Constraints on new physics from an improved calculation of parity violation in 133 Cs
aa r X i v : . [ h e p - ph ] A ug Constraints on new physics from an improved calculation of parity violation in Cs B. K. Sahoo ∗ Atomic, Molecular and Optical Physics Division,Physical Research Laboratory, Navrangpura, Ahmedabad-380009, India
B. P. Das
Department of Physics, School of Science, Tokyo Institute of Technology,2-1-2-1-H86 Ookayama Meguro-ku, Tokyo 152-8550, JapanCentre for Quantum Engineering Research and Education,TCG Centres for Research in Science and Technology, Sector V, Salt Lake, Kolkata 70091, India (Dated: Received date; Accepted date)We report the result of our calculation of the nuclear spin-independent parity violating electricdipole transition amplitude ( E PV ) for the 6 s S / − s S / transition in Cs to an accuracyof 0.3% using a variant of the perturbed relativistic coupled-cluster (RCC) theory. In the presentwork, we treat the contributions of both the low-lying and high-lying excited states to the abovementioned amplitude on the same footing, thereby overcoming the limitations of previous highaccuracy RCC calculations. We obtain an accurate value for the vector polarizability ( β ) for theabove transition and by combining it with the results from our present calculation of E PV andthe latest measurement of Im ( E PV /β ), we extract the nuclear weak charge ( Q W ); and analyze itsdeviation from its value in the Standard Model (SM) in order to constrain certain scenarios of newphysics beyond the SM. I. INTRODUCTION
Atomic processes are usually studied by consideringthe exchange of photons ( γ ) between the bound electronsand the nucleus and the bound electrons themselves. Thelongitudinal γ s are responsible for the Coulomb interac-tion, which is the dominant contribution to electromag-netic interactions in atomic systems. However, the Breitinteraction [1] due to the transverse γ s and quantumelectrodynamics (QED) effects must also be consideredin high precision atomic calculations. All these interac-tions preserve parity symmetry, and these systems are de-scribed conveniently using spherical coordinates [2]. In-clusion of the neutral current weak interactions due tothe exchange of Z boson in atomic systems leads to par-ity violation [3], and this phenomenon has been referredto as atomic parity violation (APV). Depending on theelectron-nucleus vector–axial-vector (V-A) or the axial-vector–vector (A-V) currents, APV interactions can benuclear spin ( I ) independent or dependent. In additionto the Z exchange interactions, the possible interactionof the nuclear anapole moment with electrons can giverise to APV that depends on I [4]. The I dependentAPV contributions are relatively smaller than its NSIcounterpart, as the odd-nucleon contributes primarily toAPV. In the case of the nuclear spin independent (NSI)interaction, the nucleons which ultimately arise from theup- and down-quarks, contribute coherently, and thesecontributions are several orders of magnitude larger thanthose due to the exchange of Z between the electrons[3–6]. Nonetheless, the NSI APV interaction is too weakto be detected using typical spectroscopic measurements, ∗ [email protected] and therefore, special techniques have been developed indifferent laboratories to observe effects due to it [7–11].Heavier atomic systems are preferred for measuring thesubtle NSI APV effects owing to the fact that the weakinteraction causing them in atomic systems scales slightlyfaster than Z [3], where Z is the atomic number. Thesemeasurements in combination with high-precision atomiccalculations have the potential to probe physics beyondthe Standard Model (SM) of elementary particles [12–16].APV has been measured to an accuracy of 0.35% inthe 6 s S / − s S / transition in Cs [9]. This isthe most accurate APV measurement to date. New ex-periments have been proposed to measure APV in Cs[17, 18], which have the potential to surpass the accu-racy of this measurement. The stage is now clearly setto take the APV calculations in Cs to the next level,which would lead to an improvement in the accuracy of Q W , thereby making it possible to probe new physicsbeyond the SM. This indeed provides the motivation forour present work. The principal quantity of interest inthe APV studies is the nuclear weak charge ( Q W ), whichis a linear combination of the coupling coefficients be-tween electrons, and up- and down-quarks in an atomicsystem [3, 10]. The difference in the model independentvalue of Q W obtained from APV and that obtained fromthe SM could, in principle, shed light on new physics be-yond the SM. The APV study in Cs currently yieldssin θ exp W = 0 . θ SM W = 0 . θ W , at the zero momentum transfer in the MS scheme.The difference in the values of Q W also gives the lowermass limit of an extra Z χ boson as M Z χ >
710 GeV /c and weak isospin conserving parameter S = − . σ level [19]. It has been argued that the signature ofa dark boson ( Z d ) (also referred to as dark photon in theliterature) can be obtained from the above mentioned dif-ference [21–23]. Direct experimental signature suggestsits value to be less than 0.2 GeV [25]. A fairly recentmanuscript [24] suggests that the limit on an effectiveelectron-nucleus coupling describing new physics beyondthe SM expressed as f eff Vq Λ < . × − GeV − witha new energy scale Λ and emphasizes that the bound onthe effective couplings inferred from APV is more strin-gent than the ones from the neutrino-nucleus coherentscattering processes.State-of-the art relativistic atomic many-body theorieshave been applied to Cs APV calculations in the lastfour decades [19, 26–31]. The accuracy of the calcula-tions have steadily improved, as the theoretical methodshave been able to incorporate larger classes of higher-order effects during this time due to advances in highperformance computing. The latest two high precisioncalculations have been reported in Refs. [19, 31]. Thesecalculations divided the entire electron correlation con-tribution into three parts and they are calculated bymixed many-body methods. Further, the dominant partwas evaluated through sum-over-states approach and theother contributions were not treated on the same footing.Contributions from the Breit and QED effects were takenfrom the earlier works, but not double core-polarization(DCP) effects [32]. Some of these issues triggered dis-cussions recently [16, 33], and therefore, it is necessaryto revisit this problem. In this work, we intend to cir-cumvent the above mentioned limitations of the previ-ous calculations by solving the first-order perturbed wavefunctions due to the APV interaction for atomic states inthe framework of the relativistic coupled-cluster (RCC)theory. Thus, it considers both the electromagnetic andweak interactions simultaneously in addition to account-ing for Coulomb, Breit and QED interactions using thesame many-body method. Most importantly, it treats allthe three different parts of the total correlation contribu-tion mentioned above on the same footing.
II. THEORY
Neglecting the A-V interaction, the short range effec-tive Lagrangian corresponding to the V-A neutral weakcurrent interaction of an electron with up- and down-quarks in an atomic system is given by [6, 7] L VAeq = G F √ X u,d (cid:2) C u ¯ ψ u γ µ ψ u + C d ¯ ψ d γ µ ψ d (cid:3) ¯ ψ e γ µ γ ψ e = G F √ X n C n ¯ ψ n γ µ ψ n ¯ ψ e γ µ γ ψ e , (1)where G F = 1 . × − GeV − is the Fermi constant,sums u , d and n stand for up-quark, down-quark and nu-cleons respectively, and C i = u,d,n represent coupling co-efficients of the interaction of an electron with quarksand nucleons. Adding them coherently and taking the non-relativistic approximation for nucleons, the temporalcomponent gives the NSI weak interaction Hamiltonian H VAen = − G F √ Q W ρ ( r ) + ( N C N − ZC P )∆ ρ ( r )] γ , (2)where N and P representing for neutron and proton re-spectively, ρ ( r ) = ( ρ N ( r ) + ρ P ( r )) / ρ P ( r ) andnormalized neutron density ρ N ( r ), ∆ ρ ( r ) = ρ N ( r ) − ρ P ( r ), and Q W = 2[ ZC P + N C P ] is known as the nu-clear weak charge. In the atomic calculations, contribu-tion from ∆ ρ ( r ) is neglected at first, but is added lateras “nuclear skin” correction. The nuclear skin correctionto Q W is expressed as [34]∆ Q N − PW = 0 . N ( Zα e ) q P tr P , (3)where α e is the fine-structure constant, r i = P ( N ) are theroot mean square radius of proton (neutron), t = r N − r P is the neutron skin, and q P is defined as q P = Z d rf ( r ) ρ P ( r ) (4)with the electronic form factor f ( r ) that describes thespatial variation of the electronic axial-vector matrix el-ement over the size of the nucleus.The NSI weak interaction Hamiltonian for atomic cal-culations, thus, is given by [3] H NSIAP V = X e H AVen = − G F √ Q at W X e γ e ρ ( r e ) , (5)where Q at W = Q W − ∆ Q N − PW . It is obvious that Q W isa model dependent quantity. Thus, the difference of itsactual value from the SM, given by ∆ Q W = Q exp W − Q SM W ,can provide signatures about new physics. In the SM, C u = (cid:2) − sin θ SM W (cid:3) and C d = − (cid:2) − sin θ SM W (cid:3) [5–7, 10]. This follows C N = 2 C d + C u = − / C P = 2 C u + C d = (1 − θ SM W ) / ≈ . θ W varies with energy scale (denoted by Q ) and is parameterized in the MS scheme as [14]sin θ W ( Q ) = κ ( Q ) sin θ W ( M Z ) MS , (6)where M Z is the mass of Z -boson and κ ( Q ) denotesperturbative γ – Z -boson mixing. For the normalization κ ( Q ≡ M Z ) = 1 .
0, it corresponds to κ (0) ∼ .
03 [14].In the one-loop radiative correction, the mass of W -bosonand sin θ W ( m Z ) MS are given by [14] M W = 80 . − . S + 0 . T ] GeV /c (7)andsin θ W ( m Z ) MS = 0 . . S − . T ] , (8)where c is speed of light, and S and T are the isospinconserving and isospin breaking parameters, respectively.By comparing the above expression for M W with its ex-perimental value of 80 . c [20], it gives [22] S = 0 . ± .
09 and T = 0 . ± . . (9)Also, M Z χ can be obtained in the SO(10) model as [12]∆ Q W ≈ . × (2 N + Z )( M W /M Z χ ) . (10)In the Z and Z d mixing of two-Higgs doublet modelscenario, we get [12, 22, 23]sin θ exp W (0) − sin θ SM W (0) ≃ − . εδ M Z M Z d , (11)where ε and δ are the model dependent parameters, and M Z d is mass of Z d .Accounting for all the aforementioned possible physicsbeyond the SM, the weak charge of Cs atom can beexpressed in terms of all the combined parameters as Q W ( Cs) = Q SM W ( Cs) × [1 + 0 . S − . T − . M Z /M Z χ ) − . εδ ( M Z /M Z d )] , (12)where Q SM W ( Cs) = − N + Z (1 − θ SM W ) = − . Cs in the SM [20].In an effective description [24], ∆ Q W is encoded usinga new energy scale Λ as∆ Q W = 2 √ G F f eff V q ( Z + N ) , (13)where f eff V q = C u (2 Z + N ) + C d ( Z + 2 N )3( Z + N ) . (14)Similarly, Q W can be expressed in terms of thenucleon-electron V-A couplings as [20] Q W = − Zg ep + N g en + 0 . (cid:16) − α e π (cid:17) , (15)where g ep ( n ) are the electron-proton(neutron) couplingconstants. The SM offers 55 g ep + 78 g en = 36 . III. ATOMIC CALCULATIONSA. General aspects
The atomic wave function ( | Ψ v i ) of a state in Cs atomis calculated by dividing the total Hamiltonian as H = H em + λH w , (16)where H em represents the dominant electromagnetic in-teractions in the atom and H NSIAP V ≡ λH w with λ = G F √ Q at W . The electric dipole transition amplitude be-tween the same nominal parity states | Ψ i i and | Ψ f i statesdue to the presence of H NSIAP V can be written as E P V = h Ψ f | D | Ψ i i p h Ψ f | Ψ f ih Ψ i | Ψ i i . (17) Since the strength of H NSIAP V is much weaker than thatof the H em in an atomic system, the wave function fora state (say, | Ψ v i ) corresponding to the total Hamilto-nian H = H em + λH w and its energy (say, E v ) can beexpressed as | Ψ v i = | Ψ (0) v i + λ | Ψ (1) v i + O ( λ ) (18)and E v = E (0) v + λE (1) v + O ( λ ) , (19)where the superscripts 0 and 1 stand for the zeroth-orderand first-order contributions due to H w , respectively. Byneglecting O ( λ ) contributions, we get E P V ≃ λ h Ψ (1) f | D | Ψ (0) i i + h Ψ (0) f | D | Ψ (1) i i q h Ψ (0) f | Ψ (0) f ih Ψ (0) i | Ψ (0) i i . (20)As mentioned before, the previous two high-precision cal-culations of E P V were evaluated using the sum-over-states approach by expanding the first-order wave func-tion as | Ψ (1) v i = X I = v | Ψ (0) I i h Ψ (0) I | H w | Ψ (0) v i E (0) v − E (0) I , (21)where I denotes all possible intermediate states, thatcan be divided into core states (contributions from thesestates are designated as “Core”), low-lying bound states(contributions from these states are given as “Main”),and the remaining high-lying states including continuum(whose contributions are mentioned as “Tail”) for compu-tational simplicity. The drawback of this approach is thatin an actual calculation, it is possible to evaluate “Main”contributions from only a few low-lying valence excitedbound states accurately by calculating them individuallyusing a powerful many-body method, and the “Core”and “Tail” contributions are estimated using less rigor-ous many-body methods. Therefore, the results from thelatter two sectors are less accurate. In other words, thisapproach of evaluating correlation effects in a piecemealmanner does not take into account certain types of cor-relation effects. As a consequence, contributions fromeffects such as the DCP are completely excluded. Keep-ing in mind the high accuracy needed for APV to achieveits ultimate objective of probing new physics beyond theSM, it is desirable to include contributions from all theintermediate states on an equal footing. This can beaccomplished not by summing over intermediate states,but rather by obtaining the first-order perturbed wavefunctions for the initial and final states directly.From the equation H | Ψ v i = E v | Ψ v i , the inhomoge-neous equation for the first-order wave function is ob-tained as( H em − E (0) v ) | Ψ (1) v i = ( E (1) v − H w ) | Ψ (0) v i , (22)where E (1) v = 0 in the present case owing to the odd-parity nature of H w . Obtaining | Ψ (1) v i directly by solvingthe above equation can implicitly include contributionsfrom all the intermediate states I of Eq. (21), thereby,overcoming the problem of unequal treatment of vari-ous electron correlation effects from different sectors asmentioned above. Moreover, it is also necessary to ac-count for correlation effects involving both the weak andelectromagnetic interactions. Therefore, it is very im-portant to consider a powerful and versatile many-bodytheory to obtain both | Ψ (0) v i and | Ψ (1) v i accurately. SinceCs is a heavy atom, it is necessary to employ a rela-tivistic method for computing the wave functions of thisatom. The coupled-cluster (CC) theory is currently con-sidered to be one of the leading quantum many-bodymethods and has been referred to as the gold standard fortreating electron correlation effects in atomic and molec-ular systems [35–37]. Thus, the relativistic version ofthe CC (RCC) theory is very well suited for the accu-rate evaluation of the correlation effects in E P V for the6 s S / − s S / transition in Cs.
B. Atomic Hamiltonian
The starting point of our calculation is the Dirac-Coulomb (DC) Hamiltonian [38] representing the leadingorder contributions to H em to calculate the zeroth-orderwave functions and energies which in atomic units (a.u.)is given by H DC = X i (cid:2) c α i · p i + ( β i − c + V n ( r i ) (cid:3) + X i,j>i r ij , (23)where α and β are the usual Dirac matrices, p is thesingle particle momentum operator, V n ( r ) denotes thenuclear potential, and P i,j r ij represents the Coulombpotential between the electrons located at the i th and j th positions. It should be noted that the above Hamil-tonian is scaled with respect to the rest mass energies ofelectrons. Contributions from the Breit interaction [39]to H em is determined by including the following potential V B = − X j>i [ α i · α j + ( α i · ˆr ij )( α j · ˆr ij )]2 r ij , (24)where ˆr ij is the unit vector along r ij .Contributions from the QED effects to H em are esti-mated by considering the lower-order vacuum polariza-tion (VP) interaction ( V V P ) and the self-energy (SE)interactions ( V SE ). We account for V V P through theUehling [40] and Wichmann-Kroll [41] potentials ( V V P = V Uehl + V W K ), given by V Uehl = − X i α e r i Z ∞ dx x ρ ( x ) Z ∞ dt p t − × (cid:18) t + 12 t (cid:19) h e − ct | r i − x | − e − ct ( r i + x ) i (25)and V W K = X i . Z πc (1 + (1 . cr i ) ) ρ ( r i ) , (26) respectively.The SE contribution V SE is estimated by including twoparts [42] V efSE = A l X i πZα e r i I ef ( r i ) − B l X i α e r i I ef ( r i ) (27)known as the effective electric form factor part and V mgSE = − X k iα e γ · ∇ k r k Z ∞ dx x ρ n ( x ) Z ∞ dt t √ t − × h e − ct | r k − x | − e − ct ( r k + x ) − ct ( r k + x − | r k − x | ) i , (28)known as the effective magnetic form factor part. In theabove expressions, we use [43] A l = ( .
074 + 0 . Zα e for l = 0 , .
056 + 0 . Zα e + 0 . Z α e for l = 2 , (29)and B l = ( . − . x − . x + 0 . x for l = 0 ,
10 for l ≥ . (30)The integrals are given by I ef ( r ) = Z ∞ dx x ρ n ( x )[( Z | r − x | + 1) e − Z | r − x | − ( Z ( r + x ) + 1) e − ct ( r + x ) ] (31)and I ef ( r ) = Z ∞ dx x ρ n ( x ) Z ∞ dt √ t − (cid:26) (cid:18) − t (cid:19) × (cid:20) ln( t −
1) + 4 ln (cid:18) Zα e + 12 (cid:19)(cid:21) −
32 + 1 t (cid:9) × { α e t h e − ct | r − x | − e − ct ( r + x ) i + 2 r A e r A ct × [ E (2 ct ( | r − x | + r A )) − E (2 ct ( r + x + r A ))] (cid:27) (32)with the orbital quantum number l of the system, x =( Z − α e , r A = 0 . Z α e , and the exponential integral E ( r ) = R ∞ r dse − s /s .We have determined the nuclear potential and densityby assuming a Fermi-charge distribution given by [44] ρ n ( r ) = ρ e ( r − b ) /a (33)for the normalization factor ρ , the half-charge radius b = 5 . a = 2 . / ln
3) is related tothe skin thickness.
C. RCC theory for unperturbed wave function
In the RCC theory framework, the unperturbed wavefunction of an atomic system with a closed-core and avalence orbital like in the case of Cs atom due to H em can be expressed as [46, 47] | Ψ (0) v i = e T (0) n S (0) v o | Φ v i , (34)where | Φ v i is the reference wave function, which is ob-tained by solving Dirac-Hartree-Fock (DHF) wave func-tion of the closed-core ( | Φ i ) and then, appending thecorresponding valence orbital v to it as | Φ v i = a † v | Φ i . T (0) and S (0) v are the core and the valence excitationoperators with the superscript 0 represents absence ofany external perturbation. The amplitudes of the un-perturbed RCC operators and energies are obtained bysolving the following equations (see, e.g. [48–50]) h Φ K | ¯ H em | Φ i = δ K, E (0)0 (35)and h Φ Mv | ¯ H em { S (0) v }| Φ v i = E (0) v h Φ Mv |{ δ M,v + S (0) v }| Φ v i , (36)where ¯ H em = e − T (0) H em e T (0) , the superscripts K and M represent the K th and M th excited state determinantswith respect to their respective reference states | Φ i and | Φ v i , E is the energy of the closed-core (i.e. Cs + ) and E v is the energy of a neutral state of Cs atom. Theseenergies are determined by E (0)0 = h Φ | ¯ H em | Φ i (37)and E (0) v = h Φ v | ¯ H em n S (0) v o | Φ v i . (38)∆ E v = E (0) v − E (0)0 is the electron binding energy and isthe negative of the electron affinity (EA) for the valence v orbital. We have incorporated one-particle and one-hole(single), two-particle and two-hole (double) and three-particle three-hole (triple) excitations in our calculationsthrough the RCC operators by defining T (0) ≃ T (0)1 + T (0)2 + T (0)3 (39)and S (0) v ≃ S (0)1 v + S (0)2 v + S (0)3 v , (40)where the subscripts K and M run over 1, 2 and 3 whichare referred to as single, double and triple excitationsrespectively. To assess the importance of the triple exci-tations, we have performed calculations considering sin-gle and double excitations in the RCC theory (RCCSDmethod) after exciting all the core electrons, and thenwith single, double and triple excitations in the RCC the-ory (RCCSDT method). In addition, we have also car-ried out calculations using the second-order relativisticmany-body theory (RMP(2) method), considering two-orders of the residual interaction and only keeping linearterms from the RCCSD method (RLCCSD method) as | Ψ (0) v i ≃ n T (0) + S (0) v o | Φ v i . (41) TABLE I. Calculated EAs (in cm − ) at different levels of ap-proximations. Corrections from the Breit and QED interac-tions are given as ∆Breit and ∆QED, respectively. Extrap-olated contributions from the finite size basis functions aregiven as “Extra” and the estimated uncertainties are quotedwithin the parentheses.Method 6 S P / S P / P / Dirac-Coulomb contributionsDHF 27954.01 18790.51 12111.79 9221.90 5509.15RMP(2) 31818.40 20297.57 13026.52 9683.05 5720.11RLCCSD 31806.94 20393.25 12936.56 9681.38 5710.04RCCSD 31520.14 20248.86 12895.49 9647.42 5696.17RCCSDT 31347.68 20215.57 12859.52 9639.21 5695.68Corrections from Breit interactionDHF − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − .
25 0.61 − .
70 0.22 0.10RMP(2) − .
62 0.68 − .
82 0.25 0.12RLCCSD − .
91 0.81 − .
62 0.28 0.13RCCSD − .
81 1.25 − .
27 0.52 0.68RCCSDT − .
53 1.31 − .
09 0.57 0.71Extra 30.71 14.47 6.69 3.91 2.15Final 31357(50) 20243(20) 12861(15) 9641(10) 5697(10)NIST [61] 31406.47 20229.21 12871.94 9642.12 5698.63
Intermediate results from the RMP(2) and RLCCSDmethods can demonstrate the propagation of electroncorrelation effects from lower- to all-order methods sys-tematically in order to understand the role of electroncorrelation effects in the accurate calculations of EAs ofvalence electrons in different states of the Cs atom.
D. RCC theory for perturbed wave function
Extending the RCC theory ansatz of atomic wave func-tion, the first-order perturbed wave function due to H w can be expressed as [51–53] | Ψ (1) v i = e T (0) n S (1) v + T (1) (cid:16) S (0) v (cid:17)o | Φ v i , (42)where T (1) and S (1) v are the core and the valence excita-tion operators with the superscript 1 representing orderof perturbation in H w . After obtaining the amplitudesof the unperturbed RCC operators, we obtain the am-plitudes of their perturbed counterparts by solving thefollowing equations h Φ K | ¯ H em T (1) + ¯ H w | Φ i = 0 (43)and h Φ Mv | ( ¯ H em − E (0) v ) S (1) v + ( ¯ H em T (1) + ¯ H w ) ×{ S (0) v }| Φ v i = 0 , (44)where ¯ H w = e − T (0) H w e T (0) . Here, the subscripts K and M run again over 1, 2 and 3 which are referred to assingle, double and triple excitations respectively. Theimportant difference between the amplitude determiningequations for unperturbed and perturbed wave functionsis that the projected determinantal states (denoted bysuperscripts K and M ) have even and odd parities, re-spectively. The RCC operators representing perturbedsingle, double and triple excitations are denoted by T (1) ≃ T (1)1 + T (1)2 + T (1)3 (45)and S (1) v ≃ S (1)1 v + S (0 / v + S (1)3 v . (46)Along with the calculations using the RCCSD andRCCSDT methods, we also determine perturbed wavefunctions in the RLCCSD approximation by consideringthe expression | Ψ (1) v i ≃ n(cid:16) T (0) (cid:17) S (1) v + T (1) (cid:16) S (0) v (cid:17)o | Φ v i . (47)The differences in the results from these methods willdemonstrate the role of non-linear in T (0) terms andtriple excitations to the amplitudes of the first-order per-turbed wave functions. E. Evaluation of atomic properties
To test the accuracies of the wave functions, we alsoevaluate other relevant properties apart from the bind-ing energies and compare them with their high precisionexperimental values. The accuracies of the calculated en-ergies are sensitive to the quality of the wave functionsslightly away from the nuclear region of atomic systems.For testing the accuracies of the wave functions in thenuclear region and the far nuclear region, we evaluatethe magnetic dipole hyperfine structure constants ( A hf )and the electric dipole (E1) transition amplitudes, andcompare them with their respective experimental values.These quantities were evaluated using the expression h O i fi = h Ψ (0) f | O | Ψ (0) i i q h Ψ (0) f | Ψ (0) f ih Ψ (0) i | Ψ (0) i i = h Φ f |{ S (0) † f + 1 } O { S (0) i }| Φ i ih Φ f |{ S (0) † f + 1 } N { S (0) i }| Φ i i , (48)where O = e T (0) † Oe T (0) for the operator O represent-ing the respective property and N = e T (0) † e T (0) . Inthe evaluation of A hyf , we set | Ψ (0) f i = | Ψ (0) i i . Both O and N are the non-terminating series, which are eval-uated by adopting iterative procedures as described inRefs. [48, 49, 54]. We also present results from theRMP(2) and RLCCSD methods to make a comparativeanalysis of trend of correlation effects in the determi-nation of the aforementioned properties. We have used g I = µ I /I = 0 . TABLE II. Calculated A hyf values (in MHz) from differentapproximations in the many-body theory are given. Correc-tions from the Breit interaction, QED effect and BW effectare given as ∆Breit, ∆QED and ∆BW, respectively. Esti-mated “Extra” contributions and uncertainties to the finalcalculated values are quoted, but error bars of experimentalresults are not given because they appear beyond the inter-ested significant digits. We have used g I = 0 . S P / S P / P / Dirac-Coulomb contributionsDHF 1433.96 161.07 394.12 57.69 27.01RMP(2) 2317.02 267.09 559.62 89.08 40.95RLCCSD 2492.22 311.80 571.67 98.45 44.21RCCSD 2328.40 286.48 548.65 92.52 41.79RCCSDT 2308.52 290.21 548.48 94.03 41.65Corrections from Breit interactionDHF 0.01 − . − . − . − . − .
42 0.54 − . − . − .
09 0.75 − . − . − .
16 0.85 − . − . − .
18 0.83 − . − . − .
61 0.01 − .
18 0.004 ∼ . − .
29 0.05 − .
61 0.02 0.01RLCCSD − .
22 0.06 − .
69 0.01 ∼ . − .
58 0.05 − .
65 0.01 ∼ . − .
28 0.05 − .
51 0.01 ∼ . − . − . − . − . − . ∼ . a b c d e Refs. a [62]; b [63]; c [64]; d [65]; e [66]. µ I for the evaluation of A hyf values. We have also takeninto account the Bohr-Weisskopf (BW) effect by definingthe nuclear magnetization function ( F ( r )) in the Ferminuclear charge distribution approximation as F ( r ) = f W S N [( r/b ) − a/b )( r/b ) R (( b − r ) /a )+6( a/b ) ( r/b ) R (( b − r ) /a ) − a/b ) × R (( b − r ) /a ) + 6( a/b ) R ( b/a )] (49)for r ≤ b and F ( r ) = 1 − N [3( a/b )( r/b ) R (( r − b ) /a )+6( a/b ) ( r/b ) R (( r − b ) /a )] (50)for r > b , where N = 1 + ( a/b ) π + 6( a/b ) R ( b/a ) (51)and R k ( x ) = ∞ X n =1 ( − n − e − nx n k . (52) TABLE III. Magnitudes of the reduced E1 matrix elements in atomic units (a.u.) are given at different levels approximationsof many-body theory. Corrections from the Breit and QED interactions are given as ∆Breit and ∆QED, respectively, andextrapolated contributions are given as “Extra”. The final values are given along with the uncertainties and compared withthe extracted values from the latest experiments.Transition DHF RMP(2) RLCCSD RCCSD RCCSDT ∆Breit ∆QED Extra Final Experiment6 P / → S − . P / → S − . P / → S − . P / → S − . P / → S − . P / → S − . In Eq. (49), f W F takes into account the Woods-Saxon(WS) potential correction and is estimated after neglect-ing the spin-orbit interaction within the nucleus usingthe following expressions [55, 56] f W S = 1 − (cid:18) µ I (cid:19) ln (cid:16) rb (cid:17) (cid:20) − I − I + 1) g S + ( I − / g L (cid:21) for I = L + and f W S = 1 − (cid:18) µ I (cid:19) ln (cid:16) rb (cid:17) (cid:20) I + 38( I + 1) g S + I (2 I + 3)2( I + 1) g L (cid:21) for I = L − with the total orbital angular momentum L of the nucleus. We have used the nuclear parameters g L = 1 and g S = 4 .
143 for
Cs atom [55].
F. Evaluation of E PV In the RCC theory framework, Eq. (20) is given by E P V ≃ h Φ f |{ S (1) f + ( S (0) † f + 1) T (1) † } D { S (0) i }| Φ i ih Φ f |{ S (0) † f + 1 } N { S (0) i }| Φ i i + h Φ f |{ S (0) † f + 1 } D { T (1) (1 + S (0) i ) + S (1) i }| Φ i ih Φ f |{ S (0) † f + 1 } N { S (0) i }| Φ i i , (53)where D = e T (0) † De T (0) and N = e T (0) † e T (0) . Contribu-tions from the non-terminating expressions D and N areestimated by an iterative approach similar to that used inthe expression for evaluating properties, which is givenin Eq. (48). The “Core” contributions for the initialand final states originate from T (1) † D and DT (1) respec-tively, and the rest of the RCC terms involving S (0 / † f and S (0 / i give rise to valence contributions from the ‘fi-nal’ and ‘initial’ states, respectively. The simultaneouspresence of both the electromagnetic and NSI weak in-teractions through the RCC operators account for coreand valence correlation contributions, including the DCPcorrelation effects. G. Basis functions
We have used Gaussian type orbitals (GTOs) [57] toconstruct the single particle DHF wave functions. Theradial components for the large and small components ofDHF orbitals are expressed using these GTOs as P ( r ) = N k X k =1 c L k ζ L r l e − α β k r (54)and Q ( r ) = N k X k =1 c S k ζ L ζ S (cid:18) ddr + κr (cid:19) r l e − α β k r , (55)where P ( r ) and Q ( r ) are the large and small radial com-ponents of the DHF orbitals, l is the orbital quantumnumber, κ is the relativistic angular momentum quan-tum number, c L ( S ) k are the expansion coefficients, ζ L ( S ) are the normalization factors of GTOs, α and β are op-timized GTO parameters for a given orbital, and N k rep-resents the number of GTOs used. We have considered40 GTOs for each symmetry up to l = 6 for the RCCcalculations and up to l = 9 for analyzing results usingthe RMP(2) method. For the construction of GTOs, thevalues of α we use are 0.0009, 0.0008, 0.001, 0.004 and0.005 for the s , p , d , f and other higher angular momen-tum symmetry orbitals, respectively. The corresponding β values we have used are 2.15, 2.15, 2.15, 2.25 and 2.35for the s , p , d , f and other higher symmetry orbitals,respectively. Since our orbitals are not bounded by acavity, we carry out the numerical integration of radialintegrals up to r = 500 a.u. using a 10-point Newton-Cotes Gaussian quadrature formula on grids. Non-lineargrids are defined, as in Ref. [58], for the numerical cal-culations with the step-size 0.0199 a.u. over 1200 gridpoints. We have considered excitations from all the oc-cupied orbitals, but limited the virtual space to excita-tions of orbitals in that space with energies less than 2000a.u. This includes 1 − s , 2 − p , 3 − d , 4 − f , 5 − g ,6 − h and 7 − i -symmetry orbitals. These orbitals willbe referred to as the “active orbitals” hereafter. TABLE IV. List of the experimental values of E1 matrix ele-ments (in a.u.) for a few low-lying transitions reported overthe years using different measurement techniques.Transition Value Reference Year6 P / ↔ S P / ↔ S P / ↔ S Not available yet6 P / ↔ S P / ↔ S P / ↔ S Not available yet
IV. RESULTS AND DISCUSSIONA. Cs APV calculations & context of present work
The main thrust of our present work is a high-precisioncalculation of E P V for the 6 s S / − s S / in Cs;the transition on which the most accurate APV measure-ment (0.35% accuracy) has been carried out to date [9].As mentioned earlier, a considerable amount of effort hasalso gone into performing very accurate calculations on E P V using state-of-the art relativistic many-body the-ories (e.g. see [19, 31] and references therein ). At thetime of the last Cs APV measurement, the accuracies ofthe atomic calculations were about 1% [26, 27]. Later byusing the amended values of the E1 matrix elements in-ferred from the high precision measurements of lifetimesand polarizabilities of atomic states, the uncertainty inthe calculation was reduced to 0.4% [59]. This yieldeda Q at W that disagreed with its SM value by 2.5 σ . Sub-sequently, the leading order relativistic correction fromthe Breit interaction and the lower-order QED effectsand the neutron skin were included in the atomic calcu-lations (refer to [28–30] for discussions). As pointed outbefore, there has been a renewed interest in the inclusionof the neglected correlation effects in Cs APV since abouta decade. (e.g. see discussions in [16, 33]). The latestcalculations including the effect of the valence triple exci-tations were investigated by employing the RCC theory,and it was found that their contributions to the atomicproperties of Cs were relatively important in reducingthe uncertainty in the E P V amplitude to 0.27% [31].This result is in good agreement with the SM, howeverthe calculation on which it is based had used a sum-over-states approach in which the leading contributions fromthe excited states up to the principal quantum number n = 9 were estimated by using matrix elements, calcu-lated using the RCC theory and referred to as “Main”contribution. The rest were classified into “Core” and“Tail”, and they were evaluated using mixed many-body methods [31]. Later, Dzuba et al. reported another re-sult with 0.5% accuracy by evaluating the “Main” con-tribution, again, using a sum-over-states approach butwith different “Core” (opposite sign than [31]) and “Tail”contributions by taking into account certain sub-classesof correlation effects [19]. This resulted in a differenceof about 0.8% between the E P V calculations of Porsev et al [31] and Dzuba et al . [19]. Following these works,Roberts et al. have reported the contributions from QEDand DCP effects [32, 60]. There are still unresolved is-sues in the determination of electron correlation in CsAPV due to the disparate approaches that have beenused in the treatment of different physical effects in thelow- and high-lying excited states. In other words, the“Main”, “Tail” and “Core” contributions have not beenevaluated on par with each other. Also, the Breit inter-action and the effective QED interactions have not beentreated at the same level as the DC interaction in Refs.[19, 31]. The contributions from the triple excitationsinvolving core orbitals were not determined in Ref. [31].In contrast to the previous previous works, our calcula-tion of the E P V amplitude adopts an approach basedon the perturbed RCC theory as outlined above. Weexcite all the core electrons in our RMP(2), RLCCSDand RCCSD calculations to account for the electron cor-relation effects. However, we correlate all the electronsexcept the 1 − s , 2 − p , and 3 d occupied orbitals andbeyond n = 15 virtual orbitals for triple excitations dueto limitations in the available computational resources. B. Ancillary Properties
At the outset, we would like to reemphasize that itis customary to compare the results of the calculationsof energies, E1 matrix elements and A hyf values basedon a many-body theory with the available experimentaldata to assess the accuracy of the E P NC amplitude. Wegive values for all these quantities by taking into accountcontributions from the DC Hamiltonian, the Breit inter-action, and the QED effects at different levels of approx-imation in the many-body methods systematically. Wehave also estimated the contributions to different proper-ties by extrapolating our basis functions to infinite-size,which we have referred to as “Extra”, and given their val-ues. The uncertainties in our calculations are estimatedby analyzing the optimized GTOs used in the calcula-tions and contributions from the higher level excitationsthat are neglected here.In Table I, we give the final EA values from our cal-culations and these values are compared with the precisemeasurements listed in the National Institute of Scienceand Technology (NIST) database [61]. Following this,we have given the A hyf values in Table II using dif-ferent methods. After adding up all the contributionsalong with corrections from the BW effect, the final val-ues are compared with the high-precision experimentalvalues [62–66]. It can be seen that the triple excitations TABLE V. Magnitudes of the H NSIAPV matrix elements in − i ( Q W /N ) × − from different methods. Corrections from the Breitand QED interactions are given as ∆Breit and ∆QED, respectively. The final values after including “Extra” contributions aregiven along with the uncertainties in the parentheses. The quantity X is defined as X = | [ R th / R ex ] − | × h Ψ (0) k || H NSIAPV || Ψ (0) v i for the corresponding theoretical (denoted with subscript th ) and experimental (denoted with subscript ex ) values, where R = q A khyf A vhyf with superscripts k and v designated for the states with valence orbitals k and v , respectively. | Ψ (0) k i → | Ψ (0) v i DHF RMP(2) RLCCSD RCCSD RCCSDT ∆Breit ∆QED Extra Final X P / → S − . − . P / → S − . − . P / → S − . − . P / → S − . − . P / → S − . − . P / → S − . − . E PV amplitude(in − i ( Q W /N ) ea × − ) of the 6 s S / − s S / tran-sition in Cs from different terms of the RLCCSD, RCCSDand RCCSDT methods. ‘Others’ are the terms including cor-rection due to normalization of wave functions that are notmentioned explicitly. Contributions corresponding to “Core”and “Valence” correlations are given separately to distinguishthem. D ≡ D in the RLCCSD method approximation.RCC term RLCCSD RCCSD RCCSDTCore contributions DT (1)1 − . − . − . T (1) † D − . − . ∼ . − . − . − . DS (1)1 i − . − . − . S (1) † f D S (0) † f DS (1)1 i − . − . − . S (1) † f DS (0)1 i − . − . − . DS (1)2 i − . − . − . S (1) † f D T (0) † DS (1)3 i − . S (1) † f DT (0)2 − . T (1) † DS (0)3 i − . S (0) † f DT (1)2 − . S (0) † f DS (1)3 i − . S (1) † f DS (0)2 i − . − . − . improve the A hyf results of the P / states quite signif-icantly. Contributions from the Breit and QED interac-tions are non-negligible for achieving high precision re-sults. We give the values of the reduced matrix elementsof D of important transitions along with their error barsin Table III. The extracted E1 matrix elements from thelatest precise measurements of lifetimes and Stark shiftsare given in the same table. The agreement between our calculations and the experimental values [69–72] is foundto be quite good. However, we would like to mention thatthe experimental values of these matrix elements havebeen reported differently over the time [67–76]; sometimethey do not even agree within the quoted error bars ascan be found from the list given in Table IV. Nonetheless,it can be seen from Table III that the DHF values of E1matrix elements are large in magnitude and they reducesuccessively after the inclusion of the correlation effectsat the RCCSD and RCCSDT levels. The triples contribu-tions to the E1 matrix elements are more significant thanthose in the case of other properties for Cs. Similarly,the matrix elements of H NSIAP V are given in Table V. As canbe seen from this table, the correlation trends in the ma-trix elements of H NSIAP V are completely different than thosefor the E1 matrix elements but almost similar to thoseof A hyf . We analyze the accuracies of R = q A khyf A vhyf ,the superscripts k and v denoting for states with valenceorbitals k and v respectively, by comparing our theoreti-cal values with the experimental results. This is used todetermine the accuracy of the h Ψ k | H NSIAP V | Ψ v i matrix el-ements and their accuracies are quantified by evaluating X = | [ R th / R ex ] − | × h Ψ (0) k || H NSIAP V || Ψ (0) v i values, withsubscripts th and ex referring to our theoretical valuesand experimental results respectively, for important low-lying states. These values are found to be very small,implying that the H NSIAP V matrix elements are obtainedquite accurately by us. C. E PV results In Table VI, we present and compare our E P V re-sults for the 6 s S / − s S / transition in Cs fromdifferent terms of the RLCCSD, RCCSD and RCCSDTapproximations. For the sake of brevity, we present con-tributions from terms representing “Core” correlationsand valence correlations separately in the same table. Itshould be noted that these valence correlation contribut-ing terms contain both “Main” and “Tail” contributionsof the sum-over-states approach implicitly. As can be0
TABLE VII. Contributions from the ‘Core’ and ‘Valence’ cor-relations to the E PV amplitude (in − i ( Q W /N ) ea × − )using the Dirac-Coulomb Hamiltonian in the DHF, RCCSDand RCCSDT methods. Valence contributions are given intwo parts as ‘Main’ by considering contributions only fromthe np P / states with n = 6, 7 and 8, while ‘Tail’ refer tothe contributions from the remaining bound states and con-tinuum. Contributions from the extrapolated basis function,“Extra” and neutral weak interactions among electrons ( e − e )are also quoted.Method Core Main Tail Extra e − e [77]DHF − . − . − . † − . † − . † . † Contains additional contribution from the 9 p P / state. seen from the table, the RLCCSD result seems to be rel-atively large, but the rather small difference between theRCCSD and RCCSDT values suggests the convergenceof the results after the inclusion of higher level particle-hole excitations. The fairly large RLCCSD value is notentirely surprising, given that in this method there havebeen quite significant deviations of various spectroscopicproperties from their experimental values as discussed inthe previous subsection. The differences in the spectro-scopic properties at the RCCSD and RCCSDT levels aresomewhat large, and their trends are nonuniform. Forexample, it can be seen from Tables I and III that thecalculated energies and E1 matrix elements decrease ingoing from the RCCSD method to the RCCSDT method,while the matrix elements of H NSIAP V , given in Table V,increase. This is the reason for the small difference be-tween the RCCSD and RCCSDT E P V values. It can beseen from Table VI that there are significant changes inthe core contributions through the individual RCC termsin the RLCCSD and RCCSD methods, but the differ-ences in the RCCSD and RCCSDT methods are negligi-bly small. However, we find that changes in the valencecorrelations from different RCC terms in all the threeapproximations are relatively large. Compared to contri-butions from the first-order perturbed DS (1)1 i term of theground state, the perturbed S (1) † f D term of the excited7 S / state contributes predominantly, which correspondto contributions mainly from the one-particle one-holeexcitations. The contributions from the two-particle two-hole excitations to E P V are found to be small, which arerepresented by DS (1)2 i and S (1) † f D for the perturbed wavefunctions of the ground and excited states, respectively.As mentioned above, there is a small difference betweenthe final results from the RCCSD and RCCSDT methods.However, a comparison of the contributions of the indi- vidual terms obtained from both these methods revealsthat there are significant differences among them. This isbecause the RCCSD wave functions change when tripleexcitations are added, due to the change in the coupledcluster amplitudes. However, this change leads to largecancellations among the net contributions of the individ-ual terms arising through the initial and final perturbedwave functions resulting in a small difference in their fi-nal values. This is also in accordance with our analysisof energies and E1 matrix elements changing differentlythan the matrix elements of H NSIAP V in both the methods,which are manifested in the contributions from the indi-vidual RCC terms in a different form. Nonetheless, theconvergence of E P V amplitude with the higher-level ex-citations in the framework of the RCC theory stronglysuggests that the neglected correlation effects are indeedsmall.By using the calculated energies, E1 matrix ele-ments and amplitudes of H NSIAP V for the intermediate n (= 6 , , P / states at different levels of approxima-tions in the tables previously discussed, we estimatedthe “Main” contributions for a qualitative comparison ofits value with other results reported using the sum-over-states approach. Combining the “Main” contributionswith the “Core” contributions, contributions from the“Tail” are estimated in the DHF, RCCSD and RCCSDTmethods. This breakdown from the DHF, RCCSD andRCCSDT methods are given in Table VII and comparedwith the previously reported values from the sum-over-states approach. Our core contributions are in agreementwith the values reported in [27, 31], but it differs fromthe latest calculation reported in [19]. Since the con-tribution from the the 9 P / state is not included in our“Main” contribution and contained in the“Tail”, it wouldbe more appropriate to make comparison among the to-tal valence correlation contributions (“Main+Tail”) fromdifferent calculations. We find that our valence cor-relation contributions are 0.8980 and 0.8985 from theRCCSD and RCCSDT methods, respectively, against thevalues 0.911 [27], 0.9018 [31], and 0.8916 [19] in units of × − i ( − Q W /N ) ea . This shows that our valence cor-relation contribution is closer to that of [19]. In TableVII, we also present contributions from the extrapolatedbasis functions, denoted as “Extra”, and a small contri-bution to E P V from Ref. [77] due to possible neutralweak interactions among electrons ( e − e ) that was notincluded in our calculation.In Table VIII, we also give contributions from the Breitand QED interactions using the RCCSDT method andcompare them with the values reported by other ap-proaches earlier [28–30, 42, 60, 78]. We have also men-tioned the many-body method employed by other worksin the same table to estimate contributions from the Breitand QED interactions to E P V of the 6 s S / − s S / transition in Cs. We find consistency in the resultsobtained from various works. This means that these rel-ativistic corrections are not influenced significantly bythe electron correlation effects. Nonetheless, our method1
TABLE VIII. Comparison of contributions from the Breit andQED interactions to the E PV amplitude (in − i ( Q W /N ) ea × − ) of the 6 s S / − s S / transition in Cs fromvarious methods employed by different works.Breit QED Method Reference − . − . − . − . − .
004 Optimal energy Ref. [29] − . − . − . − . is more rigorous than the previous calculations of thesecorrections to the above E P V amplitude.After taking into account contributions from the DCHamiltonian, Breit interaction and QED effects from theRCCSDT method, the estimated value of “Extra” andsmall correction from the e − e contribution, we obtainthe E P V amplitude of the 6 s S / − s S / transitionin Cs as 0 . × − i ( − Q W /N ) ea . To estimate itsuncertainty, we adopt the following approach: We havetaken the difference between the RCCSD and RCCSDTvalues to estimate the uncertainties in the core and va-lence contributions to E P V . The major source of errorfor this transition amplitude comes from the finite size ofthe basis used in our calculation, which is extrapolated tobe 0 . × − i ( − Q W /N ) ea . We assume this as themaximum uncertainty arising from the incomplete basisfunctions. This approach to the estimation of the erroris more rigorous than the one adopted in Ref. [31]. Inthe latter work, an uncertainty of 10% is assigned to the“Core” and “Tail” contributions based on the spread oftheir results in different approximations, and the uncer-tainty in “Main” is taken to be 0.18% by analyzing resultsfrom a calculation using an ab initio calculation and an-other obtained from scaled wave functions. We have alsoestimated the uncertainties from the Breit and QED con-tributions. Adding all the uncertainties mentioned abovein quadrature, we find that the final uncertainty in E P V is 0 . × − i ( − Q W /N ) ea .We have given a list of the calculated E P V amplitudeof the 6 s S / − s S / transition in Cs over theyears in Table IX. We also mention the approaches usedin the previous works to determine this quantity. As canbe seen, apart from a few calculations, most of the previ-ous results were reported either using the sum-over-statesapproach or by considering mixed many-body methods.The last two high-precision calculations were carried outby adopting the sum-over-states approach, and estimat-ing “Core” and “Tail” contributions using different typesof many-body methods. Our ab initio calculation hassimilar accuracy to those are obtained using the sum-over-states approach, but our error estimation is morerigorous than that of the latter. The most important
TABLE IX. Progresses in the atomic calculation of the E PV amplitude (in − i ( Q W /N ) ea × − ) of the 6 s S / − s S / transition in Cs over the years by adopting variousapproaches.Year Result Approach Reference1989 0.908(9)
Ab initio
Ref. [26]1990 0.909(4) Sum-over-states Ref. [27]2000 0.8991(36) Ref. [27] + Breit Ref. [28]2001 0.901 Scaled optimal energy Ref. [29]2002 0.904(5)
Ab initio
Ref. [79]2005 0.904 Ref. [79]+QED corr. Ref. [30]2009 0.8906(24) Sum-over-states Ref. [31]2012 0.8977(40) Ref. [31]+core corr. Ref. [19]2020 0.8914(27)
Ab initio
This workTABLE X. Contributions to the scalar dipole polarizability( α ) of the 6 s S / − s S / transition in Cs using themost precise E1 matrix element amplitudes from the availablemeasurements and our calculations. We have used experi-mental energies from the NIST database [61] to reduce theuncertainty in the result. Estimated uncertainties from theE1 matrix matrix elements are quoted within the parentheses.Intermediate Initial state Final state Contributionstate 6 s S / s S / (in a.u.) → p P / − . − . → p P / a . b − . → p P / c − . → p P / c − . − . → p P / . − . → p P / − . − . → p P / − . − . − . → p P / − . − . n > − . − . a [67]; b [69]; c [68]. feature of our work is that it treats correlation contribu-tions from the “Core”, “Main” and “Tail” sectors at parwith each other, thereby resolving the large discrepancyin the “Core” contribution between the works reportedin Refs. [19] and [31] in an unambiguous manner. D. Vector polarizability
An accurate determination of the vector ( β ) dipole po-larizability of the 6 s S / − s S / transition in Cs isimperative so that it can be combined with the measuredvalue of Im ( E P V /β ) and our high accuracy calculationof E P V to extract Q at W . A very precise measurement of α/β = − . α is the scalar dipole polarizability of the transi-2tion. The α value for the transition | Ψ i i → | Ψ f i can beexpressed as [27] α = X k h Ψ (0) f | D | Ψ (0) k ih Ψ (0) k | D | Ψ (0) i i q h Ψ (0) f | Ψ (0) f ih Ψ (0) i | Ψ (0) i i× " E (0) f − E (0) k + 1 E (0) i − E (0) k . (56)As in the case of E P V , the contributions to α comefrom the “Core”, “Main” and “Tail” regions. We haveincluded the E1 matrix elements up to the 9 P states inthis estimation. Most of these matrix elements were cal-culated in the present work using the RCCSDT method,except a few for which very accurate experimental dataare available [67–69]. We have also used measured val-ues of the energies in our calculations. The contributionsfrom the “Core” and “Tail” were estimated to be smallusing the RMP(2) method. The individual contributionsfrom “Main” that come from the low-lying intermedi-ate states, “Core” and “Tail” are given in Table X. Thematrix elements used from different works are presentedin the same table. As can be seen from the table, themaximum contribution to α of the 6 s S / − s S / transition in Cs comes from the 7 p P / state fol-lowed by the 6 p P / state. The contributions fromthe 8 P state onwards are found to be small. Our fi-nal value is α = − . ea . Another recent studyhas found this value to be − . ea [80], wherecontributions from many matrix elements were includedexplicitly by analyzing them from the literature. Theyhad estimated the “Core” and “Tail” contributions us-ing the DHF method, whereas we have done so using theRMP(2) method. Nonetheless, we find very good agree-ment between both the results. By combining our valuefor α with the measured ratio of α/β = − . β = 27 . ea . The accuracy of this quantity is about0.15%; even better than the accuracy of our calculated E P V for the above transition. In Ref. [80], a summaryof the results for β have been presented, the variation inthese values covers a wide range. Our result is in agree-ment with all those values, but with a precision similarto the most accurate one [80]. E. Inferred Q W value and its implications Combining our results of E P V and β withthe precisely measured Im ( E P V /β ) = 1 . Im means imaginary part, for the6 s S / − s S / transition in Cs, we get Q at W = − . ex (22) th . After accounting for the nuclear skineffect [34], we get Q W = Q at W + ∆ Q N − PW = − . ex (22) th + 0 . − . ex (23) th . (57) −4−3−2−1 0 1 0.01 0.1 1 10 100 D s i n q W ( Q ) i n % Energy scale (Q ) in GeVRunning from the SMThis work FIG. 1. Plot demonstrating deviation of ∆ sin θ W ( Q ) (inpercentage) in the SM from sin θ W ( m Z ) MS = 0 . Q ) in GeV. The value obtained using thepresent APV study in Cs is shown at Q = 30 MeV, whichshows good agreement with the SM. This results in the difference between the value of Q W obtained from our calculation and the SM value Q SM W = − . Q W ≡ Q W − Q SM W = − . σ level.From the relation Q W = − N + Z (1 − θ W ), wecan derive as Q W ≈ − N + Z [1 − θ SM W + ∆ sin θ W )]= Q SM W − Z ∆ sin θ W . ⇒ ∆ Q W ≈ − Z ∆ sin θ W . (58)This gives change in sin θ W as ∆(sin θ W ) = 0 . θ SMW = 0 . θ W (0) =0 . θ W ( Q ) with respect to sin θ W ( m Z ) MS from theSM and the deviation obtained in this work at Q = 30MeV corresponding to the experiment on Cs [9, 77].It can be seen that the ∆ sin θ W value obtained fromthe present study agrees quite well with the SM.From the above ∆ Q W value, we constrain the isospinconserving parameter S ≃ . T fromthe relation ∆ Q W ≈ − . S − . T [12]. Furthermore,in the SO(10) model [12]∆ Q W ≈ . × (2 N + Z ) M W M Z x , (59)we get a lower limit M Z x >
961 GeV/ c compared to 3.5TeV/ c from the observation using the ATLAS detector[82]. Furthermore, ∆ Q W can be expressed in the darkphoton model characterized by U (1) d gauge symmetry as[21] ∆ Q W = 220 (cid:18) εε z (cid:19) sin θ W cos θ W δ − Q SMW δ , (60)3where ε is a dimensionless parameter, ε z = δM Z d /M Z ,and δ is a model dependent quantity. Substituting theaforementioned SM values, we get (cid:20) . (cid:18) εε z (cid:19) − (cid:21) δ ≃ . . (61)Using the effective field theory, suggested in Ref. [24],we obtain f effV q / Λ ≃ − × − GeV − . (62)Similarly, in terms of the nucleon-electron V-A couplings,defined in Ref. [20], it yields 55 g ep + 78 g en = 36 . V. SUMMARY
We have revisited the calculation of electric dipole am-plitude due to the nuclear spin independent neutral weakinteraction for the 6 s S / − s S / transition in Csby employing the relativistic coupled-cluster theory. Inour approach, we solve an inhomogeneous equation toobtain the first-order perturbed wave function due tothe weak interaction in order to account for the corre-lation effects of the electrons from the occupied, valenceand virtual orbitals on an equal footing. This resolvesthe large discrepancy, including sign, for the core elec-tron correlation contribution to the above amplitude be-tween the two latest high accuracy calculations. More-over, it includes contributions from correlation effects due to the double core-polarization, the Breit interac-tion and lower-order quantum electrodynamics effects bythe same method used to incorporate contributions fromthe Dirac-Coulomb atomic Hamiltonian. Relevant spec-troscopic properties have been evaluated at different lev-els of many-body approximations and the role of elec-tron correlation effects arising from higher-level particle-hole excitations, in particular the triple excitations, havebeen demonstrated to be non-negligible. By analyzingthe differences between these calculated results and theirrespective high-precision experimental values, the accu-racy of the above electric dipole transition amplitude isestimated and found to be of the order of 0.3%. This isslightly better than the reported accuracy of the corre-sponding measurement. We have determined the vectorpolarizability of the above transition with an accuracyof 0.15% . Combining all our calculated values with themeasurement, we have obtained the nuclear weak charge Q W = − . ex (23) th for Cs , which differs fromthe Standard Model value by − . ACKNOWLEDGEMENT
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