Exploring a nonperturbative method for calculation of the anomalous magnetic moment of the electron

Transcription

1 Exploring a nonperturbative method for calculation of the anomalous magnetic moment of the electron Universit of Minnesota-Duluth, USA and Southern Methodist Universit, USA sophia@phsics.smu.edu The electron s anomalous magnetic moment is computed in a light-cone Hamiltonian approach, with Pauli-Villars (PV) particles as UV regulators. The eigenstate is obtained b diagonalizing a matrix that represents the discretization of 48 coupled integral equations for the one-photon/oneelectron wave functions. This generalizes earlier work on a one-photon truncation and extends the eigenstate expansion to two photons. In addition to the phsical particles, one PV electron flavor and two PV photon flavors are included in the basis. The second PV photon allows the solution to have a ver smooth, slowl varing dependence on the PV electron mass. This in turn allows the numerical approximation to use smaller mass ratios, which reduces round-off errors. An intermediate calculation that retains the one-photon contribution and the two-photon selfenerg contributions ields a much improved result. Where the older, one-photon calculations of the anomalous moment b S.J. Brodsk et al. differed b 14% from the correct value, inclusion of the self- energ contribution reduces the discrepanc to less than %. PoS(LC008)00 LIGHT CONE 008 Relativistic Nuclear and Particle Phsics Jul 7-11, 008 Mulhouse, France Speaker. c Copright owned b the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence.

2 1. Introduction The anomalous magnetic moment of the electron is predicted b perturbative QED to ver high precision as [1] a e (g )/ = α π ( α π ) ( απ ) ( απ ) 4. This is in excellent agreement with experiment and provides an important check for nonperturbative methods. Here we describe progress toward such a check for a light-cone Hamiltonian method that uses Pauli Villars (PV) particles as the ultraviolet regulators. The electron eigenstate is obtained within a truncated basis, and the anomalous moment is computed from the wave functions. The one-photon truncation has been solved analticall []. The new work involves inclusion of twophoton states, as well as additional analsis of the one-photon truncation at finite PV masses. The regularization is done b including enough PV particles in the Lagrangian. The formulation is then covariant and, at least for QED, gauge invariant. These smmetries are then broken b the truncation of the Fock-state expansion, but the effect of the breaking is expected to be small. The existence of a meaningful Fock-state expansion, with well defined wave functions as the coefficients, is made possible b the choice of light-cone quantization [3]. The vacuum is then simple and the projection of the lower eigenstates onto the higher Fock sectors can be small. This is not the case for equal-time quantization [4]. Usuall in QED, light-cone gauge is chosen, to simplif constraints; however, in this case, three PV electrons were found to be not enough to properl regulate the theor []. There was a strong dependence on PV masses and a lack of equivalence between light-cone perturbation theor and standard covariant perturbation theor. To work in this gauge requires an additional PV photon and higher-order derivative counterterms []. In Fenman gauge, one PV electron and one PV photon are in principle sufficient at our level of truncation. The PV interactions can be arranged to cancel the usual instantaneous electron interactions and to simplif the fermionic constraint equations. This worked well for ver large PV electron masses []. However, if the infinite PV-electron-mass limit is not approached, there remains some strong dependence on the PV photon mass, which can be eliminated b including a second PV photon. This is important for the two-photon truncation, which must be handled numericall and is vulnerable to large round-off errors in the quadratures when the PV electron mass is ver large. The renormalization is handled b requiring the mass M of the lowest eigenstate in the singleelectron sector to be the phsical mass of the electron, m e. This fixes the bare-mass parameter m 0 of the Lagrangian as a function of the PV masses. The coupling α is unrenormalized in our truncation, because there are no antifermion contributions. In practice, the eigenvalue problem is solved at fixed M = m e with α obtained from the eigenvalue, and m 0 is determined b requiring α to take the correct phsical value. This approach differs from that of sector-dependent renormalization [5, 6], where the bare parameters are given different values in each Fock sector. In [6] this has ielded wave functions that are not well defined. The probabilit of the lowest Fock sector becomes negative if the PV masses are too large, and the expectation value for the number of phsical photons is infinite for an choice of PV masses. These difficulties are overcome to some extent b onl computing quantities with respect to some external probe and separatel renormalizing the external coupling. PoS(LC008)00

3 The limitation to finite PV masses occurs also in our approach, due to uncancelled divergences [], but the wave functions alwas ield probabilities between zero and one.. Fenman-gauge light-cone QED The Lagrangian of PV-regulated QED in Fenman gauge is L = where 1 4 ( 1)i F µν i F i,µν 1 ( 1) i ( µ ) 1 A iµ + ( 1) i ψ i (iγ µ µ m i )ψ i e ψγ µ ψa µ, (.1) A µ = ξi A iµ, ψ = 1 ψ i, F iµν = µ A iν ν A iµ. (.) The subscript i = 0 denotes a phsical field and i = 1 or a PV field. The constants ξ i satisf the following constraints: ( 1) i ξ i = 0, ( 1) i ξ i µ i ln µ i = 0, (.3) with ξ 0 = 1. The first constraint guarantees that A µ is a zero-norm field. The second arranges the cancellation of the leading dependence on the PV electron mass,. The coupling of the two zero-norm fields A µ and ψ as the interaction term reduces the fermionic constraint equation to a solvable equation without forcing the gauge field A + to zero. For the null combination ψ 0 + ψ 1 that couples to A + the constraint on the nondnamical components is the same as the constraint for a free fermion. The regularization scheme does have the disadvantage of breaking gauge invariance, through the presence of flavor changing currents where a phsical fermion can be transformed to a PV fermion or vice versa. However, the breaking effects disappear in the limit of large PV fermion mass [], because the phsical fermion cannot make a transition to a state with infinite mass. Without antifermion terms, the result for the Hamiltonian is [] P = i,s + i, j,l,s,µ d p m i + p p + ( 1) i b i,s (p)b i,s(p) + l,µ d pdq dk µ l + k k + ( 1) l ε µ a lµ (k)a lµ(k) (.4) {b i,s (p) [ b j,s (q)v µ i j,s (p,q) + b j, s(q)u µ i j, s (p,q) ] ξl a lµ (q p) + h.c. }, PoS(LC008)00 where the vertex functions are given in []. We work in the frame where the total P is zero. The eigenstate with total J z = ± 1 then has the following Fock-state expansion: ψ ± (P) = z i b i± (P) 0 + dkc µ± i js (k)b is (P k)a jµ (k) 0 (.5) i i jsµ + dk 1 dk C µν± i jks (k 1 1,k ) b i jksµν 1 + δ is jk δ (P k 1 k )a jµ (k 1)a kν (k ) , µν The wave functions C µ± i js are b determined the mass eigenvalue problem P + P P = M P. As it stands, the eigenstate can lead to unphsical answers, because the PV contributions have negative norm. We extract a phsical state b means of a projection onto the phsical subspace [7]. 3

4 The projection is accomplished b expressing Fock states in terms of positivel normed creation operators a 0µ, a µ, and b 0s and the null combinations a µ = i ξi a iµ and b s = b 0s + b 1s. The b s particles are annihilated b the generalized electromagnetic current ψγ µ ψ; thus, b s and a µ create unphsical contributions that are dropped. The anomalous moment is computed onl after making this projection. 3. Solution of the eigenvalue problem and In the one-photon truncation, the wave functions satisf the coupled integral equations [M m i ]z i = dk [ ] ξl ( 1) j+l ε µ P + V µ ji+ (k,p)cµ+ jl+ (k) + P+ U µ ji+ (k,p)cµ+ jl (k) j,l,µ [ M m i + k (1 ) µ l + k [ M m i + k (1 ) µ l + k ] C µ± (3.1) il± (k) = ξ l ( 1) j z j P + V µ i j± (k,p), (3.) ] C µ± j il (k) = ξ l ( 1) j z j P + U µ i j± (k,p). (3.3) The wave functions C µ± il± are found explicitl in terms of the bare amplitudes z i, b inverting the second set of equations, and then eliminated from the first set, to obtain, with use of the definitions of the vertex functions [], Here j (M m i )z i = e ( 1) j [ M z j J + m i z j m j Ī 0 (3.4) j Ī n = ddk 16π J ddk = 16π jl jl ( 1) j+l ξ l M m j +k 1 ( 1) j+l ξ l M m j +k 1 Mz j (m i + m j )Ī 1 ]. µ l +k µ l +k (m j /M) n (1 ) n, (3.5) (m j + k )/M (1 ). (3.6) PoS(LC008)00 The form of Eq. (3.4) matches that of the equivalent eigenvalue problem in Yukawa theor [7], with M used as the mass scale instead of µ 0 and with the replacements g e and I 1 Ī 1. Consequentl, the solution for the one-photon truncated eigenvalue problem is α ± = (M ± m 0)(M ± ) 8πM( m 0 )(Ī 1 ± Ī 0 ), z 1 = M ± m 0 M ± z 0, (3.7) with z 0 determined b normalization. The plot in Fig. 1 shows α ± /α as functions of m 0. The α branch is the phsical choice, because the no-interaction limit (α = 0) corresponds to the bare mass m 0 becoming equal to the phsical electron mass. If the PV electron has a sufficientl large mass, the value of m 0 that ields 4

5 α + α α +/ /α m 0 /m e Figure 1: The two solutions of the one-photon eigenvalue problem, for PV mases = 1000m e, µ 1 = 10m e, and µ =. The horizontal line shows where α ± = α. The α branch corresponds to the phsical choice, but with m 0 less than m e PoS(LC008)00 (π/α) a e 1.0 = infinit = 10 4 m e 0.9 = 10 5 m e = 10 6 m e = 10 7 m e µ 1 /m e Figure : The anomalous moment of the electron in units of the Schwinger term ( π α ) plotted versus the PV photon mass, µ 1, for a few values of the PV electron mass,, with the second PV photon mass, µ, set to infinit. 5

6 α = α is less than m e. In this case, the integrals Ĩ n and J contain poles for j = l = 0 and are defined b a principal-value prescription []. When the second PV photon is removed via the µ limit, the results for the anomalous moment have a ver strong dependence on the PV electron mass, as can be seen in Fig.. This strong dependence arises from the sensitivit of the anomalous moment integral to the masses of the constituents, in particular the bare electron mass m 0. The leading finite- correction to the bare electron mass is of the form µ 1 ln(µ 1/ ) 8π m e. This requires m 1 to be much larger than µ 1 /m e. Such behavior comes from the contribution of Ĩ 1 to the relationship between m 0 and α in (3.7). Relative to the value Ĩ 1 ( ) at infinite, we have Ĩ 1 Ĩ 1 ( ) + j=1 ξ j ( 1) j µ j ln(µ j/ ) 8π m e. (3.8) The second constraint (.3) on the coupling factors ξ j reduces this leading correction to zero. In the two-photon truncation, we can eliminate the three-bod wave function and the barefermion amplitudes to obtain 48 coupled equations for the two-bod wave functions π α f µ± i js (,q ) = I i j,1 i(,q ) η 1 i, j (,q ) f µ± 1 i, js 1 (,q ) (3.9) 1 + ε µ d dq J(0)µµ i js,i i j s µ 0 j s (,q ;,q )η i j (,q ) f µ ± i j s (,q ) + ε µ i j s µ 1 0 d dq J()µµ i js,i j s (,q ;,q )η i j (,q ) f µ ± i j s (,q ), where I i jk is the self-energ contribution and J (n)µµ i js,i j s is the kernel with n intermediate photons. The diagonal self-energ contribution is included in the redefined wave functions f µ± i js (,q = C µ± i js (,q )/(M η i j (,q )), (3.10) PoS(LC008)00 with (1 ) η i j (,q ) = q + (m i + I i ji(,q ) + (1 )µ. (3.11) j (1 )M The full solution of this eigenvalue problem is in progress. As an intermediate step, we can solve it with onl the two-photon self-energ corrections, that is, with J () set to zero. The solution is analtic up to the evaluation of certain integrals, which can be done numericall with some care and effort, especiall with respect to poles, to an accurac of order 1%. The anomalous moment computed from this solution is shown in Fig. 3, along with the one-photon-truncation results. We see that, although the one-photon result differs b 14% [], our new result, the filled circles of Fig. 3, is consistent with perturbative QED, showing onl the variations expected from the numerical errors in calculating the integrals. Acknowledgments This work was supported in part b the US Department of Energ and the Minnesota Supercomputing Institute. 6

7 (π/α) a e µ 1 /m e = infinit =*10 4 m e =5*10 4 m e =10 5 m e =*10 4 m e, with self-energ Figure 3: Same as Fig. but with µ = µ 1 and with inclusion of the effect of keeping the two-photon self-energ corrections. References [1] M. Maggiore, A Modern Introduction to Quantum Field Theor, Oxford Universit Press, Oxford 005. [] S.J. Brodsk, V.A. Franke, J.R. Hiller, G. McCartor, S.A. Paston, and E.V. Prokhvatilov, A nonperturbative calculation of the electron s magnetic moment, Nucl. Phs. B 703, 333 (004). [3] P.A.M. Dirac, Forms of relativistic dnamics, Rev. Mod. Phs. 1, 39 (1949). [4] S.J. Brodsk, H.-C. Pauli, and S.S. Pinsk, Quantum chromodnamics and other field theories on the light cone, Phs. Rep. 301, 99 (1998). [5] R.J. Perr, A. Harindranath, and K.G. Wilson, Light-front Tamm Dancoff field theor, Phs. Rev. Lett. 65, 959 (1990); R.J. Perr and A. Harindranath, Renormalization in the light-front Tamm Dancoff approach to field theor, Phs. Rev. D 43, 4051 (1991); J.R. Hiller and S.J. Brodsk, Nonperturbative renormalization and the electron s anomalous moment in large-α QED, Phs. Rev. D 59, (1998). [6] V.A. Karmanov, J.-F. Mathiot, and A.V. Smirnov, Sstematic renormalization scheme in light-front dnamics with Fock space truncation, Phs. Rev. D 77, (008). [7] S.J. Brodsk, J.R. Hiller, and G. McCartor, The mass renormalization of nonperturbative light-front Hamiltonian theor: An illustration using truncated, Pauli Villars-regulated Yukawa interactions, Ann. Phs. 305, 66 (003). PoS(LC008)00 7