Pauli Villars regularization of field theories on the light front

Transcription

1 on the light front Universit of Minnesota-Duluth, USA Four-dimensional quantum field theories generall require regularization to be well defined. This can be done in various was, but here we focus on Pauli Villars (PV) regularization and appl it to nonperturbative calculations of bound states. The philosoph is to introduce enough PV fields to the Lagrangian to regulate the theor perturbativel, including preservation of smmetries, and assume that this is sufficient for the nonperturbative case. The numerical methods usuall necessar for nonperturbative bound-state problems are then applied to a finite theor that has the original smmetries. The bound-state problem is formulated as a mass eigenvalue problem in terms of the light-front Hamiltonian. Applications to quantum electrodnamics are discussed. PoS(LC010)009 Light Cone LC010 June 14-18, 010 Valencia, Spain Speaker. c Copright owned b the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence.

2 1. Introduction In order to solve a (3 + 1)-dimensional theor, the theor must be regulated in some wa. In doing so, one should attempt to preserve as man smmetries as possible. We do this b adding enough Pauli Villars (PV) 1] fields to regulate perturbation theor and assume that the nonperturbative eigenproblem is also regulated. Numerical methods are then applied to a finite theor, just as was the case for (1+1)-dimensional superrenormalizable theories ]. From the Hamiltonian eigenproblem, we can compute wave functions as coefficients in Fock-state expansions, and from these, compute observables. As a test of the approach, we consider QED. This is not meant to compete with perturbation theor; the numerical errors of the nonpertubative calculation are too large to resolve high-order contributions that perturbation theor computes directl. However, the method is intended for strong-coupling theories where perturbation theor is ineffective. There have been a number of applications of the method. The first being an exploration in terms of a soluble, heav-source model 3, 4]. The dressed-fermion state in Yukawa theor has been studied extensivel 5, 6, 7], as have exact solutions in the limit of equal PV masses 8]. Applications to gauge theories have been primaril to the dressed-electron state in QED 9, 10, 11, 1, 13], but also to the photon eigenstate 14]. A scheme has been proposed for QCD 15]. In order to have a well-defined Fock-state expansion, the theories are quantized on the light front 16, ]. We define light-cone coordinates of time, x + = t + z, and space, x = (x, x ), with x t z and x = (x,). The light-cone energ is p = E p z and momentum, p = (p +, p ), with p + E + p z and p = (p x, p ). These lead to the mass-shell condition, p = m, in the form p = m +p p +. We work with the standard parameterization, where the bare parameters of the Lagrangian are fixed b fits to phsical constraints. There is the alternative of sector-dependent parameterization, where the bare parameters of the Lagrangian are allowed to depend on the Fock sector(s) on which the operators act. This was originall proposed b Perr, Harindranath, and Wilson 17] and applied to QED b Hiller and Brodsk 18]. More recent work with this approach has been b Karmanov, Mathiot, and Smirnov 19, 0]. Other nonperturbative approaches include lattice theor 1, ], Dson Schwinger equations 3], effective-particle representations 4], and basis-function methods 5]. The remainder of this paper contains a brief description of light-front QED in Lorentz gauge, followed b discussion of the eigenproblem for the electron state in lowest truncation. The anomalous magnetic moment is calculated. We also compare the results of the sector-dependent approach, and then conclude with a brief summar. PoS(LC010)009. Light-front QED in Lorentz gauge The Lorentz-gauge QED Lagrangian, regulated b two PV fermion flavors and two PV photon flavors, is L = i=0 ( 1) i 1 4 F µν i F i,µν + 1 µ i A µ i A iµ 1 ] ( µ ) A iµ (.1)

3 + i=0 ( 1) i ψ i (iγ µ µ m i )ψ i e 0 ψγ µ ψa µ, where ψ = i=0 βi ψ i, A µ = i=0 ξi A iµ, F iµν = µ A iν ν A iµ. (.) A subscript of i = 0 indicates a phsical field, and i = 1 or a PV field. The i = 1 fields are chosen to have negative norm. The mass of the bare photon µ 0 is zero. The constants β i and ξ i control the coupling strengths of the various fields. We require that β 0 = 1 and ξ 0 = 1 and require the constraints i=0 ( 1)i β i = 0 and i=0 ( 1)i ξ i = 0. These guarantee the regularization and that the combinations ψ and A µ in (.) have zero norm. A third pair of constraints comes from requiring that the photon eigenstate have zero mass 14] and that the mass of the electron eigenstate becomes zero when m 0 is set to zero 10]. The nondnamical fermion fields satisf the following constraints (i = 0,1,): ] i( 1) i ψ i + e 0 A βi ψ = iγ 0 γ ( 1) i ψ i+ ie 0 βi A ψ + ( 1) i m i γ 0 ψ i+. (.3) From these we obtain a constraint on the null combination i ψ = (iγ 0 γ ) ψ + γ 0 m i βi ψ i+. (.4) i The terms containing the photon field cancel because i ( 1) i β i = 0; therefore, light-cone gauge is not necessar. The nondnamical field ψ can then be constructed from a sum of ψ i that satisf the free-fermion constraint. The mode expansion for the full Fermi field is ψ i = 1 16π 3 s dk ] b is (k)e ik x u is (k) + d k + i, s (k)eik x v is (k). (.5) PoS(LC010)009 The spinors are defined in 6], and the nonzero anticommutators are {b is (k),b i s (k )} = ( 1) i δ ii δ ss δ(k k ) and {d is (k),d i s (k )} = ( 1) i δ ii δ ss δ(k k ). The mode expansion for the ith photon flavor is A iµ = 1 16π 3 dk a iµ (k)e ik x + a (k)eik x] k + iµ, (.6) with the commutator a iµ (k),a i ν (k )] = ( 1) i δ ii ε µ δ µν δ(k k ). The metric signature is chosen to be ε µ = ( 1,1,1,1), for Gupta Bleuler quantization 7]. Because we do not use light-cone gauge, there is no constraint on A + = A, and, consequentl, there will be no instantaneous photon interaction term ] in the Hamiltonian. We can now construct the light-front Hamiltonian P from spinor matrix elements 14]: P = i,s + l,µ d p m i + p p + ( 1) i b i,s (p)b i,s(p) + d p m i + p i,s p + ( 1) i d i,s (p)d i,s(p) dk µ l + k k + ( 1) l ε µ a lµ (k)a lµ(k) (.7) 3

4 + β i β j ξ l i, j,l,s,µ d pdq {b i,s (p) b j,s (q)v µ i j,s (p,q) + b j, s (q)u µ i j, s ]a (p,q) lµ (q p) +b i,s (p) d j,s (q) V µ i j,s (p,q) + d j, s (q)ū µ i j, s (p,q) ]a lµ (q + p) d i,s (p) d j,s (q)ṽ µ i j,s (p,q) + d j, s(q)ũ µ i j, s (p,q) ] a lµ (q p) + H.c. }. The vertex functions can be found in 14]. The Hamiltonian does not contain an instantaneous fermion terms ]. The cancel between phsical and PV contributions because the are independent of the fermion mass and proportional to ( 1) i β i for the ith flavor. The sum over flavors then ields i ( 1) i β i = 0. This is independent of the gauge choice. 3. Electron eigenstate In the one-electron/one-photon truncation, the Fock-state expansion of the electron eigenstate, for total J z = ± 1, is dk i js (k)b is (P k)a jµ (k) 0. (3.1) ψ ± (P) = z i b i± (P) 0 + i i jsµ It is normalized according to ψ σ (P ) ψ σ (P) = δ(p P)δ σ σ. The second PV fermion flavor (i = ) plas no role in this sector and can be removed. In order to have a positive norm, all phsical quantities are computed from a projected state 9]: ψ phs ± (P) = ( 1) i z i b 0± (P) 0 + dk i sµ j/+1 k= j/ 1 i=0 j=0, ( 1) i+k ξk iks (k)b 0s (P k)a jµ (k) 0. ξ j (3.) PoS(LC010)009 To be an eigenstate of the light-cone Hamiltonian, the wave functions must satisf the following coupled equations, with = k + /P + : M m i ]z i = (P + ) dd k ξl ( 1) j+l ε µ (3.3) and V µ ji± M m j + k 1 µ l + k ] M m j + k 1 µ l + k j,l,µ (P k,p)cµ± jl± ] µ (k) +U ji± (P k,p)cµ± jl ], (k) jl± (k) = ξ l ( 1) i z i P + V µ ji ± (P k,p), (3.4) i jl (k) = ξ l ( 1) i z i P + U µ ji ± (P k,p). (3.5) i An index of i corresponds to the one-electron sector and j to the one-electron/one-photon sector. Therefore, in the sector-dependent approach, a mass m i in a vertex function is assigned the bare mass, and m j is the phsical mass. In the standard parameterization, all are bare masses. 4

5 The coupled equations can be solved analticall 9]. The wave functions ils and the amplitudes satisf with 9] il± (k) = j ( 1) j z j P + V µ i j± (P k,p) ξ l, M m i +k 1 µ l +k (3.6) il (k) = j ( 1) j z j P + U µ i j± (P k,p) ξ l, (3.7) M m i +k 1 µ l +k (M m i )z i = e 0( 1) i z i J + m i m i Ī 0 (m i + m i )Ī 1 ], (3.8) i Ī n (M ) = ddk 16π J(M ddk ) = 16π There is also the identit 10] J = M Ī 0. The analtic solution is 9] jl jl ( 1) j+l ξ l M m j +k 1 ( 1) j+l ξ l M m j +k 1 µ l +k µ l +k are m n j (1 ) n, (3.9) m j + k (1 ). (3.10) α 0± = (M ± m 0)(M ± m 1 ) 8π(m 1 m 0 )(Ī 1 ± MĪ 0 ), z 1 = M ± m 0 M ± m 1 z 0. (3.11) A graphical solution is given in 10]. In general, the lower sign ields the phsical answer, because m 0 then becomes the phsical mass M = m e at zero coupling. The coupling coefficient ξ is fixed b requiring that M = 0 when m 0 = 0. In this truncation, we can safel take the m 1 limit, where z 1 = 0, m 1 z 1 ±(M m 0 )z 0, and α 0± = ± M(M ± m 0) 8π(Ī 1 ± MĪ 0 ), (3.1) and the second PV photon flavor can be discarded. In the sector-dependent approach, Ī 1 and Ī 0 are independent of m 0, and the solution for α 0 can be rearranged as an explicit expression for m 0 m 0 = M + 8π α 0± M (Ī 1 ± MĪ 0 ). (3.13) The anomalous magnetic moment is computed from the spin-flip matrix element of the electromagnetic current J + 8]. In the one-photon truncation and in the limit where the PV electron mass m 1 is infinite, the expression for the anomalous moment is a e = α 0 π m ez 0 (1 )ddk (3.14) ) ( 1) k m 0 + (1 )µ k + k. m e(1 ) ( 1 k=0 PoS(LC010)009 For the sector-dependent parameterization, the product α 0 z 0 is just α, and the bare mass m 0 in the denominator is replaced b the phsical mass m e. Plots of the anomalous moment as a function of the regulator mass µ 1 can be found in 1]. 5

6 4. Sector-dependent and standard parameterizations Even though we do not include the vacuum-polarization contribution to the dressed-electron state, the sector-dependent bare coupling is not equal to the phsical coupling. Instead, the are related b e 0 = e/z 0 18], where z 0 is the amplitude for the bare-electron Fock state computed without projection onto the phsical subspace. In general, the bare coupling would be e 0 = Z 1 e/ Z i Z f Z 3 ; this includes the truncation effect that splits the usual Z into a product of different Z from each fermion leg 9]. With no fermion-antifermion loop, we have Z 3 = 1, and without a second photon, there is no vertex correction and Z 1 = 1. Also, onl the fermion leg with no photon spectator will be corrected b Z ; therefore, we find Z i Z f = z 0. In the sector-dependent approach, the bare-electron amplitude without projection is determined, in the infinite-m 1 limit, b the normalization 1 = z 0 + e 0 z 0J, with J = 1 8π ddk 1 k=0 ( 1) k ( + )m e + k k + (1 )µ k + m e]. (4.1) Replacing e 0 b e/z 0, we can solve for z 0 as z 0 = 1 e J and find e 0 = e/ ) 1 e J. For large µ 1, one finds J 1 ln µ 1µ 0. Thus, e 0 can become imaginar and Fock-sector probabilities 8π ( m 3 e range outside 0,1] due to IR and UV divergences, and consistenc then imposes limits on µ 0 and µ 1 1], as confirmed for Yukawa theor in 0]. In the standard parameterization, the bare amplitude is determined b 1 = z 0 + e z 0 J, with J = 1 8π ddk m 0 4m 0 m e (1 ) + m e(1 ) + k ] (4.) ( 1 ) ( 1) k 1 k=0 k + (1 )µ k + m 0 (1. )m e] PoS(LC010)009 Thus the bare amplitude is z 0 = 1/ 1 + e J, which is driven to zero as µ 1 and causes most expectation values also to go to zero. Therefore, in this case there is a limit on µ 1, but µ 0 can be zero. The anomalous moment in the sector-dependent case is a e = α π m e (1 )ddk (4.3) 1 ( ) ( 1) k 1 m e + (1 )µ k + k m e(1 ) k=0 In the µ 1, µ 0 0 limit, this becomes exactl the Schwinger result a e = α π m e ddq /(1 ) ] = α + q m π. (4.4) e m e +q 1 However, this limit cannot be taken without making the underling theor inconsistent. 6

7 5. Summar With use of PV regularization, one can formulate and solve nonperturbative problems in field theories. It is important to maintain smmetries, which can be done with additional PV fields, such as those introduced to maintain a zero photon mass and the chiral smmetr of the masslesselectron limit. It is best to regulate before appling numerical methods, to clearl separate limits of regulators from those of numerical convergence. The PV fields do add to the numerical load but also reduce it, b eliminating instantaneous fermion and instantaneous photon interactions. For both the standard parameterization and the sector-dependent parameterization, truncation of the Fock space results in uncancelled divergences, which require that not all PV masses be taken to infinit; however, meaningful results can be extracted at finite PV masses. For the sectordependent approach, this is complicated b infrared divergences 1]. As discussed elsewhere 13], these methods have been extended to a truncation that includes two photons. The next step is to also include an electron-positron-pair contribution to the dressedelectron state and stud charge renormalization as well as current covariance. This truncation will then include all contributions of order α to the anomalous moment. A calculation at large α, where numerical errors in the order-α contribution would be small compared to the α contributions, could be compared with higher-order perturbation theor. It would also be interesting to consider the dressed electron in a magnetic field and extract its induced magnetic moment. In addition, as a precursor to consideration of mesons in QCD, one can compute two-fermion bound states in Yukawa theor and QED. Acknowledgments The work reported here was done in collaboration with S.S. Chabsheva and supported in part b the US Department of Energ and the Minnesota Supercomputing Institute. PoS(LC010)009 References 1] W. Pauli and F. Villars, Rev. Mod. Phs. 1 (1949) 434. ] For reviews of light-cone quantization, see M. Burkardt, Adv. Nucl. Phs. 3, 1 (00); S.J. Brodsk, H.-C. Pauli, and S.S. Pinsk, Phs. Rep. 301 (1998) 99. 3] S.J. Brodsk, J.R. Hiller, and G. McCartor, Phs. Rev. D 58 (1998) ] S.J. Brodsk, J.R. Hiller, and G. McCartor, Phs. Rev. D 60 (1999) ] S.J. Brodsk, J.R. Hiller, and G. McCartor, Phs. Rev. D 64 (001) ] S.J. Brodsk, J.R. Hiller, and G. McCartor, Ann. Phs. 305 (003) 66. 7] S.J. Brodsk, J.R. Hiller, and G. McCartor, Ann. Phs. 31 (006) ] S.J. Brodsk, J.R. Hiller, and G. McCartor, Ann. Phs. 96 (00) ] S.J. Brodsk, V.A. Franke, J.R. Hiller, G. McCartor, S.A. Paston, and E.V. Prokhvatilov, Nucl. Phs. B 703 (004) ] S.S. Chabsheva and J.R. Hiller, Phs. Rev. D 79 (009)

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