Pauli form factors of electron and muon in nonlocal quantum electrodynamics
aa r X i v : . [ h e p - ph ] J a n Pauli form factors of electron and muon in nonlocal quantum electrodynamics
Fangcheng He and P. Wang
1, 2 Institute of High Energy Physics, CAS, Beijing 100049, China Theoretical Physics Center for Science Facilities, CAS, Beijing 100049, China (Dated: January 3, 2019)Pauli form factors of electron and muon are studied in nonlocal quantum electrodynamics. Wecalculate one loop QED correction to their Pauli form factors. The relativistic regulator is generatedby the correlation function in the nonlocal interaction. The cut-off parameter Λ in the regulatoris determined to get the consistent anomalous magnetic moments of electron and muon at thesame level as local QED. When momentum transfer is large, there exists obvious difference betweennonlocal and local QED.
I. INTRODUCTION
The anomalous magnetic moments of electron and muon a e and a µ are among the most precisely determinedobservables in particle physics. The most accurate measurement of a e so far has been carried out by the Harvardgroup as a e = 1159652180 . × − [1, 2]. Further improvements for the electron and positron measurementsare currently prepared by the Harvard group [3]. For the muon magnetic moment, the E821 measurement at BNL [4],corrected for updated constants [5, 6], is a µ = 116592089(63) × − . Two next generation muon g − a e = ( − ± × − , (1)which shows a 2 . σ deviation. For muon, the anomalous magnetic moment a µ has 3.7 σ discrepancy with a positivesign, opposite to the a e deviation [10, 14]. The value of ∆ a µ is∆ a µ = (2 . ± . × − . (2)As standard model predictions almost without exception match perfectly all other experimental information, thedeviation in one of the most precisely measured quantities in particle physics remains a mystery and inspires theimagination of model builders. In this paper, we will investigate the anomalous form factors of electron and muonwith the nonlocal quantum electrodynamics inspired from the nonlocal effective field theory (EFT).Nonlocal effective field theory was recently proposed to study hadron physics. Different from the traditional EFT,the interaction is nonlocal which reflects the non-point behavior of hadrons. Therefore, in the nonlocal Lagrangian,there is a correlation function F ( x − y ), where the baryon is located at x and meson at y . If F ( x − y ) is chosento be δ ( x − y ), the Lagrangian will be changed back to the local one. With the correlation function, there is noultraviolet divergence in the loop intergral. The nonlocal EFT has been applied to study the nucleon electromagneticform factors, strange form factors, parton distributions, etc [15–18].The nonlocal Lagrangian may be not only the phenomenological method to deal with the divergence, but also thegeneral property for all the physical interactions. In other words, whether there is divergence or not, the nonlocalregulator always exists. Therefore, with the same idea, the interaction between electron and photon could also benonlocal. In the classical scenario (tree level diagram), the nonlocal effect is certainly negligible for the low momentumtransfer. However, for the quantum fluctuation or loop diagram, the internal photon can detect the structure of thephysical particle since its momentum can be infinite.In this paper, we will show the deviation of the lepton Pauli form factors in nonlocal QED from local one. It canbe seen that high order QED corrections and hadronic effects do not affect the final conclusion since the deviationis significant, especially at large Q . Compared with the observables in hadron physics, where the non-perturbative p A (q) u p / FIG. 1: Feynman diagram for the one loop vertex correction to the lepton form factors. effect is very important, it is a great advantage for the lepton anomalous form factors to serve as the quantities to teststandard model. In the following, we will first derive the form factors in nonlocal QED and then show the numericalresults.
II. NONLOCAL QED LAGRANGIAN
The nonlocal Lagrangian for quantum electrodynamics is written as L nlQED = ¯ ψ ( x )( i /∂ − m ) ψ ( x ) − F µν ( x ) − e Z d a ¯ ψ ( x ) /A µ ( x + a ) ψ ( x ) F ( a ) , (3)where the electron field ψ ( x ) is located at x and the photon field A µ ( x + a ) is located at x + a . F ( a ) is the correlationfunction normalized as R d aF ( a ) = 1. If it is chosen to be a δ function, the nonlocal Lagrangian will be changedback to the local one. The above nonlocal Lagrangian is invariant under the following gauge transformation ψ ( x ) → e iα ( x ) ψ ( x ) , A µ → A µ − e ∂ µ α ′ ( x ) , (4)where α ( x ) = R daα ′ ( x + a ) F ( a ). Different from the nonlocal Lagrangian for EFT, where the gauge link is introducedto guarantee the local gauge invariance, here we need no gauge link since photon is charge neutral.With the correlation function, the lepton-photon interaction is momentum dependent as eγ µ ˜ F ( q ), where ˜ F ( q ) isFourier transformation of the correlation function F ( a ) and q is photon momentum. Ward-Takahashi identity becomes − iq µ Γ µ ( p + q, p ) = ˜ F ( q )( S − ( p + q ) − S − ( p )) , (5)where Γ µ ( p + q, p ) is the vertex. S ( p + q ) and S ( p ) are the lepton propagators with wave function renormalization S ( p + q ) = iZ /p + /q − m , S ( p ) = iZ /p − m , (6)where Z is lepton wave function renormalization factor. At q = 0, ˜ F ( q ) is 1 due to the normalization of F ( a ) andEq.(5) can be written as Z Γ µ ( p, p ) = γ µ ˜ F (0) . (7)The lepton form factors is defined as [19] Z Γ µ ( p + q, p ) = γ µ F ( q ) + iσ µν q ν m F ( q ) . (8)Therefore, we have F (0) = ˜ F (0) which is consistent with that the renormalized lepton charge is 1.The one loop Feynman diagram for the lepton form factors is plotted in Fig. 1. At one loop level, the vertex iswritten as¯ u ( p ′ )Γ µloop ( p ′ , p ) u ( p ) = ¯ u ( p ′ ) Z d k (2 π ) ˜ F ( q ) ˜ F ( k ) ( − ieγ ν ) i/p ′ − /k − m γ µ i/p − /k − m ( − ieγ ρ ) − ig νρ k u ( p ) (9)From Eq. (8), one can get F loop ( q ) = − ie ˜ F ( q )(4 m − q ) Z d k (2 π ) ˜ F ( k ) (cid:8) − m (( k · p ) + ( k · p ′ ) ) + 8 m k (4 m − q )(( p ′ − k ) − m )( p − k ) − m ) k + 2(2 m − q )(4 m − q ) − m + 2 q )( k · p )( k · p ′ ) − k · p + k · p ′ )(8 m − m q + q )(( p ′ − k ) − m )( p − k ) − m ) k (cid:9) (10)and F loop ( q ) = − ie ˜ F ( q )8 m q (4 m − q ) Z d k (2 π ) F ( k ) (cid:8) (4 m + 2 q )(( k · p ) + ( k · p ′ ) ) − m − q )( k · p )( k · p ′ )(( p ′ − k ) − m )( p − k ) − m ) k + ( q − m q )( k · p + k · p ′ + k )(( p ′ − k ) − m )( p − k ) − m ) k (cid:9) . (11)In the above expressions, the momentum dependent vertexes ˜ F ( q ) and ˜ F ( k ) appear. For ˜ F ( q ), if the externalmomentum q is much smaller than the scale of electron, the size of electron can be neglected and ˜ F ( q ) ≃
1. However,for ˜ F ( k ), the internal momentum k varies from 0 to infinity. The regulator is very important and it makes the loopintegral for F and F both convergent.For the Dirac form factor F ( q ), there is a contribution from tree level as Z ˜ F ( q ). The wave function renormal-ization constant Z is obtained as [19] Z − d Σ( /p ) d/p (cid:12)(cid:12)(cid:12)(cid:12) /p = m , (12)where the lepton self-energy is expressed asΣ( /p ) = − Z d k (2 π ) ˜ F ( k )( ieγ ν ) 1 /p − /k − m ( − ieγ ρ ) − ig νρ k . (13)It is straightforward to find d Σ( /p ) d/p = − F loop (0). Again, we have F (0) = Z + F loop (0) = 1.In the following, we focus on the Pauli form factor F ( q ) and F (0) gives the anomalous magnetic moment. Forthe numerical calculation, the regulator ˜ F ( k ) is chosen to be a dipole form as˜ F ( k ) = Λ ( k − Λ ) . (14)The Pauli form factor at one loop level can be obtained as F nl ( Q ) = α π ˜ F ( − Q ) Z dx Z − x dy m ( x + y )(1 − x − y ) ( Q xy + m ( x + y ) )( m ( x + y ) + Q xy + (1 − x − y )Λ ) . (15)When Λ goes to infinity, the local result will be recovered as F lo ( Q ) = α π Z dx Z − x dy m (1 − x − y )( x + y ) m ( x + y ) + Q xy , (16)which results in the well known anomalous magnetic moment at one loop level a l = F lo (0) = α π [20]. In the aboveequations, we replaced q by − Q for convenience. With the results from nonlocal and local QED, we can get thediscrepancy between them as ∆ F = F lo − F nl . The relative discrepancy is defined as R = ∆ F /F lo . The loopintegral for Pauli form factors are both ultraviolet convergent in local and nonlocal cases. We should mention that - - - - Q ( TeV ) F e ( Q ) - - - - - - - - - - Q ( TeV ) F e ( Q ) FIG. 2: Electron Pauli form factor F e ( Q ) versus momentum transfer Q . The solid, dashed and dotted lines are for Λ = 0 . . F ( Q ) at low Q up to 0 .
001 TeV . the regulator is not introduced phenomenologically to deal with the divergence. This is different from the originalfinite-range-regularization [21–36]. The regulator is naturally generated from the nonlocal Lagrangian with the naiveidea that the interaction between photon and lepton does not necessary take place at one point. For the ultravioletdivergent integral at local case, the regulator will make the integral convergent. For the integral which is convergent atlocal case, the regulator also exists and will give obvious deviations from the local result, especially at large momentumtransfer. III. NUMERICAL RESULTS
In the numerical calculation, there is one free parameter Λ in the regulator needs to be determined. Λ is order of1 GeV for nucleon. For leptons, Λ could be much larger since their sizes are much smaller. Certainly, on the onehand, the smaller the value of Λ, the larger the deviation from the standard model. On the other hand, Λ shouldbe large enough and make the nonlocal results consistent with the experiments at the same level as standard model.When Λ = 0 . a nle in nonlocal QED is 0.00116171491307, which is 2 . × − deviation fromthe corresponding value in standard model. Considering the experimental accuracy 2 . × − [8] and discrepancybetween experiments and SM prediction 8 . × − , the choice of Λ = 0 . a µ in nonlocalQED is 8 . × − deviation from that in local case. Comparing with ∆ a µ is 2 . × − , Λ = 0 . . . Q . We will show that whatever Λ is, the relative discrepancy between local and nonlocalQED is always significant if momentum transfer Q is large enough.In Fig. 2 we plot the Pauli form factor of electron F e ( Q ) versus Q . The solid, dashed and dotted lines are forΛ = 0 .
2, 0 . Q . Itcan be seen that the form factor decreases very fast with the increasing Q due to the fact that the electron mass isvery small. When Q is small, the discrepancy between nonlocal QED and SM is much smaller than the form factorsthemselves. For example, at Q = 0, the discrepancy ∆ F e is at least 10 times smaller than the anomalous magneticmoments. With the increasing Q , the discrepancy is clearly shown in the figure when its value is comparable withthe form factors. Due to the nonlocal effect, the form factor in nonlocal QED is smaller than that in SM at any Q .The result for muon is shown in Fig. 3. Similarly as for electron, the form factor drops quickly with the increasing Q . Since the mass of muon is larger than that of electron, the form factor drops slower than electron. At Q = 0,the form factors of electron and muon are close to each other (In SM, they are exactly the same at leading order).However, at finite Q , they have huge difference. For example, for 0 . < Q < .
01 TeV , the muon form factor isabout 4 − − R in Fig. 4. Thesolid and dashed lines are for Λ = 0 . . F and the form factor F are - - - - - Q ( TeV ) F μ ( Q ) - - - - - - - - - Q ( TeV ) F μ ( Q ) FIG. 3: Same as Fig.2 but for muon Pauli form factor. Q ( TeV ) R ( Q ) FIG. 4: The relative deviation R ( Q ) for electron and muon versus Q . The solid and dashed lines are for Λ = 0 . . both much larger for muon than that for electron. The relative deviation R are almost the same for electron andmuon. This can be seen from Eqs. (15) and (16). The lepton mass dependence in the numerator cancels each other for F nl /F lo . The mass in the denominator is much smaller than Λ which makes R nonsensitive to the mass. At Q = 0,the relative deviation is very small and is order of 10 − and 10 − for electron and muon. This is more or less thesame order for the relative deviation of SM from the experiments. Therefore, to confirm this discrepancy for electron,the experiment should be very accurate to get 10 more effective digits. The relative deviation R increases with theincreasing Q . For example, at Q = 0 .
01 TeV , R is about 0.37 and 0.08 for Λ = 0 . Q is larger than 0.1 TeV , R is larger than 0.5 for both Λs. This means the nonlocal value of F nl is less thanhalf of the SM value. For even larger Q , say larger than 0.2 TeV , F nl could be one magnitude smaller than theSM value. We can see though the absolute value of the form factor is small, the relative deviation of nonlocal QEDfrom SM is very large. Even if the experiment can only measure the form factor with one effective digit at finite Q ,one can still conclude whether there is physics beyond SM. Since the deviation is so large at finite Q , the conclusionis not changed by the high order QED correction or hadronic effect, as we know for both electron and muon, thesecorrections are less than one percent. In summary, we list the discrepancy between nonlocal QED and SM and therelative deviation at some Q for electron and muon in Table I. TABLE I: The discrepancy between nonlocal QED and SM and the relative deviation for electron and muon. Q (TeV ) 0 0 .
001 0 .
01 0 .
05 0 . . F e (Λ = 0 . . × − . × − . × − . × − . × − . × − ∆ F e (Λ = 0 . . × − . × − . × − . × − . × − . × − R e (Λ = 0 . . × − R e (Λ = 0 . . × − F µ (Λ = 0 . . × − . × − . × − . × − . × − . × − ∆ F µ (Λ = 0 . . × − . × − . × − . × − . × − . × − R µ (Λ = 0 . . × − R µ (Λ = 0 . . × − IV. SUMMARY
We studied the anomalous form factors of electron and muon in nonlocal QED inspired by the nonlocal effective fieldtheory for hadron physics. The interaction between lepton and photon is describe by the nonlocal QED. The regulatoris generated from the correlation function in the nonlocal Lagrangian. For the Dirac form factor of leptons andelectromagnetic form factors of nucleon, the ultraviolet divergence of loop integral in local interaction will disappearwith the regulator. For the Pauli form factors of electron and muon, the loop integrals are both convergent fornonlocal and local QED. The parameter Λ is chosen to make the nonlocal result of lepton anomalous magneticmoments consistent with the experimental data at the same level as SM. The form factors of the electron and muondecrease very fast with the increasing Q because of the small lepton masses. At Q = 0, the absolute discrepancybetween nonlocal and local results is more or less similar as the discrepancy between experiments and SM predictions.However, the relative deviation from the SM is large at finite Q . Since the high order QED correction and hadroniceffect is less than one percent, the large relative deviation is not affected. If this deviation can be measured, it willdefinitely indicate new physics beyond SM. It will also imply that nonlocal behavior could be the general propertyfor all the interactions and as a result, we will have to reconsider the regularization and renormalization. Acknowledgments
This work is supported by the National Natural Sciences Foundations of China under the grant No. 11475186,the Sino-German CRC 110 “Symmetries and the Emergence of Structure in QCD” project by NSFC under the grantNo.11621131001, and the Key Research Program of Frontier Sciences, CAS under grant No. Y7292610K1. [1] D. Hanneke, S. Fogwell, and G. Gabrielse
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