lasinio, Phys. Rev., 345 (96 8 U. Vogl, M. Lutz, S. Klimt and W. Weise, Nucl. Phys. A 56, 469 (99; J. Singh, Phys. Rev. D 3, 97 (985. 9 Y. Nambu, G. Jona-Lasinio, Phys. Rev. 4, 46 (96; V. Bernard, Phys. Rev. D 34, 6 (986. Y. Dai, Z. Huang and D. Liu, Phys. Rev. D 43, 77 (99..4...8.6.4.. 3 4 5 6 G a M -<q q> /3 FIG.. Functions of the quark dynamical mass M, anomalous magnetic moment a and quark condensate as functions of the adimensional couling G for model III 5

.4. M..8 -<q q> /3.6.4. a. 3 4 5 6 G