EEPJ manuscript No. (will be inserted by the editor)
Nuclear Currents in Chiral Effective Field Theory
Hermann Krebs
Institute for Theoretical Physics II,Faculty of Physics and Astronomy,Ruhr-Universit¨at Bochum, D-44780 Bochum, GermanyReceived: date / Revised version: date
Abstract.
In this article, we review the status of the calculation of nuclear currents within chiral effectivefield theory. After formal discussion of the unitary transformation technique and its application to nuclearcurrents we will give all available expressions for vector, axial-vector currents. Vector and axial-vectorcurrents will be discussed up to order Q with leading-order contribution starting at order Q − . Pseudoscalarand scalar currents will be discussed up to order Q with leading-order contribution starting at order Q − .This is a complete set of expressions in next-to-next-to-next-to-leading-order (N LO) analysis for nuclearscalar, pseudoscalar, vector and axial-vector current operators. Differences between vector and axial-vectorcurrents calculated via transfer-matrix inversion and unitary transformation techniques are discussed. Theimportance of consistent regularization is an additional point which is emphasized: lack of consistentregularization of axial-vector current operators is shown to lead to a violation of the chiral symmetry inthe chiral limit at order Q . For this reason a hybrid approach at order Q , discussed in various publications,is non-applicable. To respect the chiral symmetry the same regularization procedure needs to be used in theconstruction of nuclear forces and current operators. Although full expressions of consistently regularizedcurrent operators are not yet available an isoscalar part of the electromagnetic charge operator up to order Q has a very simple form and can be easily regularized in a consistent way. As an application, we reviewour recent high accuracy calculation of the deuteron charge form factor with a quantified error estimate. PACS. β decay; double β decay; electron and muon captures The internal structure of nucleons and nuclei can be stud-ied by probing them with electromagnetic, weak, or evenscalar probes. Scalar probes play an important role in be-yond the standard model search of dark matter. The in-teractions of hadrons with the external probes are wellapproximated by one photon, W ± , Z . In this case, thefull scattering amplitude factorizes in a leptonic and ahadronic part. In the case of electroweak interaction, theamplitude can be written as a multiplication of leptonicand hadronic four-current operators. Leptonic four-currentcan be well approximated by perturbative calculationswithin the standard model. Hadronic four-current is lessknown. Electroweak nuclear currents have been extensivelystudied in the last century within boson-exchange (pionsand heavier mesons) and soliton models, see [1,2] for re-cent and [3,4] for earlier reviews on this topic. Electro-magnetic nuclear currents have been reviewed in [5,6,7,8]. One of the simplest approximations of the nuclear cur-rent operator is Impulse approximation (IA) where onlyone nucleon in a nucleus is probed by an external source Send offprint requests to : and other nucleons act as spectators. IA can be expectedto work well at higher energies. However, this approxima-tion is not satisfactory in the low-energy sector. Riska andBrown showed in their seminal paper [9] on radiative cap-ture of a thermal neutron on a proton, n + p → γ + d , that10% discrepancy between the IA prediction and experi-ment can be explained by taking into account the lead-ing pion exchange electromagnetic current between twonucleons which was calculated by Villars [10] and tookadditionally ∆ (1232) resonance and ω → π + γ chan-nel in to account [11]. This was a start for the devel-opment of more sophisticated meson exchange currentswhere heavier mesons and nucleon resonances have beentaken into account. The currents have been studied bothin relativistic and non-relativistic formalisms. Relativis-tic approach is more complicated than a non-relativisticone and is reviewed e.g. in [12,13], see also [14] for rel-ativistic Hamiltonian approach. In a non-relativistic for-malism one usually performs a Foldy-Wouthousen unitarytransformation [16] and eliminates in this way antinucleoncontributions. In practical calculations, relativistic correc-tions are then treated in terms of one-over-nucleon-massexpansion. Based on the studies of Poincare algebra [17,18] one can give a systematic one-over-nucleon-mass ex- a r X i v : . [ nu c l - t h ] A ug Hermann Krebs: Nuclear Currents in Chiral Effective Field Theory pansion of wave functions and currents [19,20,21]. Onecan even block-diagonalize the full Poincare algebra si-multaneously reducing in this way, quantum field theo-retical problem to quantum mechanical one [22,23] . Inthis way one can either keep everything relativistic or per-form a large nucleon mass expansion of block-diagonalizedoperators. Through the phenomenological studies of thenuclear currents of the last century, one could gain veryimportant insights into a general construction of nuclearcurrents. The interrelation between nuclear forces and cur-rents was clearly emphasized to keep gauge symmetryexact [24]. Gauging technique of nuclear forces were de-veloped to derive consistent nuclear currents out of nu-clear forces which respect explicitly the gauge symme-try [25,26]. Off-shell and energy-dependence of the nu-clear forces and currents had been extensively studied.Block-diagonalization techniques were developed to con-struct energy-independent nuclear forces [28,29]. Exten-sion of these techniques, in particular unitary transforma-tion technique, to a construction of nuclear currents hadbeen presented in [32]. The advantage of the procedurepresented in [32] is a systematic construction of the nu-clear currents if perturbation theory would work. Withinthis procedure, a vector current has been studied up toone-loop level in a meson exchange model [33] which is ofcomparable complexity as the state of the art calculationsof nuclear currents in chiral effective field theory.Already in the early studies of the nuclear current op-erators, the prominent role of the chiral symmetry (sym-metry of QCD if the quark masses are set to zero) inthe nuclear forces and currents was well appreciated [27].Basically in all realistic models the longest range inter-actions are governed by one-pion-exchange. For this rea-son, the chiral symmetry was respected in lowest orderapproximation in the low energy-momentum expansion.How to further systematically improve phenomenologicalmodels and in particular their connection to QCD wasrather unclear. A groundbreaking idea that made system-atically improvable calculations of nuclear forces and cur-rents possible came with the birth of the chiral perturba-tion theory [35,34]. Gasser and Leutwyler showed in [34]that perturbative expansion in small momenta and massesof pions divided by the chiral symmetry breaking scale Λ χ can be systematically performed beyond a tree-levelapproximation [35]. To organize the infinite number ofpossible interactions they used naive dimensional analysis(power counting scheme) which was proposed by Wein-berg [35]. The price which one has to pay is the appear-ance of more and more complicated Lagrangians with un-known coefficients, so-called low energy constants (LEC),if higher precision is required. The procedure in [34] al-lows one to approximate Green functions of QCD in thepionic sector by chiral perturbation theory in a system-atically improvable way [36]. Degrees of freedom in chiralperturbation theory are pointlike pions which gain theirstructure at higher orders in the chiral expansion (loop This statement was proven by Gl¨ockle and M¨uller in [22]only for a restrictive model. The proof for a general field theorywas given only recently [23]. effects). Only a few years later chiral perturbation the-ory was formulated in the presence of matter field allow-ing to extend the formalism to nucleon degrees of free-dom [37]. Nucleon states appeared in [37] as initial andfinal states which are on-shell. Strictly speaking, the for-malism does not allow to make any statement about off-shell dynamics of the nucleons with a clear connection toQCD. However, within QCD calculated matrix elementswith on-shell nucleons in the initial and final states canbe approximated in a systematically improvable way bychiral perturbation theory. One technical difficulty whicharises with the description of nucleons within chiral per-turbation theory is the appearance of the nucleon masswhich is a hard scale. As a consequence nucleon mass di-vided by chiral symmetry breaking scale is not small butof the order one. Naive application of dimensional regular-ization in loop diagrams would generate also terms pro-portional to positive powers of nucleon mass and woulddestroy in this way a power counting. There are two so-lutions to this problem: the first one is to perform a fieldredefinition and eliminate nucleon mass from the nucleonpropagator on the path integral level reducing the theoryto a non-relativistic approach. Poincar´e invariance is re-stored order by order in the form of a systematic large nu-cleon mass expansion. The method is called heavy-baryonapproach [38,39] and was successfully applied to variousscattering observables in the single-nucleon sector [40].Another method, called infrared regularization, respectsthe Lorentz-invariance of the theory resuming the wholelarge nucleon mass expansion without violation of powercounting [41]. In this formulation, one introduces non-physical cuts far away from the applicability region of thetheory. Nevertheless, in practical calculations, these cutsmight have long tails such that it is advantageous not tohave them. Another formulation of the relativistic theorywithout violation of the power counting scheme can be re-alized by modification of the subtraction scheme. In thismodified scheme all power counting violating terms whichare caused by hard nucleon mass scale are absorbed intoavailable LECs. The method is called extended-on-mass-renormalization-scheme [42,43]. Applications of relativis-tic and non-relativistic chiral perturbation theory meth-ods in the single-nucleon sector are reviewed in [44].Extension of chiral perturbation theory to two- andmore-nucleon sector was pioneered by Weinberg [45,46,47]. The difficulty in the two- and more-nucleon sectorsis the existence of bound states which makes the pertur-bative approach impossible. As a way out of this Wein-berg suggested using chiral perturbation theory for thecalculation of an effective potential, which is called nu-clear force. Observables like nuclear spectra can be ex-tracted out of the non-perturbative numerical solution ofthe Schr¨odinger equation with chiral nuclear forces as in-put. The effective potential was originally defined as a setof time-ordered diagrams without two-nucleon or more-nucleon intermediate states. The absence of these statesmakes a perturbative approach applicable. This idea wasfollowed by several groups. Already one year after origi-nal publication [46] nuclear forces have been studied up to ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 3 next-to-leading-order (NLO) in chiral expansion by [48].Soon after this publication next-to-next-to-leading-order(NNLO) corrections have been calculated in [49,50,51].At this order, one has to take two-pion-exchange correc-tions into account. For two-nucleon operators, they ap-pear as one loop corrections and for three-nucleon forcesas tree-level diagrams. Time-ordered perturbation theory(TOPT) gives a nice graphical interpretation of the forcesbut introduced a drawback of energy-dependence in nu-clear forces. This makes it difficult to apply them in a few-and many-body simulations. This drawback, however, wascured with the application of unitary transformation tech-nique for construction of nuclear forces [52,53] and leadto properly normalized energy-independent nuclear forces.Next-to-next-to-next-to-leading-order (N LO) correctionsto two-nucleon forces have been calculated more than adecade ago [56,57,58,59]. Numerical studies of these con-tributions, including fits of various short-range LECs whichappear at this order, have been performed by Bonn-Bochum[60] and Idaho group [61]. We call these forces as first-generation nuclear forces in further discussion. At the sameorder, there are corrections to leading three-nucleon forceswhich have been calculated in [62,63]. At N LO also four-nucleon forces start to contribute. Their analytical expres-sions can be found in [68,69]. Density-dependent interac-tions which are needed for applications in nuclear matterstudies have been derived from the N LO three-nucleonforces in [64,65] and from the N LO four-nucleon forcesin [66], see [67] for a review on this direction. A first nu-merical estimate of He expectation values of four-nucleonforces has been performed in [70]. Numerical implemen-tations of N LO three- and four-nucleon forces in a few-nucleon sector are non-trivial and still under investigation.Only exploratory studies have been presented in [71,72,73]and various perturbative applications in many-body sec-tor have been considered in [74,75,76,77]. In these stud-ies, however, one did not pay any attention to consistencyissues of regularization between the two-, three- and four-nucleon forces. Nowadays, we know that a mismatch ofdimensional and cut-off regularizations leads to a viola-tion of the chiral symmetry at a one-loop level in three-nucleon forces which is N LO, the same is true for theaxial vector currents [78]. So a more careful investigationis needed which is work in progress. Construction and ap-plication of nuclear forces in chiral EFT are reviewed inseveral comprehensive review articles, see e.g. [79,80,81,82,83]. Three-nucleon forces within chiral EFT have beenreviewed in [85,84]. By now two-nucleon forces have beencalculated up to next-to-next-to-next-to-next-to-leading-order (N LO) for the two-nucleon forces [86,87,88,89,90].Even partial N LO contributions have been considered [91].First applications of these second-generation chiral two-nucleon forces can be found in [92,93,94,95,96]. N LOcorrections to three-nucleon forces have been consideredonly partly [97,98]. Longest and intermediate-range con-tributions have been calculated. Various short-range in-teractions, however, are still under construction. Their nu-merical implementations are under construction like in thecase of N LO three-nucleon forces. In parallel to chiral EFT activities where numericalcalculations are performed within a finite cut-off range,there was activity on non-perturbative renormalizationof the theory for arbitrary values of cut-offs. A pioneer-ing work towards this direction was published by Ka-plan, Savage and Wise (KSW) [100,99]. Based on un-naturally large nucleon-nucleon scattering length the au-thors suggested using a different power counting and toreorganize a resummation of the effective potential. Intheir power counting pion physics and higher-order short-range interactions are treated perturbatively. Only theleading-order short-range interactions are resumed. Al-though this approach leads to a non-perturbatively renor-malizable theory it showed a poor convergence in descrip-tion of S − D channel in nucleon-nucleon scattering[101], see also [102] for recent discussion. In the sameframework, electromagnetic form factors of the deuteron[103] and radiative capture n + p → d + γ were analyzed upto next-to-leading-order. KSW power counting is also usedin a pionless EFT where pions are treated as heavy degreesof freedom and are integrated out, see [104] and referencestherein. The expansion is performed around the unitarylimit where two-nucleon scattering length diverges. We arenot going to discuss in this review all important develop-ments in the pionless EFT. A comprehensive review onthis topic can be found in [105].Soon after Weinberg’s seminal papers on nuclear forces[45,46] Park et al. presented the first study of nuclearelectroweak currents based on chiral EFT [106,107] upto N LO in chiral expansion. However, these first cal-culations were incomplete: only irreducible one-loop dia-grams were considered, fourth-order pion Lagrangian con-tributions were not taken into account, in the case of vec-tor current considerations of two-pion-exchange diagramswere restricted to magnetic moment operator. The calcu-lated vector currents lead to an excellent description ofthe total cross section in radiative neutron-proton cap-ture at thermal energy in the hybrid calculation with Ar-gonne v nuclear force [111,107]. They also successfullycalculated proton-proton fusion rate [112] showing thatat N LO meson-exchange currents make a 4%-effect onproton-proton fusion rate compared to the leading single-particle Gamow-Teller matrix element. Polarized neutron-proton capture within N LO currents was presented laterin [113] where the authors included also a short-range cur-rent contribution which was ignored in [111,107]. Deuteronelectromagnetic form factors have been studied in [108,109,110]. Magnetic moment and radiative capture of ther-mal neutrons for three-nucleon observables had been stud-ied in [114]. After fitting short-range current to the mag-netic moment of the deuteron the cut-off dependence of Note that there is no contribution at the order Q − forvector- and axial-vector current operators. For this reason, or-der Q − contributions to vector- and axial-vector currents aredenoted here as NLO. This convention is similar to nuclearforces where order Q and Q contributions are denoted asleading-order (LO) and NLO, respectively. One advantage ofthis convention is that axial-vector current at NNLO dependson the same LECs as the three-nucleon force at NNLO. Hermann Krebs: Nuclear Currents in Chiral Effective Field Theory the results was significantly reduced. Application of thecurrents to the solar hep process followed where the au-thors calculated S-factor with an accuracy smaller than20% [135,136]. Application to muon capture on deuteroncan be found in [137]. Although the absorption of themuon by the deuteron leads to the energetically higher re-gion, the part of the capture rate where two neutrons carryhigher energy is known to be small such that the dominantcontribution comes from the energy region where two out-going neutrons carry low energy such that the formalismis still applicable. Also contributions of meson-exchangecurrents to triton β -decay were studied in [138] wherethe authors tried to extract the low energy constant D which governs chiral three-nucleon force at NNLO. Devel-opment of the second-generation of chiral EFT currentsstarted with the work [108,139,140], where also reducible-like diagrams had been taken into account. These dia-grams show up when one defines an effective potentialas a transition amplitude with subtracted iterated parts.In this way, one gets an energy-independent nuclear forcewhich is much easier to deal with in calculations of three-and more-nucleon observables. In [140] (TOPT currents)the authors also considered chiral nuclear forces at NLOlevel in order to derive consistent chiral forces and cur-rents using only chiral EFT and leaving in this way ahybrid approach. In parallel to these activities chiral nu-clear vector currents have been derived by using unitarytransformation technique [141,142] (UT currents)wherethe same off-shell scheme had been used as in [60]. Var-ious applications of the second-generation currents fol-lowed: Deuteron electromagnetic form factors have beenstudied with UT currents in [143]. Application to H and He photodisintegration with UT currents has been stud-ied in [144]. TOPT currents have been applied to ther-mal neutron captures on deuteron and He in [115]. Tosolve the three- and four-body problem the authors usedhyperspherical-harmonics technique, see e.g. [116] for areview. Electromagnetic form factors of deuteron and Hand He and deuteron photo(electro)-disintegration havebeen studied in [117,118]. Electromagnetic moments andtransitions have been studied for nuclei with A ≤ β -decay has been discussed in [126] and [134]. Inclusiveneutrino scattering off the deuteron has been analyzedin [127] where the authors find that the predicted cross-sections are consistently larger by a couple of percentsthan those given in phenomenological analysis of Naka-mura et al. [128,129]. They also found a very tiny cut-off dependence of the cross sections. QMC calculation ofweak transitions for A = 6 −
10 have been presentedin [130] where the authors calculated β -decays of He and C and electron capture in Be. They found an excel-lent agreement with experimental data for the electroncaptures in Be and an overestimate of the He and Cdata by ∼
2% and ∼ C decreases from 10% to4%. In more recent QMC studies of weak transitions in A ≤
10 nuclei [121] the authors find in most cases anagreement with experimental data. As input they usedN LO axial-vector currents and chiral EFT wave func-tions [131,132,133]. Two-body currents contribute at the2 −
3% level with exception of Li, B, and He β -decays.In the latter cases, the contribution of the impulse ap-proximation of the Gamow-Teller transition operator (LOapproximation) is suppressed. Two-body currents providea 20 −
30% correction which is, however, insufficient toachieve the agreement with experimental data. Extensive β -decay studies from light-, medium-mass nuclei to Snhave been presented in [122]. The authors used interac-tions and currents from chiral EFT [123,124] in combi-nation with no-core shell model, valence-space in-mediumsimilarity renormalization group, and coupled-cluster ap-proaches to cover the whole light- and medium-mass nu-clei sectors. They found an overall good description of ex-perimental data for light nuclei. Similar to QMC stud-ies, they found for A ≤ HeGamow-Teller matrix elements due to two-body currents.For medium-mass nuclei the authors found a remarkablygood agreement of Gamow-Teller matrix elements. The in-clusion of the two-body currents and three-nucleon forceswas essential for the description of the data in the medium-mass nuclei sector.The purpose of this work is to review the construc-tion of nuclear currents within chiral EFT. Electroweak, aswell as pseudoscalar and scalar two-nucleon current opera-tors, will be discussed up to one-loop (two-pion-exchange)approximation. In the main part of this manuscript, wewill concentrate on the unitary transformation technique.Gauge and chiral symmetries as well as four-vector rela-tions will be discussed. Another purpose of this work is toquantify the differences between the unitary transforma-tion technique used in the derivation of all currents by ourgroup and the currents derived by time-ordered pertur-bation theory in combination with subtraction techniqueby JLab-Pisa group. In the last part of this review, wewill concentrate on the symmetry preserving regulariza-tion. We will show that keeping the chiral symmetry atone-loop level will require a consistent regularization ofnuclear forces and currents which is still work in progress.In all available calculations, sofar dimensional regulariza-tion has been used in the construction of the current. Inthe practical calculations of observable, the current opera-tors are usually multiplied with a cut-off regulator. We willshow that this mismatch of the regularizations leads to thechiral symmetry violation at one-loop order. To cure thisit is necessary to calculate both nuclear forces and cur-rents with the same regulator which respects gauge andthe chiral symmetry by construction.We start in Sec. 2 with the presentation of the uni-tary transformation technique for nuclear forces. Exten- ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 5 sion of this technique to nuclear currents will be discussedin Sec. 3. In Sec. 4 we will discuss consistency checks like afour-vector relation or continuity equations. Those have tobe satisfied with any effective current operators. Sec. 5 isdevoted to current operators within chiral EFT where welist all expressions for vector, axial-vector, pseudoscalar,and scalar current operators which are obtained within aunitary transformation technique up to N LO. In Sec. 6we compare our results with those obtained by JLab-Pisagroup via using time-ordered perturbation theory in acombination with a transfer-matrix inversion technique.In Sec. 7 we discuss a path towards construction of con-sistently regularized nuclear forces and currents. We willdemonstrate that a naive use of the dimensional regular-ization for currents in combination with a cutoff regular-ization of all operators in the Schr¨odinger or Lippmann-Schwinger equations leads to a violation of the chiral sym-metry at N LO. For this reason at this level of precision,we have to use a consistent regularization with the samesymmetry-preserving regulator in nuclear forces and cur-rents. In Sec. 8 we will discuss a deuteron charge opera-tor which is calculated within a consistent and symmetry-preserving higher derivative regulator. A high precisiondetermination of the deuteron charge form factor allowsfor very precise extraction of the neutron radius. Lengthyexpressions for two-pion exchange vector and scalar cur-rents as well as technical details about folded-diagramtechnique, transfer-matrix with time-dependent interac-tions, and a derivation of the continuity equations aregiven in the Appendices.
The CHPT Hamiltonian up to a given order in chiral ex-pansion has a rather simple form but operates on the fullFock-space which includes all possible pion-nucleon states.Nonperturbative calculations of the amplitude with its in-put requires the quantum field theoretical methods whichare very complicated. In order to reduce the complexityof the calculation it is advantageous to decompose the fullFock space F into a model space and a rest space. In ourcase the model space M will be generated by states whichinclude only nucleons. All other states like the state withone or more pions or delta resonance belong to the restspace R F = M ⊕ R . (1)Let us denote by η and λ projector operators which project F to M and R , respectively. In the absence of exter-nal sources there is no energy-momentum flow into thesystem. Translation invariance guarantees energy conser-vation. For this reason, after switching off pseudoscalar,vector and axial vector sources, the Hamilton operatorbecomes time independent and we can start with the sta-tionary Schr¨odinger equation H | ψ (cid:105) = E | ψ (cid:105) . (2) In the first step we look for a unitary transformation whichbrings the Hamilton operator H into a block diagonal formsuch that the stationary Schr¨odinger equation is restrictedto model space ηU † HU η | φ (cid:105) = Eη | φ (cid:105) , (3)where | φ (cid:105) = U † | ψ (cid:105) , (4)and due to block - diagonalization we have ηU † HU λ = λU † HU η = 0 . (5)A unitary transformation which satisfies Eq. (5) can beconstructed via an ansatz of Okubo [28] ηU η = (cid:0) η + A † A (cid:1) − / , ηU λ = − A † (cid:0) AA † (cid:1) − / ,λU η = A (cid:0) A † A ) − / , λU λ = (cid:0) λ + AA † (cid:1) − / , (6)where the operator A satisfies A = λAη, (7)and a nonlinear decoupling equation λ (cid:0) H − (cid:2) A, H (cid:3) − AHA (cid:1) η = 0 . (8)Eq. (8) can be solved within chiral perturbation theory [52,53]. The effective Hamiltonian is given by H eff = U † HU (9)It is important to note that the unitary transformation U of Eq. (6) is not unique. Any additional transformation ofthe η -space for example will not affect decoupling condi-tions of Eq. (5). This degree of freedom can be used in or-der to achieve renormalizability of the effective potential.Renormalizability of H eff means that it becomes finite af-ter performing dimensional regularization with beta func-tions taken from the pion and one-nucleon sector which arespecified in [54,177,55]. Explicit construction of the oper-ator U is reviewed in [83]. Recent calculations of nuclearforces are performed up to N LO in the chiral expansionwhich corresponds to the full two-loop calculation for NN-and full one-loop calculation for 3N-operators.
In order to construct a nuclear current we start with thechiral perturbation theory Hamiltonian in the presence ofexternal sources, see Appendix A, and define the effectiveHamiltonian in a similar way via H eff [ s, p, a, v ] = U † H [ s, p, a, v ] U, (10)where s, p, a and v denote scalar, pseudoscalar, axial-vectorand vector sources, respectively. Here we use the same uni-tary transformation U which leads to a block-diagonal ef-fective Hamiltonian in the absence of external pseudoscalar, Hermann Krebs: Nuclear Currents in Chiral Effective Field Theory axial-vector and vector sources. Scalar source is set to thelight quark mass matrix. Note that the effective Hamilto-nian of Eq. (10) is not block-diagonal such that in general ηH eff [ s, p, a, v ] λ (cid:54) = 0 (cid:54) = λH eff [ s, p, a, v ] η. (11)Only the strong part of the Hamiltonian is block-diagonal: ηH eff [ m q , , , λ = λH eff [ m q , , , η = 0 . (12)There is, however, no reason for block-diagonalization of H eff [ s, p, a, v ] since we only want to consider expectationvalues of the current operator and are not interested inits non-perturbative iterations. We can derive a currentoperator out of the effective Hamiltonian in the presenceof the external sources by J X = δδX H eff [ s, p, a, v ] (cid:12)(cid:12)(cid:12) s = m q ,p = a = v =0 , (13)where X stays for s, p, a or v depending on which kindof nuclear current we are interested in. With H eff fromEq. (10), however, we will get a singular current whichis non-renormalizable. In order to work with renormaliz-able current we need to apply further unitary transforma-tion on the effective Hamiltonian which depends explic-itly on external sources. Due to its explicit dependenceon external sources this additional unitary transforma-tion becomes time dependent. In order to understand how H eff changes under time-dependent unitary transforma-tion U ( t ) consider a state in the Schr¨odinger picture whichsatisfies a time-dependent Schr¨odinger equation i ∂∂t | φ ( t ) (cid:105) = H eff [ s, p, a, v ] | φ ( t ) (cid:105) . (14)The state | φ ( t ) (cid:105) contains all information of the quantumsystem in the presence of external sources. We can rewritethis equation by multiplying left hand side and right handside by U ( t ) † and inserting a unity operator we get i ∂∂t | φ (cid:48) ( t ) (cid:105) = H (cid:48) eff [ s, p, a, v ] | φ (cid:48) ( t ) (cid:105) , (15)where | φ (cid:48) ( t ) (cid:105) = U ( t ) † | φ ( t ) (cid:105) (16)and H (cid:48) eff [ s, ˙ s, p, ˙ p, a, ˙ a, v, ˙ v ] = U ( t ) † H eff [ s, p, a, v ] U ( t )+ (cid:18) i ∂∂t U † ( t ) (cid:19) U ( t ) . (17)The renormalizable current operator can be generated outof H (cid:48) eff . The momentum space currents are defined˜ J X ( k ) = δδ ˜ X ( k ) H (cid:48) eff , (18)where H (cid:48) eff is taken at x = 0 and s = m q , ˙ s = p = ˙ p = a = ˙ a = v = ˙ v = 0. X stays for s, p, a or v in dependencewhich current we are considering and˜ X ( k ) = (cid:90) d x (2 π ) X ( x ) e i k · x . (19) Note, that due to time derivative term in Eq. (17), thecurrent ˜ J X ( k ) becomes energy-transfer dependent. Theexplicit form of the unitary transformations U and U ( t )can be found in [23].The energy-transfer dependent current of Eq. (18) hasa specific general structure. To see this, let us parametrizethe unitary transformations from Eq. (17) by U ( t ) = exp ( i U ( t )) , (20)where the hermitian operator U has a form U ( t ) = (cid:90) d x (cid:2) v cµ ( x ) V µ,c ( x ) + a cµ ( x ) A µ,c ( x )+ p c ( x ) P c ( x ) + s c ( x ) S c ( x ) (cid:3) , (21)where we use Einstein-convention for the space-time andisospin-indices.The momentum-space form of this opera-tor is given by U ( t ) = (cid:90) d p (cid:90) d x (2 π ) e − i p · x (cid:2) ˜ v cµ ( p ) V µ,c ( x )+ ˜ a cµ ( p ) A µ,c ( x ) + ˜ p c ( p ) P c ( x ) + ˜ s c ( p ) S c ( x ) (cid:3) = (cid:90) d p e − i p t (cid:2) ˜ v cµ ( p ) ˜ V µ,c ( − p ) + ˜ a cµ ( p ) ˜ A µ,c ( − p )+ ˜ p c ( p ) ˜ P c ( − p ) + ˜ s c ( p ) ˜ S c ( − p ) (cid:3) . (22)For the current operator of Eq. (18) we get˜ J X ( k ) = δδ ˜ X ( k ) H eff [ s, p, a, v ]+ i (cid:2) H eff [ m q , , , , δδ ˜ X ( k ) U (0) (cid:3) − ∂∂t U ( t ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = δδ ˜ X ( k ) H eff [ s, p, a, v ]+ i (cid:2) H eff [ m q , , , , ˜ X ( − k ) (cid:3) − ik ˜ X ( − k ) , (23)where ˜ X stays for ˜ S , ˜ P , ˜ A or ˜ V dependent on which cur-rent we consider, and all sources are set to zero after thefunctional derivative is taken. Eq. (23) shows a generalform of the energy-transfer dependent current. In the firstline of Eq. (23) we see a current operator δδ ˜ X ( k ) H eff [ s, p, a, v ] (24)which denotes the current with all phases of the time-dependent transformations put to zero. The part propor-tional to the phases of the additional time-dependent trans-formations is in the second line of Eq. (23). We see thatthe energy-transfer dependent part of the current is alwaysaccompanied with the commutator of the same structurewith the nuclear force. An expectation value of the the sec-ond line of Eq. (23) vanishes on-shell when k = E f − E i where E f and E i are final and initial eigenenergies of thenuclear force H eff [ m q , , , ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 7 There are various consistency checks which the nuclearvector and axial-vector current have to satisfy, in general.These relations are rooted in various symmetries of thecurrents.
Vector and axial-vector currents are four-vectors and thussatisfy e − i e · K θ J Hµ ( x ) e i e · K θ = Λ ( θ ) νµ J Hν ( Λ ( θ ) − x ) , (25)where J Hµ is a (axial) vector current in Heisenberg picture, K is a boost generator, e is a boost direction, θ is a boostangle and Λ ( θ ) is a 4 × e − i e · K eff θ J H eff µ ( x ) e i e · K eff θ = Λ ( θ ) νµ J H eff ν ( Λ ( θ ) − x ) , (26)where H eff = U † HU, K eff = U † K U, (27)and J H eff µ ( x ) = e i H eff x U † J µ ( x ) U e − i H eff x . (28)In order to keep the notation short we denote from now onthe effective current U † J µ ( x ) U in the Schr¨odinger pictureby J µ ( x ) such that Eq. (28) turns in this notation to J H eff µ ( x ) = e i H eff x J µ ( x ) e − i H eff x . (29)Note that the effective boost operator K eff has a block-diagonal form like the effective Hamiltonian H eff . Thereason is that the whole Poincar´e algebra get’s a block-diagonal form after the application of unitary transforma-tion U . A perturbative proof of this statement to all orderscan be found in [22] for a special model and in [23] for anarbitrary local field theory.Expanding Eq. (26) in θ and comparing the coefficientswe get − i (cid:104) e · K eff , J H eff µ ( x ) (cid:105) = J H eff , ⊥ µ ( x ) − x ⊥ α ∂ αx J H eff µ ( x ) , (30)where we used Λ ( θ ) x = x + θx ⊥ + O ( θ ) , (31)and x ⊥ = ( e · x , e x ) , (32)where e is a unit vector which is a boost direction. In thenext step, we use the Poincar´e algebra relation to get e − i H eff x K eff e i H eff x = K eff − x P , (33) where P denotes the momentum operator. Using Eq. (33)we can rewrite Eq. (30) into a well known relation [20] − i (cid:104) e · K eff , J µ ( x ) (cid:105) = J ⊥ µ ( x ) − i e · x (cid:2) H eff , J µ ( x ) (cid:3) , (34)where we used the relation (cid:104) i e · P , J µ ( x ) (cid:105) = − e · ∇ x J µ ( x ) . (35)Now we transform Eq. (34) into momentum space and get − i (cid:104) e · K eff , ˜ J µ ( k ) (cid:105) = J ⊥ µ ( k ) − e · ∇ k (cid:2) H eff , ˜ J µ ( k ) (cid:3) , (36)where ˜ J µ ( k ) = (cid:90) d x e i k · x J µ ( x ) . (37)At this stage the effective current operator ˜ J µ ( k ) doesnot depend on energy-transfer k since sofar we did notapply any time-dependent unitary transformation. As weare interested in a more general current operator we applynow these transformations and get˜ J µ ( k ) → ˜ J µ ( k ) + i k ˜ Y µ ( k ) − i (cid:2) H eff , ˜ Y µ ( k ) (cid:3) , (38)where ˜ Y µ ( k ) is some local hermitian operator. Our goalis to derive a consistency relation for the operator ˜ J µ ( k )which should be a generalization of Eq. (36). The onlyinformation about an operator ˜ Y µ ( k ) which we will use isits locality property (cid:2) P , ˜ Y µ ( k ) (cid:3) = − k ˜ Y µ ( k ) . (39)In particular, we do not require the operator ˜ Y µ ( k ) to bea four-vector. In order to derive the relation we rewritethe commutator (cid:2) K eff , (cid:2) H eff , ˜ Y µ ( k ) (cid:3)(cid:3) = (cid:2) H eff , (cid:2) K eff , ˜ Y µ ( k ) (cid:3)(cid:3) + (cid:2)(cid:2) K eff , H eff (cid:3) , ˜ Y µ ( k ) (cid:3)(cid:3) . (40)Using Poincar´e algebra relation (cid:2) K eff , H eff (cid:3) = − i P , (41)we get (cid:2) K eff , (cid:2) H eff , ˜ Y µ ( k ) (cid:3)(cid:3) = (cid:2) H eff , (cid:2) K eff , ˜ Y µ ( k ) (cid:3)(cid:3) + i k ˜ Y µ ( k ) . (42)Using this result in combination with Eq. (36) we get − i (cid:104) e · K eff , ˜ J µ ( k ) (cid:105) = ˜ J ⊥ µ ( k ) − e · ∇ k (cid:104) H eff , ˜ J µ ( k ) (cid:105) − e · k ∂∂k ˜ J µ ( k ) + i (cid:104) H eff , ˜ X µ ( k ) (cid:105) − ik ˜ X µ ( k ) , (43) For x = 0 e.g. this relation can be found in Eqs. (93a) and(93b) of the seminal work of Friar [21], see also [19] for moregeneral case. Hermann Krebs: Nuclear Currents in Chiral Effective Field Theory where a hermitian operator X µ is defined by˜ X µ ( k ) = ˜ Y ⊥ µ ( k ) + i (cid:2) e · (cid:0) K eff + i H eff ∇ k (cid:1) , ˜ Y µ ( k ) (cid:3) . (44)Note, ˜ J µ ( k ) is linear in k , so we have˜ Y µ ( k ) = − i ∂∂k ˜ J µ ( k ) . (45)For this reason, we can rewrite the ˜ X µ ( k ) operator interms of ˜ J µ ( k ).Eq. (43) is a final consistency relation which buildson a four-vector property of the (axial) vector current. Asomewhat different derivation of this result can be foundin [23] where, however, we did not specify the form ofthe operator ˜ X µ ( k ). In this respect, Eq. (43) with theadditional Eq. (44) includes more information than Eq.(2.78) of [23].Eq. (43) relates the charge and current operators witheach other. In particular it allows to extract a charge op-erator out of the current operator. To see this we multiplythe Eq. (43) by (0 , − e ) and get˜ J ( k ) + (cid:20) H eff , ∂∂k ˜ J ( k ) (cid:21) − k ∂∂k ˜ J ( k ) = − i (cid:20) Z, (cid:18) − k ∂∂k (cid:19) e · ˜ J ( k ) (cid:21) + e · k ∂∂k e · ˜ J ( k ) − i (cid:20) H eff , (cid:2) Z, ∂∂k e · ˜ J ( k ) (cid:3)(cid:21) , (46)where Z = e · (cid:0) K eff + i H eff ∇ k (cid:1) . (47)Since the sum of the second and third term on the lefthand side of Eq. (46) is unobservable, (cid:104) α | (cid:18)(cid:20) H eff , ∂∂k ˜ J ( k ) (cid:21) − k ∂∂k ˜ J ( k ) (cid:19) | β (cid:105) =( E α − E β − k ) (cid:104) α | ∂∂k ˜ J ( k ) | β (cid:105) = 0 , (48)Eq. (46) determines the charge operator (modulo unob-servable off-shell effects) once the current operator is known.So for practical calculations one can always use the righthand side of Eq. (46) as an energy transfer independentcharge operator. This operator, however, can not be usedto test the continuity equation since in that case the off-shell information of the charge operator is essential, seenext paragraph for explanation. It is interesting that theboost transformation constrains the charge operator insuch a way that one can express it either by longitudi-nal current, for the choice e = k / | k | , or by transversecurrent for the choice e = k ⊥ , or by linear combination ofthem for other choices of boost direction. Chiral effective field theory is by construction invariantunder chiral SU(2) L × SU(2) R as well as U (1) V transforma-tions. As a consequence this leads to various Ward - iden-tities for amplitudes and continuity equations for current operators. Continuity equation for the currents follows di-rectly from the requirement that the Hamilton operatorin the presence of external sources is unitary equivalent tothe Hamiltonian in the presence of transformed externalsources. This means that there exists a (time-dependent)unitary transformation U ( t ) such that H (cid:48) eff [ s (cid:48) , ˙ s (cid:48) , p (cid:48) , ˙ p (cid:48) , a (cid:48) , ˙ a (cid:48) , v (cid:48) , ˙ v (cid:48) ] = (cid:18) i ∂∂t U ( t ) † (cid:19) U ( t )+ U ( t ) † H (cid:48) eff [ s, ˙ s, p, ˙ p, a, ˙ a, v, ˙ v ] U ( t ) . (49)Here, the primed external sources denote the transformedsources. Considering infinitesimal chiral or U (1) V trans-formations, expanding both sides of Eq. (49) up to thefirst order in transformation angles, and comparing thecoefficients in front of transformation angles we get thecontinuity equation. For the vector current, we get C V,A ( k ,
0) + (cid:2) H eff , ∂∂k C V,A ( k , k ) (cid:3) = 0 , (50)where for the electromagnetic vector current, the quantity C V is defined via C V ( k ) = (cid:2) H eff , ˜ V ( k ) (cid:3) − k · ˜ V ( k ) , (51)and for the axial vector current, C A is C A ( k ) = (cid:2) H eff , ˜ A a ( k ) (cid:3) − k · ˜ A a ( k ) + im q ˜ P a ( k ) . (52)Here we denote vector, axial vector and pseudoscalar cur-rents in momentum space by ˜ V µ ( k ) , ˜ A aµ ( k ) and ˜ P a ( k ), re-spectively. In the derivation of the continuity equation(50), we used the fact that the energy-transfer depen-dence of the current is at most linear. For more generalenergy-transfer dependence the continuity equation getsmore complicated form with increasing number of nestedcommutators if the power of energy-transfer dependenceincreases. There is, however, a way to give a general con-tinuity equation for currents without specification of theirenergy-transfer dependence. In Appendix B we prove thefollowing general continuity equations: for the vector cur-rent one getsexp (cid:32) H eff −→ ∂∂k (cid:33) k µ ˜ V µ ( k ) exp (cid:32) − H eff ←− ∂∂k (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k =0 = 0 , (53)and for the axial-vector currentexp (cid:32) H eff −→ ∂∂k (cid:33) (cid:104) k µ ˜ A µ ( k ) + i m q ˜ P ( k ) (cid:105) × exp (cid:32) − H eff ←− ∂∂k (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k =0 = 0 . (54)Between exponential operators in Eqs. (53) and (54), wefind structures which should vanish in the classical limit asa consequence of the continuity equation. k -derivatives in ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 9 the exponentials generate an increasing number of nestedcommutators with the effective Hamiltonian. If we sand-wich continuity equations (53) and (54) between initialand final eigenstates of the full Hamiltonian H eff | i (cid:105) = E i | i (cid:105) , H eff | f (cid:105) = E f | i (cid:105) , (55)we get the classical continuity equations k µ ˜ V µ ( k ) | k = E f − E i = 0 ,k µ ˜ A µ ( k ) + i m q ˜ P ( k ) | k = E f − E i = 0 . (56)For derivation of Eq. (56) we used the relation for energy-shift operatorexp (cid:32) E f −→ ∂∂k (cid:33) f ( k ) = f ( k + E f ) ,f ( k ) exp (cid:32) − E i ←− ∂∂k (cid:33) = f ( k − E f ) , (57)which are valid for any infinitely differentiable function f . So one can interpret the continuity equations (53) and(54) as an energy-independent form of classical continuityequation (56). The energies are replaced by correspondingeffective Hamiltonians by using energy-shift operator. In this section, we summarize all expressions of the nuclearcurrents up to order Q in chiral expansion. Q denotes mo-menta and masses which are much smaller than the chiralsymmetry breaking scale. We skip here the discussion oftheir construction. As an example, we discuss here Feyn-man diagrams which contribute to the two-nucleon vectorcurrent at leading order Q − . All other details about Feyn-man diagrams and specification of unitary phases can befound in [23,141,142,165,166]. To organize chiral EFT calculations of nuclear current op-erators we follow Weinberg’s analysis [45,46]. A Feynmandiagram contributing to the current operator counts as Q ν . To derive the expression for the chiral dimension ν ,we consider a generic Feynman diagram which is propor-tional to the integral written symbolically as δ (4) ( p ) C (cid:90) ( d q ) L q ) I p q I n (cid:89) i ( q d i ) V i , (58) This expression is valid for irreducible Feynman diagrams.Nuclear forces and currents are derived within the Hamiltonianapproach where one has to deal with time-ordered structures.Propagators are given in form of energy denominators, one hasto take into account phase space factors, and loop integralsare three-dimensional. However, dimensional counting in theHamiltonian approach leads to the same expression for ν asif one would deal with an irreducible Feynman diagram in afour-dimensional formalism [45,46]. where L is the number of loops, I p and I n are numberof internal pion and nucleon lines, respectively. d i is thenumber of derivatives or pion mass insertions in the vertex” i ”, V i denotes how many times the vertex ” i ” appears ina given diagram, and C denotes the number of connectedpieces in the diagram. From Eq. (58) we read off the index ν : ν = 4 − C + 4 L − I p − I n + (cid:88) i V i d i . (59)The couplings of external sources are not taken into ac-count in Eq. (59). We treat them as small but count themseparately. They are also not taken into account in d i .For example, d i = 0 for the leading-order photon-nucleoncoupling. Using the identities (cid:88) i V i n i = 2 I n + E n (60) (cid:88) i V i p i = 2 I p + E p , (61)where n i and p i denotes the number of nucleon and pionfields in the vertex ” i ”, respectively. E n and E p denote thenumber of external nucleon and pion lines, respectively. Awell known topological identity which connects the num-ber of loops with the number of internal lines is given by L = C + I p + I n − (cid:88) i V i . (62)Using Eqs. (60), (61) and (62) we get ν = 4 − E n − E p + (cid:88) i V i ˜ κ i , (63)where ˜ κ i is given by˜ κ i = d i + 32 n i + p i − . (64)We are not interested here in the pion production, so thenumber of external pions E p = 0. The number of nucleonsis always conserved and we denote it by N = 2 E n . (65)The expression for the chiral dimension is then given by ν = 4 − N + (cid:88) i V i ˜ κ i . (66)As was pointed out in [68], Eq. (66) is inconvenient since itdepends on the total number of nucleons N . For example,one-pion exchange diagram in the two-nucleon system hasthe chiral order ν = 0 since N = 2, ˜ κ = 1 and V = 2. Inthe presence of a third nucleon which acts as a spectator,it has chiral order ν = − N = 3. The origin of this discrepancy lies in the differentnormalization of two- and three-nucleon states:2 N : (cid:104) p p | p (cid:48) p (cid:48) (cid:105) = δ (3) ( p (cid:48) − p ) δ (3) ( p (cid:48) − p ) , N : (cid:104) p p p | p (cid:48) p (cid:48) p (cid:48) (cid:105) = δ (3) ( p (cid:48) − p ) δ (3) ( p (cid:48) − p ) δ (3) ( p (cid:48) − p ) . (67) One can circumvent this if one assigns a chiral dimensionto the transition operator rather than to its matrix ele-ment in N -nucleon system. In this case, we have to modifythe expression for ν by adding 3 N to Eq. (66), accountingin this way for the normalization of the N -nucleon system.The expression for ν becomes independent of N but gives ν = 6 for one-pion-exchange. As was proposed in [68], weadjust the final expression for ν by subtracting from it 6to get ν = 0 for one-pion-exchange, which is a conven-tion. The final expression for the chiral dimension of thetransition operator becomes ν = − (cid:88) i V i ˜ κ i . (68)Similar to[148], we can also express the chiral dimension ν in terms of the inverse mass dimension of the couplingconstant at a vertex ” i ” κ i = d i + 32 n i + p i + s i − , (69)where s i is the number of external sources which for thecurrent operators can be only 0 or 1. Consistent with [148],we get ν = − (cid:88) i V i κ i . (70)for the current operator and ν = − (cid:88) i V i κ i . (71)for the nuclear force.For the counting of the nucleon mass m , we adopta two-nucleon power counting where 1 /m -contributionscount as two powers of Q [46] Qm ∼ Q Λ χ . (72)Here Λ χ ∼
700 MeV and m are the chiral symmetry break-ing scale and the nucleon mass, respectively. Q We start our discussion with electromagnetic vector cur-rent. Leading contribution to the vector current starts atthe order Q − . At this order there is only a contribution tothe single-nucleon charge operator. It is well known thatchiral expansion of the single-nucleon currents does notconverge well [5,171,172]. For moderate virtualities Q ∼ . , an explicit inclusion of ρ -meson is essential. Forthis reason the usual practice is to parametrize single-nucleon vector current by e.g. Sachs form factors and usetheir phenomenological form extracted from experimentaldata in practical calculations [194,195,196,197,198]. The general form of the single-nucleon current can be char-acterized by its non-relativistic one-over-nucleon-mass ex-pansion given symbolically by V = V + V /m + V /m , (73) V = V + V /m + V /m + V − shell . In terms of Sachs form factors, the non-relativistic chargeis parametrized by V = eG E ( Q ) ,V /m = i e m k · ( k × σ ) G M ( Q ) ,V /m = − e m (cid:2) Q + 2 i k · ( k × σ ) (cid:3) G E ( Q ) , (74)and the non-relativistic current is given by V = − i e m k × σ G M ( Q ) , V /m = em k G E ( Q ) , V /m = e m (cid:20) i k × σ (2 k + Q ) + 2 i k × k k · σ + 2 k ( i k · ( k × σ ) + Q ) − k k · k + 6 i k × σ k · k (cid:21) G M ( Q ) , (75)where k is a photon momentum, k = ( p (cid:48) + p ) /
2, and p (cid:48) ( p ) are outgoing (incoming) momenta of the single–nucleon current operator. Virtuality in our kinematics isgiven by Q = k . Note, that the form factors G E ( Q )and G M ( Q ) in Eqs. (74) and (75) are operators in isospinspace. The proton and neutron electromagnetic form fac-tors can be extracted out of them by projecting these tothe corresponding state.As already briefly explained in Sec. 3 (see [148] formore comprehensive discussion) we apply unitary trans-formations on the Hamilton operator which explicitly de-pends on an external source and thus on time. Theseunitary transformations generate off-shell contributions tothe longitudinal component of the current which dependon energy transfer k and additional relativistic 1 /m cor-rections. This contribution can also be parametrized bySachs form factors via V − shell = k (cid:18) k − k · k m (cid:19) eQ (cid:20)(cid:0) G E ( Q ) − G E (0) (cid:1) + i m k · ( k × σ ) (cid:0) G M ( Q ) − G M (0) (cid:1)(cid:21) . (76) We switch now to a discussion of the two-nucleon vectorcurrent operator. Various contributions can be character-ized by the number of pion exchanges and/or short-rangeinteractions V µ = V µ π + V µ π + V µ cont . (77) ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 11 Fig. 1.
Feynman diagrams which contribute to the two-nucleon vector current operator at order Q − . One-pion-exchange vector current
Leading contributionto the one-pion-exchange (OPE) current shows up at theorder Q − . The corresponding Feynman diagrams are listedin Fig. 1. At this order we get a well known result for cur-rent operator V ( Q − )2N:1 π = i eg A F π [ τ × τ ] q · σ q + M π (cid:18) q q · σ q + M π − σ (cid:19) + 1 ↔ , (78)and for the charge operator V , ( Q − )2N:1 π = 0 . (79)Here e is electric coupling, g A axial vector coupling tothe nucleon, F π pion decay constant, M π pion mass, σ i and τ i are Pauli spin and isospin matrices with label i =1 , q , aredefined by q i = p (cid:48) i − p i , (80)where i = 1 , p (cid:48) i or p i are outgoing or incomingmomenta of the i -th nucleon, respectively.There is no contribution at order Q such that thenext correction starts at the order Q which are leadingone-loop contributions in the static limit and/or leadingrelativistic correction to OPE current V µ, ( Q )2N:1 π = V µ, ( Q )2N:1 π, static + V µ, ( Q )2N:1 π, /m . (81)The corresponding set of diagrams can be found in [142].The explicit form of static contributions can be given interms of scalar functions f ... ( k ). The vector contributionis given by [142] V ( Q )2N:1 π, static = σ · q q + M π q × q (cid:2) [ τ ] f ( k )+ τ · τ f ( k )] + [ τ × τ ] σ · q q + M π (cid:26) k × [ q × σ ] f ( k ) + k × [ q × σ ] f ( k )+ σ · q (cid:18) k k − q q + M π (cid:19) f ( k )+ (cid:20) σ · q q + M π q − σ (cid:21) f ( k ) (cid:27) +1 ↔ , (82) where the scalar functions f i ( k ) are given by f ( k ) = 2 ie g A F π ¯ d , f ( k ) = 2 ie g A F π ¯ d ,f ( k ) = − ie g A F π π (cid:2) g A (2 L ( k ) −
1) + 32 F π π ¯ d (cid:3) ,f ( k ) = − ie g A F π ¯ d ,f ( k ) = − ie g A F π π (cid:20) M π + k ) L ( k ) + (cid:18) l − (cid:19) k − M π (cid:3) ,f ( k ) = − ie g A F π M π ¯ d . (83)Here ¯ d i are low-energy constants (LEC) from the order Q pion-nucleon Lagrangian [178]. ¯ l is a LEC from Q pion-Lagrangian [37]. Their values can be fixed from pion-nucleon scattering and pion-photo- or electroproduction.The charge contribution is given by V , ( Q )2N:1 π, static = σ · q q + M π [ τ ] (cid:20) σ · k q · k f ( k )+ σ · q f ( k ) (cid:21) +1 ↔ , (84)where f ( k ) = e g A F π π (cid:20) A ( k ) + M π − M π A ( k ) k (cid:21) ,f ( k ) = e g A F π π (cid:2) (4 M π + k ) A ( k ) − M π (cid:3) . (85)The loop function L ( k ) and A ( k ) are defined L ( k ) = 12 (cid:112) k + 4 M π k log (cid:32) (cid:112) k + 4 M π + k (cid:112) k + 4 M π − k (cid:33) ,A ( k ) = 12 k arctan (cid:18) k M π (cid:19) . (86)Relativistic corrections for the vector operator vanish V ( Q )1 π :1 /m = 0 . (87)Relativistic corrections for the charge operator are V , ( Q )2N:1 π, /m = eg A F π m q + M π (cid:26) (1 − β ) × (cid:0) [ τ ] + τ · τ (cid:1) σ · kσ · q − i (1 + 2 ¯ β ) [ τ × τ ] × (cid:20) σ · k σ · q − σ · k σ · q − σ · q q + M π σ · q × q · k (cid:21)(cid:27) + eg A F π m σ · q σ · q ( q + M π ) (cid:20) (2 ¯ β − × ([ τ ] + τ · τ ) q · k + i [ τ × τ ] (cid:0)(cid:0) β − (cid:1) q · k − (cid:0) β + 1 (cid:1) q · k (cid:1)(cid:21) + 1 ↔ . (88) Here ¯ β and ¯ β are phases from unitary transformationswhich are not fixed. The same phases show up in nuclearforces. Usually they are fixed by requirement of minimalnon-locality of the OPE NN potential. Two-pion-exchange vector current
Contributions to two-pion-exchange (TPE) vector current start to show up atorder Q . They are parameter-free. The corresponding di-agrams can be found in [141]. Due to the coupling ofthe vector source to two pions there appear loop func-tions which depend on three momenta k , q and q whichare momentum transfer of the vector source, momentumtransfer of the first and second nucleons, respectively. Thisleads to a somewhat lengthy expression which have beenderived in [141] and are listed in Appendix D for com-pleteness: Short-range vector current
The first contribution to short-range two-nucleon current shows up at the order Q . Thediagrams with short-range interactions at this order canbe found in [142]. There are two contributions [142] V µ, ( Q )2N: cont = V µ, ( Q )2N: cont , tree + V µ, ( Q )2N: cont , loop . (89)The current contribution coming from tree-diagrams isgiven by V ( Q )2N: cont , tree = e i
16 [ τ × τ ] (cid:20) ( C + 3 C + C ) q − ( − C + C + C ) ( σ · σ ) q + C ( σ · q σ + σ · q σ ) (cid:21) − e C i
16 [ τ ] [( σ + σ ) × q ]+ ieL [ τ ] [( σ − σ ) × k ]+ ieL [( σ + σ ) × q ] . (90)As can be seen from Eq. (90) there are C i LECs which alsocontribute to the two-nucleon potential and appear heredue to the minimal coupling, and there are two additionalconstants L , which describe entirely electromagnetic ef-fects. Charge short-range contribution from tree diagramsat order Q vanishes V , ( Q )2N: cont , tree = 0 . (91)There are also contributions from one-loop diagrams whichinclude one leading-order two-nucleon contact interactionand two-pion propagators. They only contribute to chargeoperator V , ( Q )2N: cont , loop = C T [ τ ] [ σ · k σ · k f ( k )+ σ · σ f ( k )] , (92) Different conventions are being used in the literature forthe leading-order two-nucleon contact interactions ∝ C S,T . Tomatch the convention of Refs. [88,90,89], the factors of 32 F π in Eq. (5.7) of [142] should be replaced by 16 F π . where f ( k ) = e g A F π π (cid:18) A ( k ) + M π − M π A ( k ) k (cid:19) ,f ( k ) = e g A F π π (cid:0) M π − (4 M π + 3 k ) A ( k ) (cid:1) . (93)Corresponding contributions to the current operator van-ish V ( Q )cont: loop = 0 . (94) At the order Q there are first contributions to three-nucleonvector current [148]. There are no contributions to the vec-tor operator V ( Q )3N = 0 . (95)Contributions to the charge operator can be parametrizedin the form V , ( Q )3N = (cid:20) ( q + q ) · σ ( q + q ) + M π + q · σ q + M π (cid:21) × (cid:0) v + v (cid:1) + 5 permutations . (96)where the long- and short-range contributions are givenby v = e F π q · σ q + M π (cid:20)(cid:0) τ · τ [ τ ] − τ · τ [ τ ] (cid:1) × (cid:18) − g A q + q · q ( q + q ) + M π + g A (cid:19) − i [ τ × τ ] g A q + q · q ( q + q ) + M π (cid:21) , (97) v = − [ τ × τ ] e g A C T F π ( q + q ) · ( σ × σ )( q + q ) + M π . (98)Due to the approximate spin-isospin SU(4) Wigner sym-metry [149], C T appears to be small such that we do notexpect large contributions from short-range part of thethree-nucleon vector current. Q The weak sector of nuclear physics can be probed by anuclear axial-vector current. We give here its expressionsup to order Q in chiral expansion. The leading-order contribution to an axial vector currentshows up at order Q − where axial-vector source couples ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 13 directly to a single-nucleon or to a pion, which itself prop-agates and couples to a single-nucleon generating in thisway a pion-pole term. It is convenient to parametrize thesingle-nucleon current by the axial and pseudoscalar formfactors. Up to the order Q the parametrization is given by A µ,a = A µ,a − shell + A µ,a − shell . (99)The charge operator is parametrized by A ,a − shell = [ τ ] a (cid:20) − k · σ m G A ( t ) + k · k m (cid:18) k · σ m − k · σ ( k + 4 k ) + 4 k · k k · σ m (cid:19) × G P ( t ) (cid:21) ,A ,a − shell = 0 . (100)The current operator is parametrized by A a − shell = [ τ ] a (cid:20)(cid:18) − σ + 18 m (cid:16) σ k − k k · σ +2 i k × k + k k · σ (cid:17)(cid:19) G A ( t ) + k (cid:18) k · σ m − m (cid:16) k · σ k + 4 k · σ k + 4 k · σ k · k (cid:17)(cid:19) G P ( t ) (cid:21) , A a − shell = k (cid:18) k − k · k m (cid:19) [ τ ] a m (cid:104) − (1 + 2 β ) × k · σ G P ( t ) + (1 + 2 β ) k · k k · σ G (cid:48) P ( t ) (cid:105) , (101)Chiral EFT results are given by chiral expansion of theaxial and pseudoscalar form factors. Corresponding ex-pressions are worked out in [23]. To make this review self-consistent we will briefly discuss them here.The well-known leading-order result for the axial chargeand current operators have the form A ,a ( Q − )1N: static = 0 , A a ( Q − )1N: static = − g A τ i ] a (cid:18) σ i − k k · σ i k + M π (cid:19) . (102)There are only vanishing contributions at order Q − . Atorder Q − , we encounter three kinds of corrections. First,there are terms emerging from the time-dependence of uni-tary transformations which have the form A ,a ( Q − )1N: UT (cid:48) = g A k k + M π k · σ i [ τ i ] a , (103) A a ( Q − )1N: UT (cid:48) = 0 . (104)and contribute to A µ,a − shell . Secondly, at order Q − there are static limit contributions to A µ,a − shell whichare given by A ,a ( Q − )1N: static = 0 , A a ( Q − )1N: static = 12 ¯ d (cid:16) σ i k − k k · σ i (cid:17) [ τ i ] a − ¯ d M π [ τ i ] a k k · σ i k + M π . (105) Finally, there are leading relativistic 1 /m -corrections whichin our counting scheme start contributing at order Q − to A µ,a − shell and read A ,a ( Q − )1N: 1 /m = − g A m [ τ i ] a σ i · k i , (106) A a ( Q − )1N: 1 / m = 0 , (107)where k = p i (cid:48) − p i , k i = p i (cid:48) + p i . (108)There are no corrections to the 1N charge and currentoperators at the order Q . Finally, there are various con-tributions at order Q . The off-shell contributions comingfrom time-dependent unitary transformations give differ-ent contributions. One of them is coming from relativisticcorrections which are proportional to k /m and are givenby A ,a ( Q )1N: 1 /m, UT (cid:48) = 0 , A a ( Q )1N: 1 /m, UT (cid:48) = − g A k m k k + M π [ τ i ] a (cid:18) β ) σ i · k i − (1 + 2 ¯ β ) k · σ i p (cid:48) i − p i k + M π (cid:19) . (109)They explicitly depend on unitary phases ¯ β and ¯ β . Notethat these are the same unitary phases that influence anon-locality degree of relativistic 1 /m corrections of theone-pion-exchange nuclear force. For the static part whichis proportional to k , we get nonvanishing contributions A ,a ( Q )1N: static , UT (cid:48) = − k [ τ i ] a k · σ i (cid:20) ¯ d + 2 ¯ d M π k + M π (cid:21) , A a ( Q )1N: static , UT (cid:48) = 0 . (110)The second class of order- Q contributions involves rela-tivistic 1 /m -corrections: A ,a ( Q )1N: 1 /m = 0 , A a ( Q )1N: 1 / m = g A m [ τ i ] a (cid:18) k k · σ i (1 − β ) ( p (cid:48) i − p i ) ( k + M π ) − k ( p (cid:48) i + p i ) k · σ i − β ( p (cid:48) i − p i ) k i · σ i k + M π + 2 i [ k × k i ] + k k · σ i − k i k i · σ i + σ i (cid:16) p (cid:48) i + p i ) − k (cid:17)(cid:19) . (111)These are a linear combination of on-shell and off-shellcontributions. The third kind of order- Q contributionsemerges from relativistic 1 /m -corrections to the leadingone-loop terms which contributes to A µ,a − shell : A ,a ( Q )1N: 1 /m = ¯ d k i · σ i [ τ i ] a k m , (112) A a ( Q )1N: 1 / m = 0 . (113) Finally, static two-loop contributions to the on-shell cur-rent are given by A ,a ( Q )1N: static = 0 , A a ( Q )1N: static = −
12 [ τ i ] a σ i (cid:16) − f A M π k + f A k + G ( Q ) A ( − k ) (cid:17) + 18 k k · σ i [ τ i ] a (cid:16) − f A M π − f P k + G ( Q ) P ( − k ) (cid:17) . (114)Here we perform the chiral expansion of the axial formfactor which can be found e.g. in [150,151], see also [152,153] for results obtained within Lorentz-invariant formu-lations. Rewritten in our notation, the chiral expansion ofthe axial form factor is given by G A ( t ) = g A + ( ¯ d + f A M π ) t + f A t + G ( Q ) A ( t )+ O ( Q ) , (115)where f Ai are LECs of dimension GeV − and G ( Q ) A ( t ) = t π (cid:90) ∞ M π Im G ( Q ) A ( t (cid:48) ) t (cid:48) ( t (cid:48) − t − i(cid:15) ) dt (cid:48) , (116)with the imaginary part calculated utilizing the Cutkoskyrules [150]Im G ( Q ) A ( t ) = g A π F π (cid:90) z < dω dω (cid:20) g A ( √ t ω − M π ) × (cid:16) l l + z (cid:17) arccos( − z ) √ − z + 2 g A (cid:16) M π − √ t ω − ω (cid:17) + M π − √ t ω + 2 ω (cid:21) , (117)where ω i = (cid:113) l i + M π with i = 1 , , and z = ˆ l · ˆ l = ω ω − √ t ( ω + ω ) + ( t + M π ) l l . (118)Here and in what follows, l i ≡ | l i | , while ˆ l i ≡ l i /l i . Thepseudoscalar form-factor up to order Q is given by [158] G P ( t ) = 4 mg πN F π M π − t − g A m (cid:104) r A (cid:105) + m f P t + m G ( Q ) P ( t ) + O ( Q ) , (119)where f Pi denotes the corresponding linear combinationsof the LECs of dimension GeV − from L (5) πN and G ( Q ) P ( t ) = t π (cid:90) ∞ M π Im G ( Q ) P ( t (cid:48) ) t (cid:48) ( t (cid:48) − t − i(cid:15) ) dt (cid:48) , (120) with the imaginary part calculated using the Cutkoskyrules [158]Im G ( Q ) P ( t ) = Im G (1) P ( t ) + Im G (2) P ( t ) (121)andIm G (1) P ( t ) = g A π F π (cid:90) z < dω dω (cid:20) − M π t − M π ) + 4 ω − M π t + ω (3 M π − t )( t − M π ) + 2 M π ω ω zt ( t − M π ) l l (cid:21) , Im G (2) P ( t ) = g A π F π t (cid:90) z < dω dω (cid:20) ( M π − √ tω ) × (cid:18) z + l l (cid:19) arccos( − z ) √ − z + l t M π t − M π (cid:18) √ t − ω − ω (cid:19)(cid:18) ω z l l + √ t + (cid:16) ( t + M π )(4 ω − √ t ) − √ tω ω (cid:17) × arccos( − z )2 l l √ − z (cid:19)(cid:21) . (122)It is important to note that the induced pseudoscalar formfactor is related to the induced pseudoscalar coupling con-stant g P = M µ m G P ( t = − . M µ ) , (123)which is measured in muon capture experiment [154]. Fortheoretical determination of g P by using chiral Ward iden-tities of QCD we refer to a groundbreaking work [155], seealso [156,157].In practical calculations, alternatively to the chiral ex-pansion of the axial and pseudoscalar form factors, one cantake their empirical parametrization [156]. This is in par-ticular reasonable if we would like to consider electroweakprobes of nuclei without being affected by the convergenceissue of the chiral expansion of electroweak single-nucleoncurrents. We now switch to a discussion of the two-nucleon ax-ial vector current operator. Various contributions can becharacterized by the number of pion exchanges and/orshort-range interactions A µ,a = A µ,a π + A µ,a π + A µ,a . (124) One-pion-exchange axial vector current
Leading contri-bution to the one-pion-exchange (OPE) current shows up ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 15 at the order Q − . At this order we get a well known re-sult [159,160,137] A ,a ( Q − )2N: 1 π = − ig A q · σ [ τ × τ ] a F π ( q + M π ) + 1 ↔ , (125) A a ( Q − )2N: 1 π = 0 . (126)At the order Q there are only contributions to the vectoroperator A a ( Q )2N: 1 π = g A F π σ · q q + M π (cid:26) [ τ ] a (cid:20) − c M π k k + M π +2 c (cid:18) q − k k · q k + M π (cid:19)(cid:21) + c [ τ × τ ] a (cid:18) q × σ − k k · q × σ k + M π (cid:19) − κ v m [ τ × τ ] a k × σ (cid:27) +1 ↔ , (127)where c i denote the LECs from L (2) πN and κ v is the isovectoranomalous magnetic moment of the nucleon. At the order Q there are leading one-loop contributions in the staticlimit and/or leading relativistic correction to OPE current A µ,a ( Q )2N:1 π = A µ,a ( Q )2N:1 π, static + A µ,a ( Q )2N:1 π, /m + A µ,a ( Q )2N:1 π, UT (cid:48) . (128)The explicit form of static contributions can be given interms of scalar functions h ... ( q ). The vector contribu-tion is given by A a ( Q )2N:1 π, static = 4 F π g A q · σ q + M π (cid:110) [ τ × τ ] a (cid:16) [ q × σ ] h ( q ) + [ q × σ ] h ( q ) (cid:17) +[ τ ] a (cid:0) q − q (cid:1) h ( q ) (cid:111) + 4 F π g A q · σ k ( k + M π )( q + M π ) (cid:110) [ τ ] a h ( q ) + [ τ × τ ] a q · [ q × σ ] h ( q ) (cid:111) + 1 ↔ , (129)and the charge contribution is given by A ,a ( Q )2N: 1 π, static = i F π g A q · σ q + M π (cid:110) [ τ × τ ] a (cid:0) h ( q )+ k h ( q ) (cid:1) + [ τ ] a q · [ q × σ ] h ( q ) (cid:111) + 1 ↔ , (130) where the scalar functions h i ( q ) are given by h ( q ) = − g A M π πF π ,h ( q ) = g A M π πF π + g A A ( q ) (cid:0) M π + q (cid:1) πF π ,h ( q ) = g A (cid:0) g A + 1 (cid:1) M π πF π + g A A ( q ) (cid:0) M π + q (cid:1) πF π ,h ( q ) = g A πF π (cid:16) A ( q ) (cid:0) M π + 5 M π q + 2 q (cid:1) + (cid:0) g A + 1 (cid:1) M π + 2 (cid:0) g A + 1 (cid:1) M π q (cid:17) ,h ( q ) = − g A πF π (cid:16) A ( q ) (cid:0) M π + q (cid:1) + (cid:0) g A + 1 (cid:1) M π (cid:17) ,h ( q ) = g A (cid:0) (cid:0)
64 + 128 g A (cid:1) M π + 8 (cid:0) g A + 5 (cid:1) q (cid:1) π F π − g A π F π L ( q ) (cid:0) (cid:0) g A + 4 (cid:1) M π + (cid:0) g A + 1 (cid:1) q (cid:1) + ¯ d g A M π F π − ¯ d g A M π F π − g A (2 ¯ d + ¯ d ) (cid:0) M π + q (cid:1) F π ,h ( q ) = g A (2 ¯ d − ¯ d )16 F π ,h ( q ) = − g A ( ¯ d − d )8 F π . (131)Here ¯ d i are low-energy constants (LEC) from Q pion-nucleon Lagrangian. Their values can be fixed from pion-nucleon scattering and axial-pion-production.The relativistic corrections for the charge operator vanish A ,a ( Q )2N: 1 π, /m = 0 , (132)and for the current operator are given by A a ( Q )2N: 1 π, /m = g A F π m (cid:26) i [ τ × τ ] a (cid:20) q + M π ) (cid:18) B − k k · B k + M π (cid:19) + 1 q + M π (cid:18) B ( k + M π ) + B k + M π + B (cid:19)(cid:21) +[ τ ] a (cid:20) q + M π ) (cid:18) B − k k · B k + M π (cid:19) + 1 q + M π (cid:18) B ( k + M π ) + B k + M π + B (cid:19)(cid:21)(cid:27) +1 ↔ , (133) where the vector-valued quantities B i depend on variousmomenta and the Pauli spin matrices and are given by B = g A q · σ [ − β ) q k · q − (1 − β )(2 q k · q − i q × σ k · q ] , B = (1 − β ) g A k k · q q · σ [2 k · k − i k · q × σ ] , B = 2 k (cid:104) − g A ((1 + 2 ¯ β ) k · q k · σ + (1 − β ) q · σ ( k · k + k · q ))+ q · σ ( k · k + i k · q × σ − k · q + k · q ) (cid:105) , B = g A [2(1 + 2 ¯ β ) q k · σ + (1 − β ) q · σ (2 k − i k × σ )] − q · σ ( i q × σ − i k × σ + 2 k ) , B = g A q · σ (cid:104) (1 − β )( q k · q − i q × σ k · q ) − i (1 + 2 ¯ β ) q × σ k · q (cid:105) , B = − (1 − β ) g A k q · σ [( k · q ) − i k · k k · q × σ ] , B = g A k (cid:104) (1 − β ) q · σ ( − i ( k · k × σ + k · q × σ ) + k + q ) − i (1 + 2 ¯ β ) k · σ k · q × σ (cid:105) , B = − g A [(1 − β ) q · σ ( k − i k × σ ) − i (1 + 2 ¯ β ) q × σ k · σ ] . (134)Finally, there are also energy-transfer dependent contri-butions to OPE axial vector current at order Q which aregiven by A ,a ( Q )2N: 1 π, UT (cid:48) = 0 , (135) A a ( Q )2N: 1 π, UT (cid:48) = − i g A F π k k q · σ ( k + M π )( q + M π ) (cid:18) [ τ × τ ] a (cid:18) − g A k · q k + M π (cid:19) − g A [ τ ] a k · [ q × σ ] k + M π (cid:19) +1 ↔ . (136)It is important to note that it is not enough to know thecurrents at vanishing energy-transfer k = 0. As will bedemonstrated later the knowledge of the slope in energy-transfer k is essential for checking the continuity equa-tions. All expressions proportional to the energy-transfer k are off-shell effects which disappear in the calculationof on-shell observables. Energy-transfer contributions arealways accompanied by the commutator with the effectiveHamiltonian. On-shell a linear combination of k -term andthe commutator with the effective Hamiltonian k X − (cid:2) H eff , X (cid:3) (137)vanishes. Here X stays for some operator. More on thiswill be discussed in Sec. 5.6. Two-pion-exchange axial vector current
Contributionsto the two-pion-exchange axial vector current start to showup at order Q . These contributions are parameter-free.The final results for the two-pion exchange operators read A a ( Q )2N: 2 π = 2 F π g A k k + M π (cid:26) [ τ ] a (cid:16) − q · σ q · k g ( q )+ q · σ g ( q ) − k · σ g ( q ) (cid:17) + [ τ ] a (cid:16) − q · σ q · k g ( q ) − k · σ g ( q ) − q · σ q · k g ( q ) + q · σ g ( q )+ k · σ q · k g ( q ) − k · σ g ( q ) (cid:17) + [ τ × τ ] a (cid:16) − q · [ σ × σ ] q · k g ( q )+ q · [ σ × σ ] g ( q ) − q · σ q · [ q × σ ] g ( q ) (cid:17)(cid:27) + 2 F π g A (cid:26) q (cid:16) [ τ ] a q · σ g ( q )+ [ τ ] a q · σ g ( q ) (cid:17) − [ τ ] a σ g ( q ) − [ τ ] a σ g ( q ) − [ τ ] a σ g ( q ) (cid:27) + 1 ↔ , (138) A ,a ( Q )2N: 2 π = i F π g A (cid:26) [ τ × τ ] a q · σ g ( q )+ [ τ ] a q · [ σ × σ ] g ( q ) (cid:27) + 1 ↔ , (139)where the scalar functions g i ( q ) are defined as g ( q ) = g A A ( q ) (cid:0)(cid:0) g A − (cid:1) M π + (cid:0) g A + 1 (cid:1) q (cid:1) πF π q − g A M π (cid:0)(cid:0) g A − (cid:1) M π + (cid:0) g A − (cid:1) q (cid:1) πF π q (4 M π + q ) ,g ( q ) = g A A ( q ) (cid:0) M π + q (cid:1) πF π + g A M π πF π ,g ( q ) = − g A A ( q ) (cid:0)(cid:0) g A − (cid:1) M π + (cid:0) g A − (cid:1) q (cid:1) πF π − (cid:0) g A − (cid:1) g A M π πF π ,g ( q ) = − g A A ( q )128 πF π ,g ( q ) = − q g ( q ) ,g ( q ) = g ( q ) = g ( q ) = g ( q ) = 0 ,g ( q ) = g A A ( q ) (cid:0) M π + q (cid:1) πF π + (cid:0) g A + 1 (cid:1) g A M π πF π ,g ( q ) = g A M π πF π , ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 17 g ( q ) = − g A A ( q ) (cid:0) M π + q (cid:1) πF π − g A M π πF π ,g ( q ) = − g A A ( q )128 πF π ,g ( q ) = g A A ( q ) (cid:0)(cid:0) g A − (cid:1) M π + (cid:0) g A + 1 (cid:1) q (cid:1) πF π q + g A M π (cid:0)(cid:0) − g A (cid:1) M π + (cid:0) − g A (cid:1) q (cid:1) πF π q (4 M π + q ) ,g ( q ) = g A A ( q ) (cid:0)(cid:0) g A − (cid:1) M π + (cid:0) g A − (cid:1) q (cid:1) πF π + (cid:0) g A − (cid:1) g A M π πF π ,g ( q ) = g A A ( q ) (cid:0) M π + q (cid:1) πF π + g A M π πF π ,g ( q ) = − g A q A ( q )128 πF π ,g ( q ) = g A L ( q ) (cid:0)(cid:0) − g A (cid:1) M π + (cid:0) − g A (cid:1) q (cid:1) π F π (4 M π + q ) ,g ( q ) = g A L ( q )32 π F π . (140) Short-range axial vector current
The first contributionto short-range two-nucleon current shows up at the order Q . A ,a ( Q )2N: cont = 0 , A a ( Q )2N: cont = − D [ τ ] a (cid:18) σ − k σ · k k + M π (cid:19) + 1 ↔ , (141)where D denote the LEC from L (1) πNN . At the order Q we decompose the short-range current in three differentcomponents A µ,a, ( Q )2N:cont = A µ,a, ( Q )2N:cont , static + A µ,a, ( Q )2N:cont , /m + A µ,a, ( Q )2N:cont , UT (cid:48) . Static contributions are given by A a ( Q )2N: cont , static = 0 , (142) A ,a ( Q )2N: cont , static = iz [ τ × τ ] a σ · q + iz [ τ × τ ] a σ · q + iz [ τ ] a q · σ × σ + z ([ τ ] a − [ τ ] a )( σ − σ ) · k + 1 ↔ , (143)where LECs z i are unknown coefficients and have to befitted to experimental data. Relativistic corrections aregiven by A a ( Q )2N: cont , /m = − g A m k k + M π [ τ ] a (cid:26) (1 − β ) (cid:16) C S q · σ + C T ( q · σ + 2 i k · σ × σ ) (cid:17) − − β k + M π (cid:16) C S k · q k · σ + C T ( k · q k · σ +2 i k · k k · σ × σ ) (cid:17)(cid:27) + 1 ↔ . (144) Finally, the energy-transfer dependent contributions aregiven by A ,a ( Q )2N: cont , UT (cid:48) = 0 , A a ( Q )2N: cont , UT (cid:48) = − i k k g A C T k · σ [ τ × τ ] a ( k + M π ) + 1 ↔ . (145) At the order Q there are first contributions to three-nucleonaxial vector current. We decompose it into long and short-range contributions A µ,a ( Q )3N = A µ,a ( Q )3N: π + A µ,a ( Q )3N: cont (146)We start with the long-range part. There are no chargecontributions such that A ,a ( Q )3N: π = 0 . (147)Current contributions are given by A a ( Q )3N: π = − F π g A (cid:88) i =1 C ai + 5 permutations , (148)where spin-isospin dependent vector structures, which in-clude up to four pion propagators are given by C a = g A F π q · σ (cid:20) [ τ ] a (cid:104) q (( q · q + q )( τ · τ − σ · σ ) + q · σ ( q · σ + q · σ )) − q ( q · σ ( q · σ + q · σ ) − ( q · q + q ) σ · σ − ( q · q + q ) τ · τ ) − σ (( q · q + q ) q · σ − ( q · q + q ) q · σ ) (cid:105) − [ τ × τ ] a ( q × σ + q × σ )( q · q + q ) − ( q + q ) × ([ τ × τ ] a q · q × σ + [ τ ] a τ · τ ( q · q + q )) (cid:21) × q + M π ][( q − k ) + M π ] , C a = g A F π q + M π ][( q − k ) + M π ] × q · σ (cid:2) [ τ × τ ] a ( k × σ − q × σ )+ ( k − q )([ τ ] a τ · τ − [ τ ] a τ · τ ) (cid:3) , C a = g A F π q + M π ][ q + M π ][ q + M π ] × ( k − q )[ τ ] a τ · τ q · σ q · σ × ( k · σ − q · σ ) , C a = g A F π σ q · σ q · σ ([ τ ] a τ · τ − [ τ ] a τ · τ )[ q + M π ][ q + M π ] , C a = g A F π k q · σ [ k + M π ][ q + M π ][( q + q ) + M π ] × (cid:20) − [ τ ] a q · q × σ ( k · q × σ + k · q × σ ) − [ τ × τ ] a q · q × σ ( k · q + k · q )+ [ τ × τ ] a ( q · q + q )( k · q × σ + k · q × σ ) − ( τ · τ [ τ ] a − τ · τ [ τ ] a ) × ( q · q + q )( k · q + k · q ) (cid:21) , C a = − g A F π k q · σ [ k + M π ][ q + M π ][( q + q ) + M π ] × (cid:20) ( k · q × σ + k · q × σ ) [ τ × τ ] a − ( k · q + k · q + q · q + q ) × ( τ · τ [ τ ] a − τ · τ [ τ ] a ) − q · q × σ [ τ × τ ] a (cid:21) , C a = − g A F π k q · σ q · σ q · σ τ · τ [ τ ] a × (cid:0) M π + 2 q · q + q + q (cid:1) × k + M π ][ q + M π ][ q + M π ][ q + M π ] , C a = − g A F π k q · σ q · σ [ k + M π ][ q + M π ][ q + M π ] × ( τ · τ [ τ ] a ( q · σ + q · σ ) − τ · τ [ τ ] a ( q · σ + q · σ )) . (149)The short-range contributions to the charge operator van-ish A ,a ( Q )3N: cont = 0 , (150)and the short-range vector contributions are given by A a ( Q )3N: cont = − F π g A (cid:88) i =1 D ai + 5 permutations , (151)with D a = − g A C T F π (cid:2) ( k − q )[ τ × τ ] a ( k · σ × σ − q · σ × σ ) − ([ τ ] a − [ τ ] a )(( k − q ) σ · σ × ( k · σ − q · σ ) + σ (( k · σ − q · σ ) × ( k · σ − q · σ ) + (2 k · q − k − q ) σ · σ )) (cid:3) × q − k ) + M π ] , D a = g A C T F π k [ k + M π ][( q + q ) + M π ] ( q · σ × σ + q · σ × σ ) (cid:0) [ τ ] a k · q × σ + [ τ × τ ] a ( k − k · q )) , D a = − g A C T F π k [ k + M π ][( q + q ) + M π ] ( q · σ × σ + q · σ × σ ) [ τ × τ ] a . (152)Due to the smallness of C T , these contributions are ex-pected to be small. Q Approximate chiral symmetry leads to relations betweenpseudoscalar current and axial-vector current. In the fol-lowing, we list all expressions for pseudoscalar current upto order Q . Single-nucleon pseudoscalar current can be parametrizedby pseudoscalar form factors and their derivatives via i m q P a = i m q P a − shell + i m q P a − shell , (153)where i m q P a − shell = [ τ i ] a (cid:18) G A ( t ) + t m G P ( t ) (cid:19)(cid:18) k · σ (cid:18) − k + k m (cid:19) − k · k k · σ m + O (1 /m ) (cid:19) ,i m q P a − shell = − M π k k · A a − shell . (154)Chiral EFT results for pseudoscalar current are given bychiral expansion of the axial and pseudoscalar form fac-tors. Corresponding expressions are worked out in [23].Here we will briefly discuss them.The single-nucleon pseudoscalar current starts to con-tribute at the order Q − and is given by P a ( Q − )1N: static = i M π g A m q [ τ i ] a k · σ i k + M π = − i m q k · A a ( Q − )1N: static , (155)with A a ( Q − )1N: static given in Eq. (102). At the order Q − , thereare only vanishing contributions. At order Q − there areonly static limit contributions which are given by P a ( Q − )1N: static = i M π ¯ d m q k [ τ i ] a k · σ i k + M π = − i m q k · A a ( Q − )1N: static , (156) ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 19 with A a ( Q − )1N: static given in Eq. (105). There are no contribu-tions to the single-nucleon pseudoscalar current at order Q − . Finally, there are various corrections at order Q :There are relativistic corrections that depend on the en-ergy transfer. Their explicit form is given by P a ( Q )1N: 1 /m, UT (cid:48) = i M π m q k k · A a ( Q )1N: 1 /m, UT (cid:48) , (157)with A a ( Q )1N: 1 /m, UT (cid:48) given in Eq. (109). The second kind ofthe order- Q contributions is given by relativistic 1 /m -corrections. P a ( Q )1N: 1 / m = i M π g A m q m [ τ i ] a (cid:18) k · σ i (1 − β ) ( p (cid:48) i − p i ) ( k + M π ) − p (cid:48) i + p i ) k · σ i − β ( p (cid:48) i − p i ) k i · σ i k + M π (cid:19) . (158)Notice that this expression is related to the pion-poleterms in the corresponding axial current operator A a ( Q )1N: 1 / m whose expression is given in Eq. (111), via P a ( Q )1N: 1 / m = i M π m q k k · A a ( Q )1N: 1 / m (cid:12)(cid:12)(cid:12)(cid:12) pion − pole terms . (159)The third kind of order- Q contributions are coming fromstatic two-loop contributions and is given by P a ( Q )1N: static = i m q k · σ i [ τ i ] a (cid:16) G ( Q ) A ( − k ) − k G ( Q ) P ( − k ) (cid:17) . (160)where G ( Q ) A ( t ) and G ( Q ) P ( t ) are defined in Eqs. (116) and(120), respectively.Alternatively to the chiral expansion of the axial andpseudoscalar form factors, one can take their empiricalparametrization in practical calculations. Similar to axial vector current we can characterize two-nucleon pseudoscalar current by the number of the pionexchanges and/or short-range interactions P a = P a π + P a π + P a . (161)First contributions start to show up at order Q − . At thisorder, there are only OPE and contact contributions whichcan be expressed by the longitudinal part of axial-vectorcurrent via i m q P a, ( Q − )2N: 1 π = k · A a, ( Q )2N: 1 π , (162) i m q P a, ( Q − )2N: cont = k · A a, ( Q )2N: cont , (163)where A a, ( Q )2N: 1 π and A a, ( Q )2N: cont are defined in Eqs (127) and(141), respectively. One-pion-exchange pseudoscalar current
At the order Q leading one-loop static contributions and relativistic1 /m corrections start to show up. P a ( Q )2N: 1 π = P a ( Q )2N: 1 π, static + P a ( Q )2N: 1 π, /m + P a ( Q )2N: 1 π, UT (cid:48) . (164)The corresponding static expressions are i m q P a ( Q )2N: 1 π, static = − M π F π g A q · σ q + M π (cid:26) [ τ ] a (cid:20) h P ( q )+ h ( q ) k + M π (cid:21) + [ τ × τ ] a q · [ q × σ ] h ( q ) k + M π (cid:27) +1 ↔ , (165)where the scalar functions h ( q ) and h ( q ) are definedin Eq. (131), while h P ( q ) is given by h P ( q ) = g A πF π (cid:16) (1 − g A ) M π + (2 M π + q ) A ( q ) (cid:17) . (166)The relativistic 1 /m corrections are i m q P a ( Q )2N: 1 π, /m = − g A M π F π m (cid:26) i [ τ × τ ] a (cid:20) − B · k ( q + M π ) ( k + M π ) + 1 q + M π (cid:18) B · k k ( k + M π ) + B · k k ( k + M π ) (cid:19)(cid:21) + [ τ ] a (cid:20) − B · k ( q + M π ) ( k + M π )+ 1 q + M π (cid:18) B · k k ( k + M π ) + B · k k ( k + M π ) (cid:19)(cid:21)(cid:27) +1 ↔ , (167)where the vector quantities B i are defined in Eq. (134).Finally, the energy-transfer dependent contributions are i m q P a ( Q )2N: 1 π, UT (cid:48) = − M π k k · A a ( Q )2N: 1 π, UT (cid:48) , (168)where A a ( Q )2N: 1 π, UT (cid:48) is given in Eq. (136). Two-pion-exchange pseudoscalar current
Two-pion-ex-change contributions can be expressed in terms of the lon-gitudinal component of the axial vector current. The ex-pression is given by i m q P a ( Q )2N: 2 π = − M π k k · A a ( Q )2N: 2 π (cid:12)(cid:12)(cid:12)(cid:12) g = g = g = g = g =0 , (169)with the scalar functions g ( q ) , . . . , g ( q ) being definedin Eq. (140). Short-range pseudoscalar current
The first contributionto short-range two-nucleon current shows up at the order Q − and is given in Eq. (163). At order Q we characterizeshort-range contributions via P a, ( Q )2N: cont = P a, ( Q )2N: cont , static + P a, ( Q )2N: cont , /m + P a, ( Q )2N: cont , UT (cid:48) . (170)The static contributions at the order Q vanish P a, ( Q )2N: cont , static = 0 . (171)The relativistic corrections are given by i m q P a, ( Q )2N: cont , /m = − M π k k · A a ( Q )2N: cont , /m , (172)where A a ( Q )2N: cont , /m is specified in Eq. (144). Finally, theenergy-transfer-dependent contributions are given by i m q P a ( Q )2N: cont , UT (cid:48) = − M π k k · A a ( Q )2N: cont , UT (cid:48) , (173)where A a ( Q )2N: cont , UT (cid:48) is specified in Eq. (145). At the order Q there are first contributions to the three-nucleon pseudoscalar current. Similar to the axial-vectorcurrent we decompose the current into the long and theshort-range contributions P a ( Q )3N = P a ( Q )3N: π + P a ( Q )3N: cont . (174)The long-range contributions are given by P a ( Q )3N: π = − i F π M π g A m q (cid:88) i =5 C ai · k k + 5 permutations , (175)where C ai are defined in Eq. (149). Short-range contribu-tions are given by P a ( Q )3N: cont = − i F π M π g A m q (cid:88) i =2 D ai · k k + 5 permutations , (176)with D ai defined in Eq. (152). Q Nuclear scalar current is important for the dark mattersearches. A scenario that dark matter is realized by weaklyinteracting massive particles (WIMPs) can be tested vianuclear recoil produced by the scattering of WIMPs offatomic nuclei (see [161] and the references therein). If WIMP, denoted by χ , is a spin-1 / L χ = 1 Λ (cid:88) q (cid:20) C SS q ¯ χχm q ¯ qq + C PS q ¯ χiγ χm q ¯ qq + C SP q ¯ χχm q ¯ qiγ q + C PP q ¯ χiγ χm q ¯ qiγ q (cid:21) + 1 Λ (cid:88) q (cid:20) C VV q ¯ χγ µ χ ¯ qγ µ q + C AV q ¯ χγ µ γ χ ¯ qγ µ q + C VA q ¯ χγ µ χ ¯ qγ µ γ q + C AA q ¯ χγ µ γ χ ¯ qγ µ γ q (cid:21) , (177)where Λ is a beyond standard model scale and various Wil-son coefficients C q are dimensionless. Indices S, P, V and A stay for scalar, pseudoscalar, vector and axial-vectorquantum numbers, respectively. q -field is a quark field and m q is a quark mass. Eq. (177) can be considered as astarting point for chiral EFT analysis where one needs tostudy scalar-, pseudoscalar-, vector- and axial-vector cur-rents within chiral EFT. While vector- and axial-vectorcurrents have been extensively studied within standardmodel, scalar-current in chiral EFT appears first in be-yond standard model physics and is much less known. Pi-oneering work towards this direction was the leading-ordercalculation of the scalar current by Cirigliano et al. [163]and Hoferichter et al. [165]. Recently we derived leadingone-loop corrections to these results [166]. Although theinformation about short-range physics at this order needsto be fixed these calculations give long-range contributionsin a parameter-free way. In the following all contributionsto scalar current up to order Q . Scalar current on a single-nucleon can be parametrizedby a scalar form factor which in principle can be calcu-lated within chiral perturbation theory. Due to the lackin convergence [41,150] one prefers to use dispersion re-lation techniques where ππ scattering channel dominates t -dependence of the scalar form factor [162]. Two- and three-nucleon scalar current will be formulatedhere within chiral EFT. In the following, we characterizea two-nucleon scalar current by the range of interaction S = S π + S π + S . (178) One-pion-exchange scalar current
First contributions tothe one-pion-exchange current show up at the order Q − and are given by m q S ( Q − )2N: 1 π = − g A M π F π q · σ q · σ τ · τ ( M π + q ) ( M π + q ) . (179) ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 21 Next correction shows up first at the order Q . The resultis given by m q S ( Q )2N: 1 π, = q · σ q + M π (cid:20) q · σ (cid:18) o ( k ) q + M π + o ( k ) (cid:19) + k · σ (cid:18) o ( k ) + q o ( k ) (cid:19)(cid:21) + 1 ↔ , (180)where the scalar functions o i ( k ) are given by o ( k ) = g A M π π F π (cid:2) π ¯ d F π M π + g A k ¯ l − g A L ( k ) (cid:0) k + M π (cid:1) + g A (cid:0) k + M π (cid:1) (cid:3) ,o ( k ) = g A M π π F π (cid:20) π F π (cid:0) d − ¯ d (cid:1) − g A ¯ l − g A L ( k ) (cid:0) k + 3 M π (cid:1) k + 4 M π (cid:21) ,o ( k ) = − g A M π π F π k (cid:20) π ¯ d F π k + g A (cid:0) − k + M π (cid:1) + 2 g A k − g A L ( k ) k + 4 M π (cid:0)(cid:0) g A + 1 (cid:1) k + (cid:0) g A + 4 (cid:1) k M π + g A M π (cid:1) (cid:21) ,o ( k ) = − g A M π π F π k k + 4 M π (1 − L ( k )) k + 4 M π . (181)The relativistic corrections and the energy-transfer depen-dent contributions to one-pion-exchange at the order Q vanish. Two-pion-exchange scalar current
Two-pion-exchangecurrent shows up first at order Q . Due to the appearanceof three-point functions the results are lengthy. Their ex-plicit form can be found in [166] and, for completeness, isgiven in Appendix E. Short-range scalar current
Short-range two-nucleon cur-rent starts to contribute at order Q . The relevant LECscome from L (0) NN = − C S N † N ) + 2 C T N † S µ N N † S µ N, (182) L (2) NN = − D S (cid:104) χ + (cid:105) ( N † N ) + D T (cid:104) χ + (cid:105) N † S µ N N † S µ N , where N is the heavy-baryon notation for the nucleonfield with velocity v µ , S µ = − γ [ γ µ , γ ν ] v ν / χ + = 2 B (cid:0) u † ( s + ip ) u † + u ( s − ip ) u (cid:1) , B , C S,T and D S,T are LECs. (cid:104) . . . (cid:105) denotes the trace inthe flavor space. All the non-vanishing diagrams whichcontribute at order Q are listed in [166]. After the renor-malization of the short-range LECs we got m q S ( Q )2N: cont = σ · σ s ( k ) + k · σ k · σ s ( k ) + s ( k ) , (183) with the scalar function s i ( k ) defined by s ( k ) = − M π π F π m q (cid:20) g A C T − π ¯ D T F π + g A C T L ( k ) (cid:0) k + 4 M π (cid:1) k + 4 M π (cid:21) ,s ( k ) = 3 g A C T M π π F π k m q k − M π ( L ( k ) − k + 4 M π ,s ( k ) = M π π F π m q (cid:20) g A C T + 8 π ¯ D S F π − g A C T L ( k ) (cid:0) k + 8 M π (cid:1) k + 4 M π (cid:21) . (184)The renormalized, scale-independent LECs ¯ D S , ¯ D T arerelated to the bare ones D S , D T according to D i = ¯ D i + β NN i F λ + β NN i π F ln (cid:18) M π µ (cid:19) , (185)with the corresponding β -functions given by β NN S = 12 (cid:0) g A − g A + 24 F g A C T (cid:1) ,β NN T = 14 (cid:0) g A − g A + 48 F g A C T (cid:1) , (186)and the quantity λ defined as λ = µ d − π (cid:18) d − (cid:0) γ E − ln 4 π − (cid:1)(cid:19) , (187)where γ E = − Γ (cid:48) (1) (cid:39) .
577 is the Euler constant, d thenumber of space-time dimensions and µ is the scale ofdimensional regularization.Notice that the LECs C S , C T , ¯ D S and ¯ D T also con-tribute to the 2N potential. The experimental data onnucleon-nucleon scattering, however, do not allow one todisentangle the M π -dependence of the contact interac-tions and only constrain the linear combinations of theLECs [167] C S = C S + M π ¯ D S , C T = C T + M π ¯ D T . (188)The LECs ¯ D S and ¯ D T can, in principle, be determinedonce reliable lattice QCD results for two-nucleon observ-ables such as e.g. the S and S scattering lengths atunphysical (but not too large) quark masses are available,see Refs. [168] and references therein for a discussion ofthe current status of research along this line. There are only vanishing 1 /m -corrections and the energy-transfer dependent contributions at the order Q . At the order Q there are no three-nucleon current contri-butions. Nonvanishing contributions start to show up firstat order Q . Scalar current at vanishing momentum transfer
At thevanishing momentum transfer, one can relate the scalarcurrent to a quark mass derivative of the nuclear forces.On the mass-shell one gets (cid:104) f | S (0) | i (cid:105) = (cid:28) f (cid:12)(cid:12)(cid:12)(cid:12) ∂H eff ∂m q (cid:12)(cid:12)(cid:12)(cid:12) i (cid:29) , (189)where the states | i (cid:105) and | f (cid:105) are the eigenstates of theHamiltonian H eff . The on-shell condition requires thatthe corresponding eigenenergies E i and E f are equal. Foreigenstates | Ψ (cid:105) corresponding to a discrete energy E , H eff | Ψ (cid:105) = E | Ψ (cid:105) , (190)the Feynman-Hellmann theorem allows one to interpretthe scalar form factor at zero momentum transfer in termsof the eigenenergy slope in the quark mass: (cid:104) Ψ | m q S (0) | Ψ (cid:105) = m q ∂E ( m q ) ∂m q . (191)In particular, for | Ψ (cid:105) being a single-nucleon state at rest,the expectation value on left-hand side of Eq. (191) isnothing but the pion-nucleon sigma-term [169] (cid:104) Ψ | m q S (0) | Ψ (cid:105) = m q ∂m ( m q ) ∂m q ≡ σ πN , (192)and for an extension to resonances | R (cid:105) , see e.g. Ref. [170].As was demonstrated in [166] we explicitly verified therelation in Eq. (189) up to order Q . To get a slope inthe quark mass of the nuclear force at NLO we used theexpressions from [167] where the authors discussed thenuclear force at NLO in the chiral limit. ¯ β and ¯ β After renormalizability and matching constraints appliedto various nuclear currents we get in the static limit aunique result. In the relativistic corrections of the vectorand axial-vector current, however, there remains a unitaryambiguity which is parametrized by unitary phases ¯ β and¯ β . In Appendix C we give their explicit form and brieflyreview all other transformations which are introduced torenormalize nuclear currents. It is instructive to unravelhow this dependence disappears if we calculate the ex-pectation values of the corresponding currents. There aretwo different mechanisms that we are going to discuss. Inthe first case, the dependence on unitary phases is com-pensated by the wave function of initial and final states.In the second case, which we call a k -dependent off-shelleffect, the dependence on unitary phases is not compen-sated by wave functions. However, it is proportional to k − E β + E α , where E α and E β correspond to the energiesof the initial and final states, respectively. Since on-shellwe have k = E β − E α , (193) the unitary ambiguity disappears for observable quanti-ties. We can explicitly disentangle contributions which aregoing to be compensated by the wave functions and whichare to disappear if the on-shell condition of Eq. (193) issatisfied. In the case of the vector current, there are onlycontributions proportional to ¯ β , which are to be compen-sated by the wave functions, see [192] for an explicit verifi-cation. The reason is that k -dependent terms in Eq. (76)do not depend on ¯ β , . Strictly speaking, there still re-mains a residual dependence on the unitary phases ¯ β , even if we calculate the expectation value of the currentoperator. The reason is that the transformation associatedwith phases ¯ β , is only approximately unitary modulo ef-fects of higher order in the chiral expansion. Due to theexpected higher order suppression, the dependence on ¯ β , should be weak in practical calculations .Energy-transfer k -dependent terms generated by time-dependent unitary transformations are, in general, can-celled on-shell by an accompanied commutator with thenuclear force (see Eq. (23)) i k y µ ( k ) − i [ H eff , y µ ( k )] . (194)In the case of the vector current the operator y µ ( k ) canbe read off from Eq. (76) y ( k ) = 0 , y ( k ) = − i k e k (cid:20)(cid:0) G E ( k ) − G E (0) (cid:1) + i m k · ( k × σ ) (cid:0) G M ( k ) − G M (0) (cid:1)(cid:21) . (195)Actually, Eq. (194) describes only the single-nucleon con-tribution to the corresponding current operator i k y µ ( k ) − i [ H , y µ ( k )] . (196)The remaining two- and more-nucleon contributions arecoming from the commutator[ H eff − H , y µ ( k )] , (197)and is perturbatively taken into account in V ( Q )2N:1 π, static given in Eq. (82). In particular, the commutator[ H ( Q )1 π + H ( Q )cont , y ( k ) [ Q ] ] (198)contributes to the order Q vector current which is a partof V ( Q )2N:1 π, static . Here the operator y ( k ) [ Q ] is given by y ( k ) [ Q ] = − i k e k (cid:20)(cid:0) G E ( k ) [ Q ] − G E (0) [ Q ] (cid:1)(cid:21) , (199) Provided that one works with consistent wave functionswhich incorporate the same dependence on unitary phases asdoes the vector current operator. Hybrid approach, where phe-nomenological wave functions (coming not from chiral EFT,like e.g. AV18 [190] or CD Bonn [191] potentials) are used,should not be used for this current operator.ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 23 where the index in the square brackets denotes the chi-ral order of the operator. Explicit expressions for the chi-ral expansion of the electromagnetic form factors can befound e.g. in [148], and the leading order nuclear forceoperator is given by H ( Q )1 π = − g A F π τ · τ σ · qσ · q q + M π ,H ( Q )cont = C S + σ · σ C T , (200)where q denotes momentum transfer between nucleons. Inpractical applications one may include higher-order oper-ators to the two- and more-nucleon current by using thewhole commutator of Eq. (197), instead of the commu-tator of Eq. (198). In this case, the expectation value ofthe operator of Eq. (194) would not contribute on-shelland, for this reason, does not need to be calculated ex-plicitly. Obviously, if one would neglect the operator ofEq. (194) one has to subtract the operator of Eq. (198)from V ( Q )2N:1 π, static .Similar to the vector current, we can write an axialvector version of Eq. (194) which is given by i k z aµ ( k ) − i [ H eff , z aµ ( k )] , (201)where z a ( k ) = 0 , z a ( k ) = i k [ τ ] a m (cid:16) (1 + 2 ¯ β ) k · σ G P ( − k ) − (1 + 2 ¯ β ) k · k k · σ G (cid:48) P ( − k ) (cid:17) . (202)Replacing H eff by H in Eq. (201) we reproduce the off-shell part of Eq. (101). The leading two-nucleon contribu-tion of Eq. (201) is given by the commutator − i (cid:104) H ( Q )eff − H , z µ ( k ) (cid:105) which depends on the unitary phases ¯ β , . Itsexplicit form is given by A a π, off − shell = − i (cid:104) H ( Q )1 π , z ( k ) (cid:105) = k g A F π m × q + M π (cid:20) [ τ × τ ] a (cid:16) − β ) k · q q · σ × G P ( − k ) + (1 + 2 ¯ β ) k · q (2 k · k − i k · ( q × σ )) × q · σ G (cid:48) P ( − k ) (cid:17) + [ τ ] a (cid:16) − (1 + 2 ¯ β )(2 k · ( q × σ )+ i q ) q · σ G P ( − k ) + (1 + 2 ¯ β )( i ( k · q ) +2 k · k k · ( q × σ ) (cid:17) q · σ G (cid:48) P ( − k ) (cid:21) + 1 ↔ . (203)In addition, at order Q , one also needs to take into accountthe relativistic corrections to the OPE 2N current that arenot associated with the terms in Eq. (201) and have the form δ A a ( Q )2N: 1 π, /m = g A F π m (cid:26) i [ τ × τ ] a (cid:20) q + M π ) (cid:18) B − k k · B k + M π (cid:19) + 1 q + M π (cid:18) δ B ( k + M π ) + δ B k + M π + B (cid:19)(cid:21) +[ τ ] a (cid:20) q + M π ) (cid:18) B − k k · B k + M π (cid:19) + 1 q + M π (cid:18) δ B ( k + M π ) + δ B k + M π + B (cid:19)(cid:21)(cid:27) + 1 ↔ , (204)where B , B , B and B are defined in Eq. (134), and δ B = − g A k σ · q k · q (cid:0) i k · ( q × σ ) − k · k (cid:1) ,δ B = − k (cid:18) g A (1 + 2 ¯ β ) k · q k · σ + σ · q × (cid:0) − i k · ( q × σ ) + (cid:0) g A (1 − β ) − (cid:1) k · k + k · q + (2 g A − k · q (cid:1)(cid:19) ,δ B = − g A k σ · q (cid:0) − i k · ( q × σ ) k · k + ( k · q ) (cid:1) ,δ B = − g A k (cid:18) i (1 + 2 ¯ β ) k · σ k · ( q × σ )+ q · σ (cid:0) i (1 − β ) k · ( k × σ )+ 4 i k · ( q × σ ) − (1 − β ) k − q (cid:1)(cid:19) . (205)Note that A a ( Q )2N: 1 π, /m = A a π, off − shell + δ A a ( Q )2N: 1 π, /m + O ( Q ) . (206)In the same way, we can decompose the relativistic cor-rections involving the contact interactions: A a , off − shell = − i (cid:104) H ( Q )cont , z ( k ) (cid:105) = − k [ τ ] a m (cid:18) − (1 + 2 ¯ β ) (cid:0) i C T k · ( σ × σ ) + C S q · σ + C T q · σ (cid:1) G P ( − k ) + (1 + 2 ¯ β ) (cid:0) k · q ( C S k · σ + C T k · σ ) + 2 i C T k · k k · ( σ × σ ) (cid:1) G (cid:48) P ( − k ) (cid:19) +1 ↔ . (207) The remaining relativistic corrections to short-range 2Ncurrent at order Q are given by δ A a ( Q )2N: cont , /m = − k g A m [ τ ] a k + M (cid:18) i C T k · ( σ × σ )+ C S q · σ + C T q · σ − k + M (cid:0) k · q ( C S k · σ + C T k · σ ) + 2 i C T k · k k · ( σ × σ ) (cid:1)(cid:19) +1 ↔ . (208)As in the case of the one-pion-exchange contributions, wehave A a ( Q )2N: cont , /m = A a , off − shell + δ A a ( Q )2N: cont , /m + O ( Q ) . (209)The unitary ambiguity in the operator A a − shell + A a − shell (210)is proportional to k − E β + E α which vanishes on-shell.Therefore, ¯ β , -dependence in this operators vanishes onlyonce k = E β − E α . On the other hand, the unitary ¯ β , -ambiguity in δ A a ( Q )2N: 1 π, /m is compensated by the sameunitary ambiguity of the initial and final state wave func-tions. Let us now summarize the status of calculations of cur-rent operators in chiral EFT. In tables 1, 2, 3, 4, 5 and6 all possible contributions up to N LO are summarizedfor vector, axial vector, pseudoscalar and scalar operators.Note that the nucleon mass m is counted as pm ∼ p Λ b , (211)where p denotes a low momentum scale and Λ b the break-down scale of the theory [88]. These are complete studiesup to the order N LO in chiral expansion.At this order, various LECs appear which need to befixed. In the following, we give a summary of contributingLECs. Single-nucleon sector is usually phenomenologicallyparametrized by form factors. For this reason, we concen-trate here on two-nucleon contributions.
Vector current : At the order Q , ¯ d , ¯ d , ¯ d , ¯ d from L (3) πN contribute to one pion-exchange current operator. Thesecan be determined from pion photo- and electroproduc-tion [44,179]. As can be seen from Eq. (83) there is also acontribution of ¯ d LEC which accounts for Goldberger-Treiman discrepancy. One can either directly determine¯ d from the Goldberger-Treiman discrepancy [40] or onecan express vector current operator in terms of effectiveaxial coupling g eff A = F π g πN m , (212) where pion-nucleon coupling constant up to given order isgiven by g πN = g A mF π (cid:18) − M π ¯ d g A (cid:19) . (213)For the most recent determination of the pion-nucleoncoupling constant with included isospin-breaking effectssee [218]. Replacement of g A by g eff A and neglect of ¯ d inEqs. (78) and (83) changes the current first at the orderhigher than Q . Finally in addition to various short-range C i LECs which appear in the nuclear forces, there arealso short-range contributions of LECs L and L whichare to be determined from isovector and isoscalar two-or three-nucleon observables like magnetic moment of thedeuteron, np → dγ radiative capture or isovector combi-nation of trinucleons magnetic moments [143,117]. Axial vector current : Apart from the ¯ d -dependence,one-pion-exchange contributions to the axial vector cur-rent depend on ¯ d , ¯ d , ¯ d and a linear combination ¯ d − d , see Eq. (131). ¯ d LECs is determined from pion-nucleon scattering [177,98]. Other LECs can appear in theweak pion production amplitude. Their numerical valuesare less known due to the lack of precise experimentaldata. A recent analysis shows that an assumption of thenatural size of these LECs leads to a satisfactory descrip-tion of the available data [174,176]. Short-range depen-dence on LECs is given in Eq. (143). Apart from LECscoming from nuclear forces there are additional contribu-tions of z , z , z , z short-range LECs. Those can/shouldbe fitted to weak reactions in two-nucleon sector. Pseudoscalar current : Due to the continuity equation,the pseudoscalar current is directly related to the longitu-dinal component of the axial-vector current. Once LECsof the axial-vector current are fixed there are no new pa-rameters in pseudoscalar current.
Scalar current : One-pion-exchange contribution to scalarcurrent is given in Eq. (181). As can be directly read offfrom Eq. (181) it depends on ¯ d , ¯ d LECs from L (3) πN and ¯ l LEC from L (4) π . Mesonic l LEC contributes to thescalar radius of the pion and can be found in [173]. ¯ d LEC contributes to the quark mass dependence of the ax-ial coupling g A . For this reason, careful chiral extrapola-tion of lattice QCD data is needed for its determination.A recent lattice QCD determination of g A can be foundin [175]. Short-range part of the scalar current dependsonly on short-range LECs that already appear in nuclearforces. However, the LECs ¯ D S and ¯ D T describe quarkmass dependence of the LECs C S and C T which appearin nuclear forces at LO. So careful studies of chiral ex-trapolations of nuclear forces are needed to get numericalvalues of ¯ D S and ¯ D T . Similar procedure was adopted in pion-nucleon scatteringamplitude [177,98,217]ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 25
Table 1.
Chiral expansion of the nuclear electromagnetic current operator up to N LO. LO, NLO, NNLO and N LO refer tochiral orders Q − , Q − , Q and Q , respectively. The single-nucleon contributions are given in Eqs. (2.7) and (2.16) of [148].order single-nucleon two-nucleon three-nucleonLO — — —NLO V + V / m V π , Eq . (78) —NNLO V — —N LO V + V / m + V − shell V π , Eq . (82)+ V π , Eq . (299)+ V , Eq . (89) — Table 2.
Chiral expansion of the nuclear electromagnetic charge operator up to N LO. LO, NLO, NNLO and N LO refer tochiral orders Q − , Q − , Q and Q , respectively. The single-nucleon contributions are given in Eq. (2.6) of [148].order single-nucleon two-nucleon three-nucleonLO V — —NLO V , — —NNLO V — —N LO V + V / m + V / m V π , Eq . (84)+ V π , Eq . (300)+ V , Eq . (89)+ V π, / m , Eq . (88) V π , Eq . (96) , (97)+ V , Eq . (96) , (98) As already mentioned previously, in our developments ofchiral nuclear currents we used the technique of unitarytransformation. In parallel to our activities there is a dif-ferent derivation of nuclear currents where time-orderedperturbation theory combined with the transfer-matrixinversion technique has been used (see [1,117] and ref-erences therein). In this chapter we would like to brieflydiscuss their method and compare vector and axial-vectorcurrents from two different derivation methods.
The JLab-Pisa-group used the inversion-of-the-transfer-matrix technique to derive energy-independent potentialand current operators. They start with the T-matrix and,by using a naive dimensional analysis (NDA), introduce apower counting for operators that appear in the T-matrix.Nucleon mass in their approach is counted as a hard chiral symmetry breaking scale m ∼ Λ χ . The starting point ofTOPT-method of JLab-Pisa-group is Eq. (6) of [182] givenby (cid:104) f | T | i (cid:105) = (cid:104) f | H ∞ (cid:88) n =1 (cid:18) E i − H + iη (cid:19) n − | i (cid:105) , (214)where E i and E f are eigenenergies of initial and finalstates | i (cid:105) and | f (cid:105) , respectively. The full Hamiltonian H is decomposed here in a free part H and an interactingpart H H = H + H . (215)For E i (cid:54) = E f this is a half-off-shell T-matrix. For E i = E f this is an on-shell T-matrix. The T-matrix of Eq. (214)can be than decomposed in chiral orders T = T (0) + T (1) + T (2) + . . . , (216)where the indices n of T ( n ) -matrix operator denote theirchiral order Q n and dots denote higher than Q order Table 3.
Chiral expansion of the nuclear axial current operator up to N LO. LO, NLO, NNLO and N LO refer to chiral orders Q − , Q − , Q and Q , respectively.order single-nucleon two-nucleon three-nucleonLO A , Eq. (102) — —NLO A , Eq . (105) — —NNLO — A π , Eq . (127)+ A , Eq . (141) —N LO A , Eq . (114)+ A /m, UT (cid:48) , Eq . (109)+ A / m , Eq . (111) A π , Eq . (129)+ A π, UT (cid:48) , Eq . (136)+ A π, /m , Eq . (133)+ A π , Eq . (138)+ A , UT (cid:48) , Eq . (145)+ A , / m , Eq . (144) A π , Eq . (148)+ A , Eq . (151) Table 4.
Chiral expansion of the nuclear axial charge operator up to N LO. LO, NLO, NNLO and N LO refer to chiral orders Q − , Q − , Q and Q , respectively.order single-nucleon two-nucleon three-nucleonLO — — —NLO A (cid:48) , Eq . (103)+ A /m , Eq . (106) A π , Eq . (125) —NNLO — — —N LO A , UT (cid:48) , Eq . (110)+ A /m , Eq . (112) A π , Eq . (130)+ A π , Eq . (139)+ A , Eq . (143) — Table 5.
Chiral expansion of the nuclear pseudoscalar operator up to N LO. LO, NLO, NNLO and N LO refer to chiral orders Q − , Q − , Q − and Q , respectively. .order single-nucleon two-nucleon three-nucleonLO P a , Eq. (155) — —NLO P a , , Eq . (156) — —NNLO — P a π , Eq . (162)+ P a , Eq . (163) —N LO P a , Eq . (160)+ P a /m, UT (cid:48) , Eq . (157)+ P a / m , Eq . (158) P a π, static , Eq . (165)+ P a π, UT (cid:48) , Eq . (168)+ P a π, /m , Eq . (167)+ P a π , Eq . (169)+ P a , UT (cid:48) , Eq . (173)+ P a , / m , Eq . (172) P a π , Eq . (175)+ P a , Eq . (176)ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 27 Table 6.
Chiral expansion of the nuclear scalar operator up to N LO. LO, NLO, NNLO and N LO refer to chiral orders Q − , Q − , Q − and Q , respectively. The single-nucleon contributions are from dispersion relation studies where ππ scatteringchannel dominates t -dependence of the scalar form factor [162].order single-nucleon two-nucleon three-nucleonLO S — —NLO S S π , Eq . (179) —NNLO S + S / m — —N LO S S π , Eq . (180)+ S π , Eq . (328)+ S , Eq . (183) — operators. The same NDA can be used for nuclear forceoperators v = v (0) + v (1) + v (2) + . . . . (217)Inversion of the Lippmann-Schwinger equation can be usediteratively to calculate the effective potential. v (0) = T (0) v (1) = T (1) − v (0) G v (0) , (218)and so on. The first iteration of leading-order potentialsleads within NDA to order Q contribution: Loop integra-tion gives order Q and a free Green function gives order Q − contributions corresponding to the inverse sum ofkinetic energies of the nucleons. For the half-off-shell T-matrix the potential from Eq. (218) is equivalent to the po-tential from the folded-diagram technique, which is mani-festly non-hermitian. Equivalence of inversion of T-matrixtechnique and folded-diagram approach is demonstratedin Appendices F and G. On top of the folded-diagramtechnique, the authors of [182] analyzed off-shell effects inwhich they allow contributions proportional to E f − E i inthe T-matrix . This has an influence on the form of theeffective potential and current operators and introducesdifferences between strict folded-diagram and JLab-Pisagroup techniques.For vector- and axial-vector currents JLab-Pisa groupproceeds in a similar way. They write a first-order pertur-bation theory for T-matrix in the presence of vector- or Adding to a T-matrix contributions proportional to [ H , X ]with X some operator will not affect the on-shell T-matrix butwill change its off-shell form. As a consequence, the effectiveHamiltonian gets transformed by a similarity transformation.In Appendix H we demonstrate on a perturbative level that aunitary transformation of time-dependent Hamiltonians doesnot affect the on-shell behavior of the T-matrix. Similar argu-ments should work for a similarity transformation. axial-vector source. For electromagnetic current, for ex-ample, the T-matrix can be organized via T γ = T ( − γ + T ( − γ + T ( − γ + . . . , (219)where T ( n ) γ is of order eQ n with e the electric charge. In-troducing v γ = V µ · j µ , (220)where V aµ is a nuclear vector current and j aµ a correspond-ing vector source they perform an inversion of a T-matrixto derive nuclear current operator order by order: v − γ = T ( − γ v − γ = T ( − γ − (cid:20) v ( − γ G v (0) + v (0) G v ( − γ (cid:21) . (221)In the following, we will discuss the differences betweenJLab-Pisa and Bochum-Bonn-Currents. The differences inthe two methods appear first at the order Q . Q At order Q , we decompose the current in the number ofpion-exchange via Eq. (77). We start with the static OPE contributions. OPE con-tributions of the JLab-Pisa group have been presentedin [140] where they give only unrenormalized results. Evenrelaxing constraints on beta functions of d i -LECs from L (3) πN Lagrangian we are unable to renormalize their Eqs.(3 . − (3 . . − (2 .
25) of [117]. Since they use this form in all their numerical calculations, we compare if our resultsare compatible with their parametrization. Replacing theform factor by unity (which does not affect order- Q re-sults) we consider the difference between our results andthat of JLab-Pisa group δV ( Q ) ,µ π, static = V (Q) ,µ π, static − V JLab − Pisa(Q) ,µ π, static . (222)The OPE contributions can be written in the form ofEq. (82) for current and Eq. (84) for charge contributions,respectively. For this reason, it is enough to give the dif-ference in terms of scalar functions δf i ( k ) = f i ( k ) − f JLab − Pisa i ( k ) , i = 1 , . . . , , (223)where the functions f i ( k ) are defined in Eq. (83) andEq. (85). f JLab − Pisa i ( k ) are extracted out of Eq. (2.22)and (2.24) of [117]. The difference is given by δf ( k ) = δf ( k ) = 0 ,δf ( k ) = − i e π F π (cid:18) g A (2 L ( k ) −
1) + (4 πF π ) g A ¯ d (cid:19) ,δf ( k ) = − i eg A F π ¯ d , δf ( k ) = − ieg A M π F π ¯ d ,δf ( k ) = − ieg A π F π (cid:18) − M π + k ( − π ¯ l )+6( k + 4 M π ) L ( k ) (cid:19) . (224)Note, that Piarulli et al. [117] discuss only tree-level con-tributions to OPE and make a phenomenological exten-sion of these results. They multiply the tree-diagram re-sults proportional to ¯ d i ’s with G γN∆ ( k ) /µ γN∆ and with G γNρ ( k ) form factors which are set here to unity. All otherterms of these form factors contribute to orders higherthan Q . In contrast to the statement of Piarulli et al. [117],the difference δ V ( Q )2N:1 π, static contributes to the magneticmoment operator µ = − ( i/ ∇ k × V | k =0 . The differencefor the magnetic moment operator is given by δ µ = − ( i/ ∇ k × δ V ( Q ) | k =0 = eg A π F π [ τ × τ ] q + M π × (cid:18) [ q × σ ] q · σ − [ q × σ ] q · σ (cid:19) . (225)This difference, however, comes from δf ( k ) and can beabsorbed into ¯ d if one makes a shift¯ d → ¯ d + g A π F π . (226)Charge operator contributions have also been discussedby JLab-Pisa group. Since they do not mention any staticcontributions to OPE charge at order Q we conclude that f JLab − Pisa7 ( k ) = f JLab − Pisa8 ( k ) = 0 , (227)in clear disagreement to our results of Eq. (85). At the same order Q there are also relativistic correc-tions to OPE charge and current operators. Both groupsgive vanishing results for relativistic corrections for cur-rent operators such that there is no disagreement on thecurrent level for relativistic corrections. Relativistic cor-rections to charge contributions agree only if one fixesunitary phases to β = ν , β = − , (228)which means that they are unitary equivalent. Parameters β and β are directly related to unitary phase parame-ters µ and ν introduced by Friar [180,21,181] to describean off-shell dependence of relativistic 1 /m corrections toOPE NN potential: µ = 4 ¯ β + 1 , ν = 2 ¯ β . (229)These parameters are usually set to¯ β = 14 , ¯ β = − , (230)to achieve a minimal non-locality form of OPE NN po-tential [89]. From Eq. (228) we conclude that the chargeoperator used by JLab-Pisa group does not correspondto the minimal-nonlocality choice even for ν = 1 /
2. Forthis reason, this charge operator should not be convolutedwith the available chiral nuclear forces where minimal non-locality is used.
At the level of the two-pion-exchange, there is an agree-ment between our and JLab-Pisa group results on the cur-rent but not on the charge operator. This disagreementwas addressed by JLab-Pisa group in [182] where theyextensively discuss the TPE charge operator. They claimthat there exists a unitary transformation that makes twocharges unitary equivalent.It is important to note that the potentials presentedin [182] are manifestly non-hermitian. One can see this di-rectly in Eq. (20) of [182] where one finds a non-hermitiancontribution to their effective potential for the off-shellparameter ν = 1. They also give a similarity transforma-tion (erroneously called a “unitary transformation”), seeEq.(25) of [182] given by H ( ν ) = e − i U ( ν ) H ( ν = 0) e iU ( ν ) (231)which transforms ν = 1 into ν = 0 potential. One canimmediately see from Eq.(28) of [182] given by i U (1) ( ν, p (cid:48) , p ) = − ν (cid:90) s v (0) π ( p (cid:48) − s ) v (0) π ( s − p )( p (cid:48) − s ) + m π (232) Piarulli et al. discuss 1 /m and 1 /m corrections to OPEcurrent operator. Although they do not vanish they contributeto higher orders in the power counting adopted by our group.ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 29 that i U (1) ( ν, p (cid:48) , p ) is not antihermitian. So H ( ν = 1) and H ( ν = 0) potentials are not unitary equivalent but canbe transformed into each other by a similarity transfor-mation .Applying the same transformations on electromagneticcurrent operator Pastore et al. [182] show that the Bochum-Bonn and JLab-Pisa charge operators can be transformedinto each other. However, since the transformation givenin Eq. (28) of [182] is not unitary the charges are not uni-tary equivalent. This is a similarity transformation thatdoes not change the spectrum of the nuclear force and theon-shell T-matrix but changes, in general, the normaliza-tion of the wave function. For axial vector current at order Q the situation is lesstransparent than for vector current operator. The differ-ences for pion-exchange contributions have been exten-sively discussed in [23,134]. The authors of [125] consid-ered only static contributions so we can not compare recoilcorrections to OPE currents. In the static limit, there is aclear disagreement between Bochum-Bonn and JLab-Pisacurrents even at vanishing momentum transfer. To clarifyif the currents are unitary equivalent we first perform acalculation of relativistic corrections to box-diagrams foreffective potential, all contributions proportional to g A /m .Starting with folded-diagram technique which is equiva-lent to the inversion of the transfer matrix technique. Theeffective potential which one gets, in this case, is non-hermitian and is in general given by [29,183] H FDeff = (1 − A ) H (1 + A ) , (233)where A satisfies a non-linear decoupling Eq. (8). Thispotential is related to a hermitian potential via a similaritytransformation H eff = S / H FDeff S − / , (234)with S = 1 + AA † + A † A. (235)The effective potential H eff is a standard hermitian po-tential of Eq. (9) which one gets via Okubo transforma-tions of Eq. (6). In order to get effective potential of JLab-Pisa group for the off-shell choice ν = 0, see in particularEqs. (19) of [182] where the pion-energy denominators ω In a later publication [117] after Eq.(2.17) the authors re-mark that “expressions for U (1) ( ν ), contain a typographicalerror: the imaginary unit on the left-hand side should be re-moved”. Without imaginary unit Eq.(28) of [182] leads indeedto a unitary transformation which, however, does not trans-form H ( ν = 1) into H ( ν = 0). This is obvious since H ( ν = 1)is non-hermitian and H ( ν = 0) is hermitian and there is nounitary transformation which transforms a non-hermitian op-erator into a hermitian one. and ω factorize, we have to perform additional unitarytransformation H JLab − Pisaeff = U † H eff U , (236)with U = exp (cid:0) α S + α S (cid:1) , (237)where antihermitian operators S and S are defined inEq. (3.25) of [68]. In order to reproduce Eq. (19) of [182]we have to fix the unitary phases to α = − , α = 14 , (238)which is a standard choice of unitary phases for renormal-izable nuclear forces. With this unitary convention, rel-ativistic corrections to box-diagrams for nuclear forces ofJLab-Pisa and Bochum-Bonn groups coincide. In this wayall transformations needed to bring folded-diagram H FDeff into the form of Bochum-Bonn nuclear force are fixed.JLab-Pisa group does not apply any time-dependent uni-tary transformations on nuclear forces to get a currentoperator. For this reason, their current operator is trans-formed in the same way as nuclear forces. For JLab-Pisagroup current operator we get j JLab − Pisa µ = U † S / j FD µ S − / U , (239)where the folded-diagram current can be extracted via aninversion of the T-matrix in the presence of an axial-vectorsource t ( E f , E i ) = (cid:0) − vG ( E f ) (cid:1) − v ( E f − E i ) × (cid:0) − ˜ G ( E i ) v (cid:1) − , (240)where the T-matrix in the presence of an axial-vectorsource is defined by T ( E f , E i ) = 2 πδ ( E f − E i ) t ( E i ) + t ( E f , E i ) . (241) E f and E i denote final and initial state energies, respec-tively. Eq. (240) is derived in Appendix I. The half-off-shelltransfer matrix t ( E i ) here does not depend on axial sourceand satisfies Lippmann-Schwinger equation t ( E i ) = v + v ˜ G ( E i ) t ( E i ) . (242)The axial-vector source coupling is within v ( E ) = (cid:90) d x e iEx j µ ( x ) · a µ ( x ) , (243)where a µ denotes the axial-vector source. In [184] we ex-plicitly performed inversion of Eq. (240) and performeda similarity transformation of Eq. (239). The outcome ofthis calculation is supposed to be JLab-Pisa group ex-pressions. However, we were unable to reproduce their re-sults concluding that either we misinterpret the methodof JLab-Pisa group or there is an error in their calculationwhich needs to be clarified in the future. Note that the discussion here is restricted to box-diagramssuch that the transformation of Eq. (239) is incomplete forother diagrams. In particular, unitary transformations propor-tional to β , β are not included in Eq. (239).0 Hermann Krebs: Nuclear Currents in Chiral Effective Field Theory So far all reported calculations of current operators havebeen performed by using dimensional regularization. Soone could take these operators and start to look at theirexpectation values to study observables. This is indeedwhat has been done by various calculations with JLab-Pisa TOPT-currents, see e.g. [1] for a review. All thesecalculations should be considered as a hybrid approachwhere no claim on consistency between nuclear forces andcurrents is made. Even if both nuclear forces and currentsare calculated from the same framework of chiral EFTthe use of different regularizations (cutoff vs dimensionalregularization) leads to chiral symmetry violation in thevery first iteration of the current with nuclear forces. Hereis the explanation:To solve the Schr¨odinger equation, nuclear forces haveto be regularized. The usual way is to use cutoff regular-ization. Let us for example choose a semi-local regulatordiscussed in [90]. The regularized form of the long-rangepart of the leading-order nuclear force, which is one-pion-exchange diagram, is given by V π,Λ = − g A F π τ · τ σ · q σ · q q + M π e − q M πΛ , (244)where q denotes momentum transfer between two nucle-ons. A nice property of this regulator is that it does notaffect long-range part of the nuclear force at any power of1 /Λ . On the other hand, a pion-pole contribution propor-tional to g A of the relativistic correction of the axial-vectortwo-nucleon current is given by A a, ( Q,g A )2N: 1 π, /m = g A F π m i [ τ × τ ] a q · σ q + M π k k + M π × (cid:104) i k · q × σ − k · q + k · ( q + k ) (cid:105) + 1 ↔ , (245)where k is the momentum transfer of the axial vector cur-rent, and other momenta are defined by q i = p (cid:48) i − p i , k i = p (cid:48) i + p i , i = 1 , , (246)and momenta p (cid:48) i and p i correspond to the final and initialmomenta of the i -th nucleon, respectively. Note that thisis not the only contribution to the relativistic correctionsof the current, but the only one proportional to g A . Com-plete expression (including terms proportional to g A ) forthe relativistic corrections can be found in [23]. After weregularized the nuclear force and the axial vector currentwe can perform the first iteration and take Λ → ∞ limit: A a, ( Q,g A )2N: 1 π, /m E − H + i(cid:15) V π,Λ + V π,Λ E − H + i(cid:15) A a, ( Q,g A )2N: 1 π, /m = Λ g A √ π / F π ([ τ ] a − [ τ ] a ) k k + M π q · σ +1 ↔ O ( Λ ) . (247) Since the one-loop amplitude should be renormalizablethere should exist a counter term which absorbs the linearsingularity in Λ . From Eq. (247) we see that this shouldbe a contact two-nucleon interaction with one-pion cou-pling to it. However, there is no counter term like thisin the chiral EFT. The counter term like this requiresderivative-less coupling of the pion which is forbidden bythe chiral symmetry: There exists only a counter term pro-portional to k · σ , but there is none which is proportionalto q · σ . Here k is the momentum of the pion coupling tothe two-nucleon interaction. If there is no counter termthat absorbs the linear cutoff singularity there should besome cancelation in the amplitude with other terms. In-deed the same singularity but with the opposite sign wewould get for the static limit of the axial vector currentof the order Q if we would calculate the current by us-ing cutoff regularization. Axial vector current at the order Q , however, is calculated by using dimensional regulariza-tion and is finite. It also remains finite if we just multiplythe current with any cutoff regulator we want. So at thelevel of the amplitude, the mismatch between cutoff anddimensional regularization used in the construction of op-erators leads to a violation of the chiral symmetry at theone-loop level which, however, is the order of accuracy ofour calculations. So we see that it is dangerous to mul-tiply the current operators calculated within dimensionalregularization by some cutoff regulator and calculate theexpectation values of this. With similar arguments one canshow that dimensionally regularized three-nucleon forcesat the level of N LO, which were published in [62,63], cannot be used in combination with the cutoff regularizedtwo-nucleon forces at the same order [79]. The mismatchbetween dimensional and cutoff regularization will leadalso in this case to a violation of the chiral symmetry atthe one loop level.To respect the chiral symmetry we need to calculateboth nuclear forces and currents with the same regulator.On top of it the regulator which we choose should be sym-metry preserving. One possibility to construct a regulator,which manifestly respects the chiral symmetry was pro-posed more than four decades ago by Slavnov [185], wherehe introduced a higher derivative regularization in a studyof the non-linear sigma model. Recently, the first applica-tions of this technique to the chiral EFT have been dis-cussed in the literature [186,187]. A basic idea of Slavnovis to change the propagator of a pion field on the La-grangian level. Since pion fields are organized in U -fieldswhich are elements of SU(2) group and all derivatives inthe Lagrangian appear as covariant derivatives to main-tain gauge and chiral symmetry, the modified chiral La-grangian should be expressed in terms of covariant deriva-tives of U -fields. In this way the gauge- and chiral symme-try are maintained by construction. For our purpose, we At higher orders one can construct derivative-less pion-four-nucleon interactions by multiplying low energy constantswith M π . They are coming from the explicit chiral symmetrybreaking by finite quark mass. However, at the order Q we cannot construct a counter term like this.ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 31 change the static pion propagator by a regularized one1 q + M π → e − q M πΛ q + M π . (248)The choice of this regulator is consistent with the semilo-cal regulator used in nuclear forces where pion propagatoris regularized via Eq. (248) and contact interactions viaa non-local Gaussian regulator. The challenge is to con-struct the modified chiral Lagrangian in terms of covari-ant derivatives and chiral U -fields which leads to modifiedpion propagator of Eq. (248), modified contact interac-tions and to regularized forces and currents. Constructionof consistent nuclear forces and currents within the higherderivative approach is a work in progress. However, thefirst results for the deuteron charge in this consistent ap-proach are already available [192] and will be briefly dis-cussed in the next section. As already mentioned in Sec. 7, at the moment we cannot take expectation values of the current operators atthe order Q without violation of the chiral symmetry atthe same order of accuracy. This happens due to the mis-match of dimensional and cutoff regularizations of nuclearforces and currents. Inconsistency between regularizationof forces and currents leads in some cases to strong cut-offdependence in the observables, see [144] for discussion ofphotodisintegration on H and He calculated with order Q vector current. In many cases, however, the inconsistent(hybrid) approach gives a satisfactory description of thedata. In the last decade, there are various application ofhybrid approach with JLab-PIsa TOPT current operators,see [1] for a recent review and [121,188,118,189] and refer-ences therein for the most recent activities in this field. Weare not going to report here on the hybrid approach activ-ities but rather concentrate on the consistent calculationof the deuteron charge form factor [192]. Deuteron form factors have been extensively studied withinEFT in pion-full and pion-less approaches, see [12,13,193]for reviews. To perform a consistent calculation of deuteronform factors, we need to construct a regularized NN poten-tial and electromagnetic current operator with the sameoff-shell properties. Up to order Q the expressions for un-regularized current have been discussed in Sec. 5.2. Atthe order Q meson-exchange contributions to the deuteroncharge form factor are given by just relativistic correctionsto the one-pion-exchange charge operator, Eq. (88). Sucha simplification for the charge operator appears since thedeuteron is an isoscalar. All complicated one-loop termsproportional to [ τ ] and [ τ ] do not contribute to ex-pectation values after convolution with the deuteron wave function. The only non-vanishing order Q pion-exchangecontribution to the deuteron charge operator is given by V , ( Q )2N:1 π, /m = − eg A F π m (cid:20) − β q + M π σ · kσ · q + 2 ¯ β − q + M π ) σ · q σ · q q · k (cid:21) + 1 ↔ . (249)For the single-nucleon charge operator, we use Eq. (74)where the charge operator is expressed in terms of electro-magnetic form factors. Due to poor convergence of chiralexpansion for electromagnetic form factors we used theirphenomenological parametrization. For the calculation ofthe deuteron charge form factor we used a global anal-ysis of the experimental data of Refs. [194,195]. To esti-mate a systematic error we also used dispersive analyses ofRefs. [196,197,198]. In derivation of Eq. (249) we appliedunitary transformations of nuclear forces on the leading-order single-nucleon charge operator. Since we work nowwith the form factor parametrization of the single-nucleoncontribution we apply the same transformation on theelectric form factor and replace Eq. (249) by V , ( Q )2N:1 π, /m = − eg A F π m (cid:20) − β q + M π σ · kσ · q + 2 ¯ β − q + M π ) σ · q σ · q q · k (cid:21) G SE ( k ) + 1 ↔ , (250)where G SE ( k ) is the isoscalar part of the electric formfactor G SE ( k ) = G pE ( k ) + G nE ( k ) , (251)and G pE ( k ) and G nE ( k ) are electric form factors of pro-ton and neutron, respectively. To regularize Eq. (250) wehave to take the same regulator which was used in the con-struction of nuclear forces. This requires to make followingreplacements in Eq. (250) for pion-exchange propagators1 q + M π → e − q M πΛ q + M π , q + M π ] → (cid:18) q + M π Λ (cid:19) e − q M πΛ [ q + M π ] . (252)Note the dependence of the deuteron charge operator inEq. (250) on unitary phases ¯ β and ¯ β . The same unitaryphases appear in relativistic 1 /m -corrections to one-pion-exchange in nuclear force. They are usually chosen as¯ β = 14 , ¯ β = − , (253)to maintain a minimal non-locality of the nuclear force.However, they might be chosen differently, and any cal-culated observable should only weakly depend on them ina consistent calculation . To test the consistency of our If unitary transformations with the phases ¯ β and ¯ β wouldbe implemented in an exact way without any approximationthe observables would be independent on them.2 Hermann Krebs: Nuclear Currents in Chiral Effective Field Theory calculation we generated two versions of nuclear forces,the non-minimal coupling choice of Eq. (253) and¯ β = ¯ β = 12 , (254)for which the deuteron pion-exchange charge contributiondisappears. Both versions lead to the same results for thedeuteron charge form factor. It is important to note thatregularization prescription in Eq. (252) works only due tothe simple structure of the charge operator in Eq. (250).For the current operator or even for the charge opera-tor for isovector observables the regularization of the cur-rent operator needs more effort and can be worked outwithin higher derivative regularization. This is still a workin progress.At the order Q there are no short-range contribution todeuteron charge form factor, as can be directly followedfrom Eq. (92). The first short-range contributions showup at the order Q and can be parametrized by threeparameters multiplied with isoscalar electric form factor V = 2 eG SE ( k ) (cid:18) A k + B k ( σ · σ )+ C k · σ k · σ (cid:19) , (255)where A , B , and C are low-energy constants (LECs) [5].Although these are Q -order effects, they catch the short-range off-shell dependence of N LO nuclear forces. Thisis a direct consequence of the fact that contributions inEq. (255) can be generated with unitary transformationsacting on the single-nucleon charge density. The same uni-tary transformations acting on the kinetic energy termproduce off-shell short-range contributions to nuclear forceat N LO. Not all three LECs contribute independently inthe deuteron charge form factor. Only a linear combina-tion A + B + C/ LO nuclear force [90] V , reg2N:cont = 2 eG SE ( k ) × (cid:20) ( A + B ( σ · σ )) F (cid:18) p − p , p (cid:48) − p (cid:48) , k (cid:19) + CF (cid:18) p − p , p (cid:48) − p (cid:48) , k (cid:19) (cid:21) , (256)where the functions F and F are defined as F i ( p , p (cid:48) , k ) := E i (cid:18) p − k , p (cid:48) (cid:19) + E i (cid:18) p + k , p (cid:48) (cid:19) + E i (cid:18) p (cid:48) − k , p (cid:19) + E i (cid:18) p (cid:48) + k , p (cid:19) , (257) E ( p , p (cid:48) ) := (cid:0) p − p (cid:48) (cid:1) e − p p (cid:48) Λ ,E ( p , p (cid:48) ) := [( σ · p )( σ · p ) − ( σ · p (cid:48) )( σ · p (cid:48) )] × e − p p (cid:48) Λ . (258) In order to give a theoretical error quantification due totruncation of chiral expansion one can use the algorithmproposed in [88]. Namely, one can estimate the truncationerror δ ( X ) ( i ) of an observable X at i -th order of the chiralexpansion, with i = 0 , , , . . . . If Q denotes the chiralexpansion parameter, the expressions for truncation errorsare δ ( X ) (0) ≥ max (cid:16) Q | X (0) | , | X ( i ≥ − X ( j ≥ | (cid:17) ,δ ( X ) (2) = max (cid:16) Q | X (0) | , Q | ∆X (2) | , | X ( i ≥ − X ( j ≥ | (cid:17) ,δ ( X ) ( i ) = max (cid:16) Q i +1 | X (0) | , Q i − | ∆X (2) | , Q i − | ∆X (3) | (cid:17) for i ≥ . (259) In the above formulas X ( i ) is a prediction for the observ-able X at i -th order, ∆X (2) ≡ X (2) − X (0) and ∆X ( i ) ≡ X ( i ) − X ( i − for i ≥
3. The algorithm of Eq. (259) isvery simple. However, it does not give a statistical in-terpretation of the error estimate. An interesting algo-rithm based on Bayesian approach was developed in [199,200,201]. Based on these studies we employed a Bayesianmodel specified in [202] to give a theoretical truncationerror estimate of the deuteron charge form factor.Deuteron charge form factor based on the consistentlyregularized charge operator, defined in Eqs. (250), (252),(256) is shown in Fig 2 for cutoff Λ = 500 MeV. A fit tothe data up to Q = 4 fm − was performed to fix a linearcombination of LECs A + B + C/
3. It was verified that thecutoff variation in the range of Λ = 400 . . . M eV leadsto the results which are lying well within the truncationerror band. Out of the calculated deuteron charge formfactor one can extract the structure radius of the deuteron r str = 1 . +0 . − . fm , (260)Individual contributions to the uncertainties are given inTable 7. This structure radius was calculated with theminimal-nonlocality choice of Eq. (253) of unitary phases¯ β and ¯ β . We repeated the calculation of the structureradius with the choice of Eq. (254) for which the deuteroncontributions of long-range charge operator vanish. Theresult for the structure radius for this choice agreed withthe one in Eq. (260) which confirms that off-shell ambi-guities do not affect the final result. With this finding,we were able to make a prediction for neutron root-mean-square (RMS) radius. The structure radius of the deuteroncan be expressed as the RMS radius of the deuteron mi-nus individual nucleon contributions and minus relativisticcorrection (Foldy-Darwin term): r = r d − r p − r n − m p , (261)where r d , r p and r n are deuteron, proton and neutronRMS charge radii, respectively. m p denotes here a protonmass.The extraction of the deuteron-proton RMS chargeradii difference can be made from the hydrogen-deuterium1S-2S isotope shift measurements [209] accompanied withan accurate QED analysis up to three-photon exchange ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 33 Table 7.
Deuteron structure radius squared predicted at N LO in χ EFT (1st column) and the individual contributions to itsuncertainty: from the truncation of the chiral expansion (2nd), the statistical error in the short-range charge density operatorextracted from G C ( Q ) (3rd), the errors from the statistical uncertainty in π N LECs from the Roy-Steiner analysis of Ref. [207,208] propagated through the variation in the deuteron wave functions (4th), the errors from the statistical uncertainty in 2NLECs from the analysis of the 2N observables of Ref. [90] (5th), the error from the choice of the maximal energy in the fit (6th)as well as the total uncertainty evaluated using the sum of these numbers in quadrature (7th). All numbers are given in fm . r truncation ρ cont2N π N LECs 2N LECs Q -range total3.8933 ± ± ± +0 . − . ± +0 . − . � � � � � � ����������������� � [ �� - � ] � � � � ������������� | � � ( � )| � � ( � ) ������ Fig. 2. (Color online) Deuteron charge form factor from thebest fit to data up to Q = 4 fm − evaluated for the cut-off Λ = 500 MeV (solid blue lines). Band between dashed(blue) lines corresponds to a 1 σ error in the determination ofthe short-range contribution to the charge density operator atN LO. Light-shaded (orange dotted) band corresponds to theestimated error (68% degree of belief) from truncation of thechiral expansion at N LO. Open violet circles, green trianglesand blue squares are experimental data from Refs. [203], [204]and [205], respectively. Black solid circles correspond to theparameterization of the deuteron form factors from Ref. [12,206] which is not used in the fit and shown just for comparison.The rescaled charge form factor of the deuteron, G C ( Q ) scaled ,as defined in Ref. [12], is shown on a linear scale. accuracy. According to [209] the deuteron-proton RMScharge radii difference is given by r d − r p = 3 . . (262)See also [210] for an earlier determination. This relationleads immediately to a very precise prediction of the neu-tron rms radius r n = − . +0 . − . fm . (263)which is 1 . σ smaller than the one given by the PDG [211].It is important to note that our results for neutron ra-dius rely on chiral nuclear forces fitted to the Granada-2013 database [212]. Isospin breaking effect analysis in the nuclear force was done along with the treatment of Ni-jmegens group [213]. Isospin-breaking in one-pion-exchangedue to different pion masses M π and M π ± as well ascharge dependence of the short-range interactions in the S wave was explicitly taken into account. These arethe dominant isospin-breaking effects that are needed fora correct description of the phase-shifts. For calculationof scattering observables in the two-nucleon system, theisospin-breaking due to long-range electromagnetic inter-action have been considered. Improved Coulomb poten-tial [214], the magnetic-moment interaction [215] as wellas the vacuum-polarization potential [216] were taken intoaccount [79]. There are, however, further corrections thatare systematically worked out within chiral EFT. Expres-sions for the leading and subleading isospin-breaking two-pion-exchange-potential and irreducible pion-photon ex-change contributions are already available. Also charge-dependence of the pion-nucleon coupling constant needsto be accounted for in a systematic treatment within chiralEFT. Chiral nuclear force which takes into account theseisospin-breaking effects was presented in [218]. It will beinteresting to repeat the calculation of the deuteron chargeform factor by using the new chiral nuclear force [218] andextract in this way the value for neutron RMS radius. As reviewed in this article, chiral EFT provides a power-ful framework for a description of nuclear forces and cur-rents in a systematically improvable way. Nuclear forcesconstructed up to N LO achieved already a tremendousprecision such that they describe two-nucleon data with χ ∼ Q with the leading-orderstarting at Q − . They have been calculated by using theT-matrix inversion technique by JLab-Pisa group and theunitary transformation technique by Bochum-Bonn group.Pseudoscalar (scalar) current operators have been workedout up to order Q with the leading-order starting at Q − ( Q − ). They have been calculated within the unitarytransformation technique by Bochum-Bonn group. To geta renormalizable current our group used time-dependentunitary transformations which explicitly depend on exter-nal sources. This leads to energy-transfer dependent cur-rents with the modified continuity equations for vectorand axial-vector currents. The behavior of four-vectorsunder the Lorentz transformation further constrains the currents. Expressions for vector as well as for axial-vectorcurrents derived by JLab-Pisa and Bochum-Bonn groupdiffer at order Q . It was shown that for the vector cur-rent, two-pion-exchange operators can be transformed intoeach other by a similarity transformation. For the axial-vector current the situation is more confusing. We recal-culated box-diagram contributions to the axial vector cur-rent, which are proportional to g A , by using the T-matrixinversion technique. At the level of Fock-space we cameto the conclusion that these currents should be unitaryequivalent to Bochum-Bonn current. However, we couldnot reproduce the final results of JLab-Pisa group. Thisissue should be clarified in the future.In most available calculations of nuclear currents, di-mensional regularization has been used to regularize loopdiagrams. Additional regularization of the current opera-tor is, however, required when it is sandwiched betweenwave functions for calculation of observables. Nuclear wavefunctions themselves are calculated from the solution ofthe Schr¨odinger equation with an input of nuclear forcewhere cutoff regularization was used. A naive procedureto multiply the current operators with some cutoff func-tion does not work at order Q . In the case of axial-vectorcurrent we have explicitly shown that this procedure leadsto violation of the chiral symmetry. This happens due todifferent divergent pieces in dimensional and cutoff reg-ularization. A completely perturbative one-loop calcula-tion of two-nucleon axial-vector current observable doesnot exist in this case in an infinite cutoff limit. To renor-malize the theory one has to include derivative-less pion-two-nucleon vertices. Chiral symmetry constraint, how-ever, does not allow the existence of such a vertex. Thisshows that regularization artifacts in such a procedure arenot under control. This happens just due to different regu-larizations used in the calculation of currents (dimensionalregularization) and in the iteration procedure (cutoff reg-ularization). If one would use the same regularization inthe calculation of forces and current operators this prob-lem would not arise. We conclude that to achieve order- Q precision, the current operators have to be calculated withthe same cutoff regularization which was used in nuclearforces. On top of this, a chosen cutoff regularization shouldpreserve the chiral symmetry. A higher derivative regular-ization approach, where cutoff regularization is introducedon the Lagrangian level with all derivative operators be-ing covariant, seems to be a promising tool to achieve thisgoal. Calculation of nuclear current operators within thisprocedure is in progress.Concerning numerical studies with nuclear current op-erators, we restricted our discussion only to consistentlyregularized current operators. It is important to mentionthat deuteron form factors and many other observables forthe vector and axial-vector current have been extensivelydiscussed in the literature within a hybrid approach, seeSec. 1 for references. We claim, however, that at the order Q the consistency issue becomes essential and should becarefully investigated.As the first application of a consistently regularizedelectromagnetic charge operator, we discussed the deuteron charge form factor. Although consistently regularized cur-rent operators are not yet available, it is already possibleto get a consistently regularized electromagnetic chargeoperator for isoscalar observables like deuteron charge formfactor. The reason is that the meson-exchange charge op-erator at the order Q has a very simple form. Only rela-tivistic 1 /m - corrections to the leading one-pion-exchangecharge operator survive at this order. A consistent regular-ization of this operator is very simple and was discussed inSec. 8.1. On top of a parameter-free long-range charge op-erator, there is one short-range operator that we fitted tothe deuteron charge form factor. The constructed chargeoperator inherits automatically unitary ambiguities of thenuclear force. Thus, a nontrivial test of the consistency isa test if the deuteron form factor is independent of thechosen unitary ambiguity in nuclear forces and currents.This consistency check was performed and we were able todescribe the deuteron form factor with a quantified trun-cation error analysis. The Bayesian approach was used togive a statistical interpretation of truncation errors. It wasdemonstrated that working with consistently regularizedforces and currents allows for very precise extraction of theRMS radius of the neutron out of the deuteron radius. Atthis level of precision isospin-breaking effects in nuclearforces play a significant role. So far they have been takeninto account only partly, in the same way as was donein most phenomenological potentials. Chiral EFT, how-ever, allows for more precise treatment. In particular, con-tributions of the leading and subleading isospin-breakingtwo-pion-exchange-potential, irreducible pion-photon ex-change and charge-dependence of pion-nucleon couplingconstant needs to be accounted for in a systematic treat-ment within chiral EFT. The construction of this force isby now finished. So it will be very interesting to repeatthe calculation of the deuteron form factor with the con-sistently regularized current operator and the state of theart chiral nuclear force. This is work in progress. Acknowledgement
I would like to express my thanks to my collaboratorsEvgeny Epelbaum and Ulf-G. Meißner as well as ArseniyFilin, Vadim Baru, Patrick Reinert and Daniel M¨oller forsharing their insight on the discussed topics. This work issupported by DFG (CRC110, “Symmetries and the Emer-gence of Structure in QCD”).
A External sources in chiral EFT
In this appendix, we briefly review the role of externalsources in chiral perturbation theory. External sourcesplayed an essential role in the original formulation of chi-ral perturbation theory by Gasser and Leutwyler [34].They allow for a systematic inclusion of chiral symme-try breaking effects, they provide a tool for generating allpossible Ward-identities, and they can be used to definenuclear current operators. Let us briefly follow the dis-cussion of [34] for two-flavour case. We start with QCD ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 35
Lagrangian in the chiral limit where up and down quarkmasses are set to zero, L = (cid:88) quark flavors ¯ qiγ µ (cid:18) ∂ µ + ig λ a G aµ (cid:19) q − G aµν G aµν , (264)with the gluon-fields G aµ and G aµν = ∂ µ G aν − ∂ ν G aµ − gf abc G bµ G cν (265)the corresponding field strength tensor. λ a are 3 × λ a λ b ) = 2 δ ab , (266) (cid:20) λ a , λ b (cid:21) = if abc λ c , (267)with f abc the totally antisymmetric SU(3) structure con-stants. It is easy to show that L QCD is invariant underthe global chiral SU(2) L × SU(2) R transformation whichcan be decomposed in vector and axial transformations.The invariance of L QCD under vector transformation isequivalent to isospin-symmetry. The invariance under ax-ial transformation is not shared by the ground state. Sothe axial symmetry of QCD is spontaneously broken. As aconsequence there exist three (number of broken genera-tors) massless Goldstone bosons which one identifies withpions. Due to their Goldstone boson nature, the interac-tion between pions vanishes if their four-momenta vanish.This gives us an expansion parameter in low energy sectorgiven by low momenta of pions divided by chiral symme-try breaking scale Λ χ . In reality, quark masses are notequal to zero but are much smaller than Λ χ . A massiveversion of QCD violates chiral symmetry. However, sincethe quark masses are small one can use perturbation the-ory for the systematic inclusion of chiral symmetry break-ing effects. Additional to quark-gluon interaction, quarksalso interact with photons and W ± , Z bosons. A con-venient method for the description of these interactionsand incorporation of quark masses is the introduction ofaxial-vector, vector, pseudoscalar, and scalar sources. Oneextends massless QCD to L QCD = L QCD + ¯ qγ µ ( v µ + γ a µ ) q − ¯ q ( s − iγ p ) q, (268)where the external fields v µ ( x ) , a µ ( x ) , s ( x ) , p ( x ) are her-mitian, color-neutral matrices in flavor space. The quarkmass matrix is included in s ( x ). If we require that the ex-ternal fields transform under a local chiral transformationas v (cid:48) µ + a (cid:48) µ = R ( v µ + a µ ) R † + iR∂ µ R † ,v (cid:48) µ − a (cid:48) µ = L ( v µ − a µ ) L † + iL∂ µ L † ,s (cid:48) + ip (cid:48) = R ( s + ip ) L † , (269) then L QCD is invariant under a local chiral transforma-tion, and the generating functionalexp( iZ [ v, a, s, p ]) = (cid:90) [ dG ][ dq ][ d ¯ q ] exp (cid:18) i (cid:90) d x L QCD (cid:19) is invariant under the transformation given in Eq. (269): Z [ v (cid:48) , a (cid:48) , s (cid:48) , p (cid:48) ] = Z [ v, a, s, p ] . (270)Eq. (270) can be used as a master equation for generat-ing all possible Ward identities of QCD. In the absence ofanomalies, the Ward identities are equivalent to the state-ment that the generating functional is invariant under thegauge transformation of the external fields [36]. In orderthat the Green functions of the effective field theory obeythe Ward-Identities of QCD it is sufficient to construct aneffective generating functional in the presence of the sameexternal fields such that it is invariant under the samegauge transformation: Z eff [ v (cid:48) , a (cid:48) , s (cid:48) , p (cid:48) ] = Z eff [ v, a, s, p ] . (271)The generating functional Z eff is built out of the point-like pion and nucleon degrees of freedom. A path-integralform of the effective generating functional is given byexp( iZ eff [ v, a, s, p ]) = (cid:90) [ dN ][ d ¯ N ][ dU ] exp (cid:18) i (cid:90) d x L eff (cid:19) , where N and U denote nucleon and pion fields, respec-tively. U field is parametrized in form of a unitary matrixwith det U = 1. The effective Lagrangian L eff in Z eff isthe most general Lagrangian compatible with Eq. (271).To achieve this, one replaces the usual derivative of pionfields by the covariant one: ∂ µ U → ∇ µ U = ∂ µ U − i ( v µ + a µ ) U + iU ( v µ − a µ ) . (272)Under local chiral transformations the above term trans-forms as ∇ µ U → R ( ∇ µ U ) L † . (273)The scalar and pseudoscalar external fields can be embod-ied in the field χ = 2 B ( s + ip ) , (274)where the constant B is related to the non vanishing quarkcondensate by (cid:104) | ¯ uu | (cid:105) = (cid:104) | ¯ dd | (cid:105) = − F π B + O ( m u , m d ) . (275)The lowest order contribution to L eff with only pionic de-grees of freedom which is invariant under a local chiraltransformation is then given by L (2) ππ = F π (cid:0) ∇ µ U ( ∇ µ U ) † + χU † + U χ † (cid:1) . (276)The only requirement for the pion field U is that it hasto be a unitary matrix with det U = 1. Their explicit parametrization affects only off-shell objects but does notaffect observables. We see that the introduction of exter-nal sources can be used as a tool for the systematic con-struction of chiral perturbation theory where explicit chi-ral symmetry breaking (due to the non-vanishing quarkmass) as well as electro-weak couplings are taken into ac-count. In the same framework, it is natural to use externalsources to derive nuclear currents where nuclei are probedby photon or W ± , Z exchange. For this purpose, onecan derive effective Hamiltonian out of the effective La-grangian. The effective Hamiltonian which we get in thisway depends on external sources and is time-dependent.Explicit expressions for the effective Hamiltonian can befound in [141,23,125]. Nuclear currents are derived out ofthe quantized effective Hamiltonian by taking functionalderivatives in external sources. B Continuity equation
Here we derive a continuity equation which is a directconsequence of chiral or U(1) V symmetry. The effectiveHamiltonian in the presence of external sources can bewritten in the form H eff [ s, p, a, v ] = H eff + (cid:90) d x (cid:104) S ( x ) f s (cid:18) i ∂∂t (cid:19) ( s ( x ) − m q )+ P ( x ) f p (cid:18) i ∂∂t (cid:19) · p ( x ) + A µ ( x ) f a (cid:18) i ∂∂t (cid:19) · a µ ( x )+ V µ ( x ) f v (cid:18) i ∂∂t (cid:19) · v µ ( x ) (cid:105) + O (source ) , (277)where “source” denotes one of the sources s, p, a or v . Thesources depend on a four-vector x = ( t, x ). Note that weallow here a dependence of the Hamiltonian not only onthe sources but also on arbitrary many time-derivatives ofthe sources which we denote by functions f s , f p , f a , and f v . Under an infinitesimal local chiral transformation, thesources transform via v µ → v (cid:48) µ = v µ + v µ × (cid:15) V + a µ × (cid:15) A + ∂ µ (cid:15) V , a µ → a (cid:48) µ = a µ + a µ × (cid:15) V + v µ × (cid:15) A + ∂ µ (cid:15) A ,s → s (cid:48) = s − p · (cid:15) A , s → s (cid:48) = s + s × (cid:15) V − p (cid:15) A ,i p → i p (cid:48) = i ( p + s · (cid:15) A ) ,i p → i p (cid:48) = i ( p + p × (cid:15) V + s (cid:15) A ) . (278)Due to the chiral symmetry, there exists a unitary trans-formation U ( t ) such that H eff [ s (cid:48) , p (cid:48) , a (cid:48) , v (cid:48) ] = U ( t ) † H eff [ s, p, a, v ] U ( t )+ (cid:18) i ∂∂t U ( t ) † (cid:19) U ( t ) . (279) We make an ansatz U ( t ) = exp (cid:32) i (cid:90) d x (cid:104) R v ( x , i∂/∂t ) · (cid:15) V ( x , t )+ R a ( x , i∂/∂t ) · (cid:15) A ( x , t ) (cid:105)(cid:33) (280)We concentrate now on the vector part and set (cid:15) A = 0.The derivation of the continuity equation for the axial-vector follows the same path. Keeping only linear termsin (cid:15) V and setting all sources to zero, besides s which isset to s = m q , we get (cid:90) d x (cid:104) V µ ( x ) f v (cid:18) i ∂∂t (cid:19) ∂ µ + i (cid:104) R v ( x , i∂/∂t ) , H eff (cid:105) − R v ( x , i∂/∂t ) ∂∂t (cid:105) · (cid:15) V ( x , t ) = 0 , (281)for arbitrary (cid:15) V ( x , t ). We can solve this equation pertur-batively in i ∂/∂t writing R v ( x , i∂/∂t ) = ∞ (cid:88) n =0 R vn ( x ) (cid:18) i ∂∂t (cid:19) n ,f v ( i∂/∂t ) = ∞ (cid:88) n =0 f nv (cid:18) i ∂∂t (cid:19) n . (282)With this ansatz we can rewrite Eq. (281) into a series ofequations (cid:90) d x (cid:104) − i V ( x ) f n − v − ∂ j V j ( x ) f nv + i (cid:104) R vn ( x ) , H eff (cid:105) + i R vn − ( x ) (cid:105) · (cid:18) i ∂∂t (cid:19) n (cid:15) V ( x , t ) = 0 , (283)with f − v = R v − ( x ) = 0. Eq. (283) gives a recursive def-inition of R vn − ( x ) operator. The vector current operatorin momentum space is given by˜ V µ ( k , k ) = (cid:90) d x V µ ( x ) f v (cid:18) i ∂∂t (cid:19) exp( − i k · x )= ˜ V µ ( k ) f v ( k ) . (284)Rewritten in momentum space Eq. (283) is given by1 n ! ∂ n ∂k n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k =0 k µ ˜ V µ ( k ) + (cid:104) H eff , ˜ R vn ( k ) (cid:105) − ˜ R vn − ( k ) = 0 , One can also write this in a following form k µ ˜ V µ ( k ) | k =0 + (cid:104) H eff , ˜ R v ( k ) (cid:105) = 0 , ˜ R v ( k ) = ∂∂k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k =0 k µ ˜ V µ ( k ) + (cid:104) H eff , ˜ R v ( k ) (cid:105) , ˜ R v ( k ) = 12! ∂ ∂k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k =0 k µ ˜ V µ ( k ) + (cid:104) H eff , ˜ R v ( k ) (cid:105) , ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 37 and so on. So altogether we get ∞ (cid:88) n =0 n ! (cid:104) H eff , ∂ n ∂k n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k =0 k µ ˜ V µ ( k ) (cid:105) n = 0 , (285)where we used a definition of n -th commutator (cid:2) A, B (cid:3) n = (cid:2) A, (cid:2) A, B (cid:3) n − (cid:3) , and (cid:2) A, B ] = B. (286)Using the Baker-Campbell-Hausdorff formulaexp( A ) B exp( − A ) = ∞ (cid:88) n =0 n ! [ A, B ] n , (287)we can write the continuity equation (285) in a more com-pact formexp (cid:32) H eff −→ ∂∂k (cid:33) k µ ˜ V µ ( k ) exp (cid:32) − H eff ←− ∂∂k (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k =0 = 0 . (288)In a similar way one can derive the continuity equationfor the axial-vector currentexp (cid:32) H eff −→ ∂∂k (cid:33) (cid:104) k µ ˜ A µ ( k ) + i m q ˜ P ( k ) (cid:105) × exp (cid:32) − H eff ←− ∂∂k (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k =0 = 0 . (289)Note that without exponential operators Eqs. (288) and(289) reduce to the on-shell continuity equations. So theexponential operators in Eqs. (288) and (289) seem toswitch on the on-shell condition. In order to prove thatthis is indeed the case we put Eqs. (288) between initialand final states which are the eigenstates of the nuclearforce H eff | i (cid:105) = E i | i (cid:105) , H eff | f (cid:105) = E f | f (cid:105) . (290)Using Eq. (290) we getexp (cid:32) E f −→ ∂∂k (cid:33) (cid:104) f | k µ ˜ V µ ( k ) | i (cid:105) exp (cid:32) − E i ←− ∂∂k (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k =0 = 0(291)Since the exponential operators in Eq. (291) are just thetranslation operators with the propertyexp (cid:32) E f −→ ∂∂k (cid:33) F ( k ) = F ( k + E f ) , (292)for any smooth function F ( k ). Applying exponential op-erators from the left and right hand sides we getexp (cid:32) E f −→ ∂∂k (cid:33) F ( k ) exp (cid:32) − E i ←− ∂∂k (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k =0 = F ( k + E f − E i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k =0 = F ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k = E f − E i . (293) Applying this result to the continuity equation for thevector current we get (cid:104) f | k µ ˜ V µ ( k ) | i (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k = E f − E i = 0 . (294)But the condition on the left-hand side of Eq. (294) isjust the on-shell condition. So we see that the exponen-tial operators in Eqs. (288) and (289) do just switch onthe on-shell condition. Note that for the energy-transferindependent currents Eqs. (288) and (289) reduce to theordinary continuity equations (cid:104) H eff , ˜ V ( k ) (cid:105) = k · ˜ V ( k ) , (cid:104) H eff , ˜ A ( k ) (cid:105) = k · ˜ A ( k ) − i m q ˜ P ( k ) , (295)For the linear dependence on the energy-transfer Eqs. (288)and (289) reduce to Eq. (50). For the quadratic and thehigher-order dependence on the energy-transfer Eqs. (288)and (289) produce an increasing number of commutatorswhich, however, is always finite, as long as the currentsare polynomials of the energy transfer. C Unitary transformations
In this appendix we review all unitary transformationswhich are needed for derivation of renormalizable nuclearforces and currents. As already explained in section 3,the unitary transformations which bring nuclear Hamilto-nian into a block-diagonal renormalizable form are time-independent. The renormalizability of the chiral nuclearforce constrains strongly the choice of the unitary phasesand leaves only two phases ¯ β and ¯ β unfixed. These uni-tary transformations can be parametrized via U η = exp (cid:32) (cid:88) i =1 α i S i (cid:33) , (296)where S i are antihermitian operators. Their explicit formas well as the values of the phases α i can be found in [68].These transformations act on η -space only, and are appliedto the effective Hamiltonian on top of Okubo transforma-tions of Eq. (6). U η are needed to make nuclear forcesrenormalizable.Additionally to these transformations there are furthertwo time-independent transformations which contributeto relativistic corrections: U η, /m = exp (cid:0) ¯ β S + ¯ β S (cid:1) , (297)where antihermitian operators S and S are given by [142,148] S = η ˜ H (2)20 ηH (1)21 λ E π H (1)21 − h.c.,S = η ˜ H (3)21 λ E π H (1)21 − h.c.. (298) The operators H ( κ ) ab refer to the vertices in the effectivechiral Hamiltonian with a nucleon and b pion fields. κ de-notes the inverse mass dimension of the coupling constantin the given vertex, see Eq. (69). The operators ˜ H ( κ ) ab re-fer to the 1 /m -correction of the corresponding vertices.The phases ¯ β and ¯ β are known to be responsible for thedegree of non-locality of the relativistic 1 /m -correctionto the one-pion exchange. Minimal nonlocality is achievedfor the values in Eq. (253).There are various time-dependent unitary transforma-tions which depend on external sources. Those are exten-sively discussed in [142,148] for vector currents and in [23]for axial-vector and pseudoscalar currents. Most of thephases of these unitary transformations are either fixedby renormalizability and matching to nuclear forces re-quirements or they do not affect the final expressions ofthe current operators. D Vector Current: Two-Pion-Exchange
In this appendix, we give two-pion-exchange contributionsto the vector current operator. Due to the coupling ofthe scalar source to two pions there appear loop func-tions which depend on three momenta k , q and q whichare momentum transfer of the vector source, momentumtransfer of the first and second nucleons, respectively. Thisleads to lengthy expressions which have been derived in[141] and are listed here for completeness: J = (cid:88) i =1 24 (cid:88) j =1 f ji ( q , q ) T i O j , (299) J = (cid:88) i =1 8 (cid:88) j =1 f jSi ( q , q ) T i O Sj , (300)where f ji ≡ f ji ( q , q ) are scalar functions and the spin-momentum operators O i and O Si are given by O = q + q , O = q − q , O = [ q × σ ] + [ q × σ ] , O = [ q × σ ] − [ q × σ ] , O = [ q × σ ] + [ q × σ ] , O = [ q × σ ] − [ q × σ ] , O = q ( q · [ q × σ ]) + q ( q · [ q × σ ]) , O = q ( q · [ q × σ ]) − q ( q · [ q × σ ]) , O = q ( q · [ q × σ ]) + q ( q · [ q × σ ]) , O = q ( q · [ q × σ ]) − q ( q · [ q × σ ]) , O = ( q + q ) ( σ · σ ) , (301) O = ( q − q ) ( σ · σ ) , O = q ( q · σ ) ( q · σ ) + q ( q · σ ) ( q · σ ) , O = q ( q · σ ) ( q · σ ) − q ( q · σ ) ( q · σ ) , O = ( q + q ) ( q · σ ) ( q · σ ) , O = ( q − q ) ( q · σ ) ( q · σ ) , O = ( q + q ) ( q · σ ) ( q · σ ) , O = ( q − q ) ( q · σ ) ( q · σ ) , O = σ ( q · σ ) + σ ( q · σ ) , O = σ ( q · σ ) − σ ( q · σ ) , O = σ ( q · σ ) + σ ( q · σ ) , O = σ ( q · σ ) − σ ( q · σ ) , O = q ( q · σ ) ( q · σ ) + q ( q · σ ) ( q · σ ) , O = q ( q · σ ) ( q · σ ) − q ( q · σ ) ( q · σ ) , and O S = ,O S = q · [ q × σ ] + q · [ q × σ ] ,O S = q · [ q × σ ] − q · [ q × σ ] ,O S = σ · σ ,O S = ( q · σ ) ( q · σ ) ,O S = ( q · σ ) ( q · σ ) ,O S = ( q · σ ) ( q · σ ) + ( q · σ ) ( q · σ ) ,O S = ( q · σ ) ( q · σ ) − ( q · σ ) ( q · σ ) . (302)As a basis for the isospin operators we choose T = [ τ ] + [ τ ] ,T = [ τ ] − [ τ ] ,T = [ τ × τ ] ,T = τ · τ ,T = . (303)The nonvanishing long-range contributions to the scalarfunctions f ji ≡ f ji ( q , q ) are given by f = ieg A L ( q )128 π F π (cid:20) g A (8 M π + 3 q )4 M π + q − (cid:21) + eπF π (cid:20) g A M π I ( d +2)(2 , , + 4 πg A M π q I ( d +4)(2 , , − πg A M π q I ( d +4)(3 , , ( q − q z ) − π g A q q zI ( d +6)(4 , , +32 π g A q q I ( d +6)(3 , , ( q z + q z + q ) − πg A q I ( d +4)(2 , , ( q + 2 q z ) − g A − g A M π × I ( d +2)(2 , , − π ( g A − g A q I ( d +4)(2 , , +8 π ( g A − g A q I ( d +4)(3 , , ( q − q z ) − π ( g A − I ( d +4)red (2 , , (cid:21) − (1 ↔ , (304) ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 39 f = ieg A ( g A + 1)256 π F π + ieg A L ( q )128 π F π (cid:20) g A (8 M π + 3 q )4 M π + q − (cid:21) + e F π (cid:20) − π g A q I ( d +4)(3 , , ( M π ( q − q z ) + q q z )+16 π g A q I ( d +4)(2 , , ( q q ( z + 1) − M π ( q − q z )) − πg A I ( d +2)(1 , , (2 M π + q q z )+ 8 πg A M π I ( d +2)(2 , , ( M π + q ( q z − q )) − π g A q q zI ( d +6)(4 , , − π g A q q I ( d +6)(3 , , × ( q z − q ( z + 1)) + 16 π g A q I ( d +4)(2 , , ( q − q z )+2 π ( g A − I ( d +2)red (1 , , − π ( g A − g A I ( d +2)(2 , , × (2 M π + q ( q z − q )) + 32 π ( g A − g A q I ( d +4)(2 , , × ( q − q z ) + 64 π ( g A − g A q I ( d +4)(3 , , ( q − q z ) − π ( g A − I ( d +4)red (2 , , + g A M π I (4)(1 , , − g A − g A M π I (4)(1 , , (cid:21) + (1 ↔ , (305) f = ieg A ( g A + 1)256 π F π + ieg A L ( q )128 π F π × (cid:20) g A (8 M π + 3 q )4 M π + q − (cid:21) + (1 ↔ , (306) f = ieg A L ( q )128 π F π (cid:20) g A (8 M π + 3 q )4 M π + q − (cid:21) − (1 ↔ , (307) f = − ieg A π F π + ieg A L ( k )32 π F π − ieg A L ( q )64 π F π + eg A π F π × (cid:20) g A ( − M π ) I ( d +2)(1 , , + 4 πg A q I ( d +4)(2 , , ( q − q z )+ ( g A − I ( d +2)red (1 , , (cid:21) + (1 ↔ , (308) f = − ieg A L ( q )64 π F π − eg A π F π I ( d +4)(2 , , q ( q + q z ) − (1 ↔ , (309) f = eg A π F π (cid:20) πg A ( q − M π ) I ( d +4)(3 , , − g A M π I ( d +2)(2 , , +192 π g A q I ( d +6)(4 , , − π g A q q zI ( d +6)(3 , , +8 πg A I ( d +4)(2 , , + ( g A − I ( d +2)(2 , , +16 π ( g A − I ( d +4)(3 , , (cid:21) − (1 ↔ , (310) f = ieg A π F π k − ieg A L ( k )32 π F π k + eg A π F π × (cid:20) πg A ( q − M π ) I ( d +4)(3 , , − g A M π I ( d +2)(2 , , (311)+384 π g A q I ( d +6)(4 , , − πg A q q zI ( d +4)(2 , , − π g A q q zI ( d +6)(3 , , + g A I ( d +2)(1 , , + 16 πg A I ( d +4)(2 , , +2( g A − I ( d +2)(2 , , + 32 π ( g A − I ( d +4)(3 , , (cid:21) + (1 ↔ ,f = eg A π F π (cid:20) g A M π I ( d +2)(2 , , − πg A q I ( d +4)(3 , , − π g A q I ( d +6)(3 , , ( q + q z ) + 4 πg A I ( d +4)(2 , , +(1 − g A ) I ( d +2)(2 , , (cid:21) − (1 ↔ , (312) f = ieg A π F π k − ieg A L ( k )32 π F π k − eg A π F π × (cid:20) − πg A I ( d +4)(2 , , (2 M π + q q z ) − g A M π I ( d +2)(2 , , +16 πg A q I ( d +4)(3 , , + 128 π g A q I ( d +6)(3 , , ( q − q z )+ g A I ( d +2)(1 , , + 8 πg A I ( d +4)(2 , , + 2( g A − I ( d +2)(2 , , +16 π ( g A − I ( d +4)(2 , , (cid:21) + (1 ↔ . (313)These two-pion exchange contributions have been calcu-lated by using dimensional regularization. In addition, thereare nonvanishing functions f j given by f = f , f = f , f = − f , f = − f ,f = f , f = f , f = f , f = f . (314)In the above equations, z ≡ ˆ q · ˆ q , q i ≡ | q i | and the loopfunctions L ( q ) and A ( q ) are defined by L ( q ) = 12 sq ln (cid:18) s + qs − q (cid:19) , with s = (cid:112) q + 4 M π ,A ( q ) = 12 q arctan (cid:18) q M π (cid:19) . (315)Further, the functions I correspond to the three-pointfunctions via I ( d )( ν ,ν ,ν ) ≡ I ( d ; 0 , q , ν ; − q , ν ; 0 , ν ) (316)with the notation for the momenta q i = (0 , q i ) (317)and a scalar one-loop three-point function in d space-timedimensions defined by I ( d ; p , ν ; p , ν ; p , ν ; p , ν ) = µ − d (cid:90) d d (cid:96) (2 π ) d (cid:96) + p ) − M π ] ν [( (cid:96) + p ) − M π ] ν [( (cid:96) + p ) − M π ] ν [ v · ( (cid:96) + p )] ν . (318) Note that in this definition higher power of covariant andheavy-baryon propagators are allowed. Here, all propa-gators are understood to have an infinitesimal positiveimaginary part. Notice that here and in what follows, wewill only need the functions I for four-momenta with van-ishing zeroth component. We further emphasize that allfunctions I ( d + n )( ν ,ν ,ν ) which enter the above equations ex-cept I ( d +2)(1 , , , I ( d +4)(2 , , and I ( d +4)(1 , , are finite in dimensionalregularization in the limit d →
4. For these functions, wedefine reduced functions by subtracting the poles in fourdimensions I ( d +2)red (1 , , = I ( d +2)(1 , , − i π L ( µ ) − i π ln (cid:18) M π µ (cid:19) , (319) I ( d +4)red (2 , , = I ( d +4)(2 , , + i π L ( µ ) + i π ln (cid:18) M π µ (cid:19) ,I ( d +4)red (1 , , = I ( d +4)(1 , , + i π L ( µ ) + i π ln (cid:18) M π µ (cid:19) , where L ( µ ) = µ d − π (cid:20) d − γ E − − ln (4 π )) (cid:21) . (320)Here, µ is the scale introduced in dimensional regulariza-tion and γ E = − Γ (cid:48) (1) (cid:39) . f S = eg A A ( k )64 πF π + iπ eg A F π (cid:20) − M π I ( d +4)(2 , , + 16 πq I ( d +6)(3 , , − πq q zI ( d +6)(2 , , + I ( d +4)(1 , , (cid:21) + (1 ↔ , (321) f S = 2 iπ eg A F π (cid:20) πq I ( d +6)(3 , , − M π I ( d +4)(2 , , (cid:21) − (1 ↔ , (322) f S = eg A M π (12 M π + 7 M π q + q q )64 πF π (4 M π + q )(4 M π + q ) − eg A A ( k )(2 M π + q )16 πF π + eg A A ( q )(2 M π + q )32 πF π + ieg A πF π (cid:20) M π ( − I ( d +2)(1 , , ) + 8 πM π q I ( d +4)(2 , , ( q − q z )+64 π q q zI ( d +6)(3 , , − π q q ( z + 1) I ( d +6)(2 , , +2 πq q zI ( d +4)(1 , , (cid:21) + (1 ↔ , (323) f S = − eg A q A ( q )64 πF π − ieg A π F π (cid:20) πq q ( z − I ( d +6)(2 , , − q q zI ( d +4)(1 , , (cid:21) + (1 ↔ , (324) f S = 2 ieg A π F π (cid:20) πq q zI ( d +6)(2 , , − I ( d +4)(1 , , (cid:21) + (1 ↔ , (325) f S = 16 iπ eg A F π q q zI ( d +6)(2 , , + (1 ↔ ,f S = eg A A ( q )128 πF π − iπ eg A F π q I ( d +6)(2 , , + (1 ↔ ,f S = − eg A A ( q )128 πF π − iπ eg A F π q I ( d +6)(2 , , − (1 ↔ ,f S = eg A M π q πF π (4 M π + q )(4 M π + q ) (326)+ eg A ( g A − A ( q )(2 M π + q )32 πF π − (1 ↔ ,f S = − eg A A ( q )128 πF π − (1 ↔ ,f S = eg A A ( q )128 πF π + (1 ↔ . (327)Note that we use here an overcomplete basis for three-point functions in d + x space-time dimensions. One couldreduce them and express everything through just one three-point function. In this case, however, we would produce alot of unphysical singularities which would cancel in thefinal result but would make expressions more lengthy. E Scalar Current: Two-Pion-Exchange
In this appendix, we give two-pion-exchange contributionsto the scalar current operator. Due to the coupling of thescalar source to two pions there appear loop functionswhich depend on the three momenta k , q and q whichdenote the momentum transfer of the scalar source, themomentum transfer of the first and the second nucleons,respectively. This leads to lengthy expressions which havebeen derived in [166] and are listed here for completeness: S ( Q )2N: 2 π = τ · τ (cid:0) q · σ k · σ t + t (cid:1) + q · σ q · σ t + q · σ q · σ t + q · σ q · σ t + σ · σ t + 1 ↔ , (328)The scalar functions t i are expressed in terms of the twoand three-point functions. The notation for the three-point function is I ( d : p , ν ; p , ν ; p , ν ; 0 , ν ) = (cid:90) d d l (2 π ) d (cid:89) j =1 l + p j ) − M π + i(cid:15) ] ν j v · l + i(cid:15) ] ν . (329)For our purpose we need only I (4 : 0 , q , k,
1; 0 ,
0) = i π (cid:90) dt (cid:90) t dy C y − y )( y − y ) , (330) ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 41 with y = D C + (cid:114) D + 4 AC C , y = D C − (cid:114) D + 4 AC C , and A = M + q (1 − t ) t , B = − q · q , C = q and D = 2 q · q + q + tB . For k = 0, the three-point functionreduces to a two-point function I (4 : 0 , q , k,
1; 0 , (cid:12)(cid:12) k =0 = − i π L ( q )4 M π + q . (331)Deploying Eq. (331) in the following expression for t i weget the scalar current at the vanishing momentum transfer k = 0. m q S ( Q )2N:2 π (cid:12)(cid:12) k =0 = M π π F π (4 M π + q ) (cid:20) L ( q )4 M π + q × (cid:16) g A (cid:0) M π + q (cid:1) (cid:0) q σ · σ − q · σ q · σ (cid:1) + (cid:0) M π (cid:0) − g A + 4 g A + 1 (cid:1) + 8 M π q (cid:0) − g A + 5 g A + 1 (cid:1) + q (cid:0) − g A + 6 g A + 1 (cid:1) (cid:1) τ · τ (cid:17) − (cid:0) M π (cid:0) g A − g A + 1 (cid:1) + q (cid:0) − g A + 17 g A (cid:1) (cid:1) × τ · τ (cid:21) . (332)Using the expressions from [167] one can evaluate the slope of the nuclear forces in respect to the quark mass. Combiningthese with expressions for the scalar current at k = 0, it is straightforward to show the validity of Eq. (189), see [166]for more details. The final expressions for t i are given by m q t = g A M π π F π k − g A M π L ( k )32 π F π k ( k + 4 M π ) ,m q t = − (cid:0) g A − (cid:1) M π F π (cid:32) (cid:0) g A − (cid:1) k q q q · q ) − q q ) + (cid:0) g A + 1 (cid:1) M π + 2 g A q (cid:33) i I (4; 0 , q , k,
1; 0 , − M π L ( q )256 π F π (( q · q ) − q q ) (cid:18) − g A (4 M π + q ) ( q q ( k + 4 M π ) − M π ( q · q ) ) (cid:16) k (cid:0) M π + 3 M π q (cid:1) − k (cid:0) M π (cid:0) q + 64 q (cid:1) + M π (cid:0) q + 36 q q (cid:1) + 4 q q (cid:1) + k (cid:0) M π (cid:0) q + 52 q q + 96 q (cid:1) + M π (cid:0) q + 35 q q + 64 q q (cid:1) + 5 q q (cid:0) q + 2 q (cid:1)(cid:1) + k Q − (cid:0) M π (cid:0) − q + 12 q q + 64 q (cid:1) + M π (cid:0) − q + 5 q q + 44 q q (cid:1) + q q (cid:0) q + 8 q (cid:1)(cid:1) + Q − (cid:0) M π (cid:0) q − q (cid:1) + M π (cid:0) q − q q (cid:1) − q q (cid:1) (cid:17) − g A (cid:0) k − k (cid:0) q + 2 q (cid:1) − q Q − (cid:1) + q (cid:0) k − Q − (cid:1)(cid:1) − M π L ( k )512 π F π (( q · q ) − q q ) (cid:18) − g A ( k + 4 M π ) ( q q ( k + 4 M π ) − M π ( q · q ) ) (cid:16) k M π + k (cid:0) M π − M π q − q q (cid:1) + 2 k q (cid:0) − M π + M π (cid:0) q + q (cid:1) + 15 q q (cid:1) + 2 k (cid:0) M π q (cid:0) q + 3 q (cid:1) + M π (cid:0) q q − q (cid:1) − q q + 8 q q (cid:1) − k M π q Q − (cid:0) M π − q + 27 q (cid:1) + 4 M π q Q − (cid:0) M π (cid:0) q − q (cid:1) + q − q q (cid:1) (cid:17) + 8 g A (cid:0) q q + q q · q − ( q · q ) (cid:1) − k q · q (cid:19) − (cid:0) g A + 1 (cid:1) M π π F π ,m q t = 3 i g A I (4; 0 , q , k,
1; 0 , M π ( q · q ) F π ( q q − ( q · q ) ) + 3 g A M π q q (cid:0) k + Q − (cid:1) L ( q ) q · q π F π ( q q − ( q · q ) ) ( q q ( k + 4 M π ) − M π ( q · q ) )+ 3 g A M π L ( k )64 π F π (cid:32) k + 4 M π − q q (cid:0) k − Q − (cid:1) q q − ( q · q ) ) ( q q ( k + 4 M π ) − M π ( q · q ) ) (cid:33) ,m q t = m q t + 3 i g A I (4; 0 , q , k,
1; 0 , M π ( q q − ( q · q ) )8 F π ( q q − ( q · q ) ) ,m q t = 3 g A M π q L ( k ) (cid:0) − k M π + k (cid:0) M π Q + 2 q q (cid:1) − k M π Q − + M π Q − Q (cid:1) π F π ( k + 4 M π ) ( q q − ( q · q ) ) ( q q ( k + 4 M π ) − M π ( q · q ) ) − g A M π q q (cid:0) k − Q − (cid:1) L ( q )64 π F π ( q q − ( q · q ) ) ( q q ( k + 4 M π ) − M π ( q · q ) ) − i g A I (4; 0 , q , k,
1; 0 , M π q q · q F π ( q q − ( q · q ) ) − g A M π q q (cid:0) k + Q − (cid:1) L ( q )64 π F π ( q q − ( q · q ) ) ( q q ( k + 4 M π ) − M π ( q · q ) ) ,m q t = − g A M π L ( k ) (cid:0) − k M π + k (cid:0) M π Q + 2 q q (cid:1) − k M π Q − + M π Q − Q (cid:1) π F π ( k + 4 M π ) ( q q ( k + 4 M π ) − M π ( q · q ) ) + 3 g A M π q q (cid:0) k + Q − (cid:1) L ( q )64 π F π ( q q ( k + 4 M π ) − M π ( q · q ) ) , (333)where Q ± ≡ q ± q . F Folded-diagram technique I
An energy-independent potential can be easily derivedwithin a Releigh-Schr¨odinger perturbation theory. We willfollow the arguments of [29]. The starting point is a time-independent Schr¨odinger equation H | Ψ (cid:105) = E | Ψ (cid:105) . (334)We project Eq. (334) to model and rest spaces and get (cid:0) E − H (cid:1) η | Ψ (cid:105) = ηV η | Ψ (cid:105) + ηV λ | Ψ (cid:105) , (cid:0) E − H (cid:1) λ | Ψ (cid:105) = λV η | Ψ (cid:105) + λV λ | Ψ (cid:105) . (335)Eliminating λ | Ψ (cid:105) from Eq. (335) we get (cid:0) E − H (cid:1) η | Ψ (cid:105) = Q ( E ) η | Ψ (cid:105) , (336)with an energy-dependent potential also called a Q -boxdefined by Q ( E ) = ηV η + ηV λ E − H − λV λ λV η. (337)For scattering observables we get | Ψ + (cid:105) = | φ (cid:105) + 1 E − H + i(cid:15) V | Ψ + (cid:105) = | φ (cid:105) + 1 E − H + i(cid:15) T ( E ) | φ (cid:105) (338)with ( H − E ) | Ψ + (cid:105) = 0 and ( H − E ) | φ (cid:105) = 0 . (339)For the initial state from the model space | φ (cid:105) = η | φ (cid:105) weproject Eq. (338) to the model space and get η | Ψ + (cid:105) = (cid:18) − E − H + i(cid:15) Q ( E ) (cid:19) − η | φ (cid:105) = (cid:18) E − H + i(cid:15) ηT ( E ) η (cid:19) η | φ (cid:105) . (340)From Eq. (340) we get the T-matrix ηT ( E ) η = ( E − H + i(cid:15) ) (cid:20)(cid:18) − E − H + i(cid:15) Q ( E ) (cid:19) − − (cid:21) = Q ( E ) + Q ( E ) 1 E − H − Q ( E ) + i(cid:15) Q ( E ) . (341)To define an energy-independent potential we introduce aMøller operator Ω which is defined by | Ψ (cid:105) = Ωη | Ψ (cid:105) , (342) with the requirement Ω = Ωη. (343)So this operator reproduces the original state out of pro-jected state . By projecting Eq. (342) to the model spaceone immediately gets ηΩ = η. (344)One applies now the operator Ω to the original Schr¨odingerequation and gets (cid:0) E − H (cid:1) Ωη | Ψ (cid:105) = V | Ψ (cid:105) , (345) (cid:0) E − ΩH (cid:1) η | Ψ (cid:105) = ΩηV | Ψ (cid:105) , (346)where H denotes a free Hamiltonian. In order to getEq. (346) one first projects the original Schr¨odinger equa-tion Eq. (334) to the model space and applies on the re-sulting equation the operator Ω . Subtracting Eq. (346)from Eq. (345) we get (cid:2) Ω, H (cid:3) η | Ψ (cid:105) = (cid:0) V − ΩηV (cid:1) | Ψ (cid:105) = (cid:0) V − ΩηV (cid:1) Ωη | Ψ (cid:105) (347)In this way we get a non-linear equation for the Ω -operator (cid:2) Ω, H (cid:3) − V Ω + ΩV Ω = 0 . (348)We can rewrite this equation into an equation for an op-erator A defined by Ω = η + A, (349)with A = λAη . The last relation follows from Eqs. (343)and (344). From Eq. (348) we get λ (cid:0) H + (cid:2) H, A (cid:3) − AV A (cid:1) η = 0 . (350)The effective energy-independent potential is defined by R = ηV Ωη. (351)If one projects the Schr¨odinger equation to the modelspace one immediately gets (cid:0) H + ηV Ω (cid:1) η | Ψ (cid:105) = (cid:0) H + R (cid:1) η | Ψ (cid:105) = Eη | Ψ (cid:105) (352)For the initial state from the model space | φ (cid:105) = η | φ (cid:105) weproject Eq. (338) to the model space and get η | Ψ + (cid:105) = (cid:18) − E − H + i(cid:15) R (cid:19) − η | φ (cid:105) = (cid:18) E − H + i(cid:15) ηT ( E ) η (cid:19) η | φ (cid:105) . (353)For the projected transfer matrix we get in this way ηT ( E ) η = ( E − H + i(cid:15) ) (cid:20)(cid:18) − E − H + i(cid:15) R (cid:19) − − (cid:21) = R + R E − H − R + i(cid:15) R. (354) This is only true in the restricted energy range whichshould be below the pion-production threshold.ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 43
Note that Eqs. (354) and (341) describe the same T-matrix.The right-hand sides of these equations are identical evenhalf-off-shell. Energy independent potential can be easilydescribed in the form of Q -boxes by using Eq. (350). Todo this we multiply the energy-independent potential R with an initial state from model space | i (cid:105) = η | i (cid:105) and get R | i (cid:105) = ηV η | i (cid:105) + ηV Aη | i (cid:105) (355)Rewriting the decoupling equation Eq. (350) into λ (cid:0) V + HA − AH − AR (cid:1) η = 0 (356)and applying it to the model state | α (cid:105) we get A | α (cid:105) = 1 E α − λHλ (cid:0) λV η − AR (cid:1) | α (cid:105) , (357)with H | α (cid:105) = E α | α (cid:105) . After multiplication of Eq. (357)with ηV from the left we get R | α (cid:105) = Q ( E α ) | α (cid:105) − ηV E α − λHλ A | β (cid:105)(cid:104) β | R | α (cid:105) . (358)Here we use Einstein-convention where we sum over dou-ble appearing states (in this case | β (cid:105)(cid:104) β | ). Eqs. (357) and(358) provide an iterative solution for energy-independentpotential. Applying e.g. Eq. (357) to Eq. (358) we caneliminate A -dependence and get iterative non-linear solu-tion for R given by R | α (cid:105) = Q ( E α ) | α (cid:105) + Q ( E α , E β ) | β (cid:105)(cid:104) β | R | α (cid:105) + ηV E α − λHλ E β − λHλ A | γ (cid:105)(cid:104) γ | R | β (cid:105)(cid:104) β | R | α (cid:105) . = Q ( E α ) | α (cid:105) + ∞ (cid:88) n =1 Q n ( E α , E β , . . . , E β n ) × (cid:104) β n | R | β n − (cid:105)(cid:104) β n − | R | β n − (cid:105) . . . (cid:104) β | R | α (cid:105) , (359)where higher Q -boxes are defined by Q n ( E , . . . , E n +1 ) = ( − n ηV n +1 (cid:89) j =1 E j − λHλ V η . (360)Up to three Q -boxes we get R | α (cid:105) = Q ( E α ) | α (cid:105) + Q ( E α , E β ) | β (cid:105)(cid:104) β | R | α (cid:105) + Q ( E α , E β , E β ) | β (cid:105)(cid:104) β | R | β (cid:105)(cid:104) β | R | α (cid:105) = Q ( E α ) | α (cid:105) + Q ( E α , E β ) | β (cid:105)(cid:104) β | Q ( E α ) | α (cid:105) + Q ( E α , E β ) | β (cid:105)(cid:104) β | Q ( E α , E γ ) | γ (cid:105)(cid:104) γ | Q ( E α ) | α (cid:105) + Q ( E α , E β , E β ) | β (cid:105)(cid:104) β | Q ( E β ) | β (cid:105)(cid:104) β | Q ( E α ) | α (cid:105) . (361)One can get the same result (but derived in a more cum-bersome way) with the folded-diagram technique via in-version of half-off-shell T-matrix, see Appendix G for deriva-tion. For application of Q -box formalism within chiralEFT see [183]. Note that the potential R is manifestly non-hermitian.Nevertheless, the half-off-shell T-matrix is exactly repro-duced by iterations of the R potential. If one prefers towork with the hermitian potential, what is usually thecase, one can derive them by applying the unitary trans-formation technique. Unitary transformations do not af-fect the spectrum of the Hamiltonian and under some as-sumptions lead to the same scattering matrix [30,31], seealso [15] for a perturbative proof. The perturbative proofof this statement is simple and we show this here on thelevel of the T-matrix. We start with the half off-shell T-matrix element and rewrite it into a shorter form (cid:104) f | T ( E i ) | i (cid:105) = (cid:104) f | (cid:18) V + V E − H + i(cid:15) V (cid:19) | i (cid:105) = (cid:104) f | (cid:18) H − E f + (cid:0) H − E f (cid:1) E i − H + i(cid:15) (cid:0) H − E i (cid:1)(cid:19) | i (cid:105) = (cid:104) f | (cid:0) H − E f (cid:1) i(cid:15)E i − H + i(cid:15) | i (cid:105) = (cid:0) E i − E f (cid:1) (cid:104) f | i(cid:15)E i − H + i(cid:15) | i (cid:105) . (362)The T-matrix from the transformed Hamiltonian U † HU ,where U is a unitary transformation is given by (cid:104) f | T U ( E i ) | i (cid:105) = (cid:0) E i − E f (cid:1) (cid:104) f | U † i(cid:15)E i − H + i(cid:15) U | i (cid:105) = (cid:0) E i − E f (cid:1) (cid:104) f | U † (cid:18) − E i − H + i(cid:15) V (cid:19) − × i(cid:15)E i − H + i(cid:15) U | i (cid:105) = (cid:0) E i − E f (cid:1) (cid:104) f | U † i(cid:15)E i − H + i(cid:15) U | i (cid:105) + (cid:0) E i − E f (cid:1) (cid:104) f | U † E i − H + i(cid:15) V × (cid:18) − E i − H + i(cid:15) V (cid:19) − i(cid:15)E i − H + i(cid:15) U | i (cid:105) (363)We require now for the unitary transformation U = 1+ δU to fulfill H δU | i (cid:105) (cid:54) = E i | i (cid:105) H δU | i (cid:105) (cid:54) = E f | f (cid:105) . (364)This is a reasonable assumption if δU is at least of thefirst order in the interaction V . It is easy to see that thefirst term of Eq. (363) does not contribute to the on-shellT-matrix, (cid:0) E i − E f (cid:1) (cid:104) f | U † i(cid:15)E i − H + i(cid:15) U | i (cid:105) = (cid:0) E i − E f (cid:1)(cid:18) (cid:104) f | U † | i (cid:105) + (cid:104) f | U † i(cid:15)E i − H + i(cid:15) δU | i (cid:105) (cid:19) = (cid:0) E i − E f (cid:1)(cid:20) (cid:104) f | U † | i (cid:105) + (cid:104) f | δU † i(cid:15)E i − H + i(cid:15) δU | i (cid:105) (cid:21) + i(cid:15) (cid:104) f | δU | i (cid:105) . (365)since the term in the rectangular bracket of Eq. (365) is byassumption non-singular at E i = E f . To see what remains from the second term in Eq. (363) on the energy-shell wenote that i(cid:15)E i − H + i(cid:15) U | i (cid:105) = | i (cid:105) + i(cid:15)E i − H + i(cid:15) δU | i (cid:105) (366)The contribution of the second term of Eq. (366) to thematrix element in Eq. (363) is non-singular at E i = E f since it hits at least once the interaction V from the left.So due to i(cid:15) in front of it this term vanishes. On the otherhand (cid:0) E i − E f (cid:1) (cid:104) f | U † E i − H + i(cid:15) V = −(cid:104) f | V + (cid:0) E i − E f (cid:1) (cid:104) f | δU † E i − H + i(cid:15) V. (367)Only the first term in Eq. (367) survives on the mass-shellsuch that we get the T-matrix equivalence (cid:104) f | T U ( E i ) | i (cid:105) = (cid:104) f | T ( E i ) | i (cid:105) + O ( E i − E f ) . (368)A more elegant proof where the authors use the Mølleroperator can be found in [15]. G Folded-diagram technique II
In this appendix, we would like to discuss the folded-diagram technique introduced by Kuo et al. in the shell-model calculations, see [146] for comprehensive introduc-tion. In particular, we show here the transfer matrix equiv-alence formulation of this technique presented in [147]. Westart with Lippmann-Schwinger equation for half-off-shellT-matrix given by T ( E i ) = V + V ˜ G + ( E i ) T ( E i ) , (369)where E i denotes the energy of initial state and the freeGreen function and its Fourier transform are defined by G + ( t ) = − i θ ( t ) e − i ( H − i (cid:15) ) t , ˜ G + ( E i ) = (cid:90) dt e i E i t G + ( t ) = 1 E i − H + i (cid:15) . (370)We can rewrite the transfer-matrix into T ( E i ) = Q ( E i ) + Q ( E i ) ˜ G η + ( E i ) T ( E i ) , (371)where Q is an energy-dependent potential which satisfies Q ( E i ) = V + ηV ˜ G λ + ( E i ) Q ( E i ) , (372)and ˜ G η + ( E ) = ˜ G + ( E ) η, ˜ G λ + ( E ) = ˜ G + ( E ) λ. (373)This result can be directly derived by rewriting Eq. (369)via (cid:2) − V ˜ G λ + ( E i ) (cid:3) T ( E i ) = V + V ˜ G η + ( E i ) T ( E i ) . (374) Multiplying both sides of Eq. (374) with (cid:2) − V ˜ G λ + ( E i ) (cid:3) − we get Eq. (371) with energy-dependent potential given by Q ( E i ) = (cid:2) − V ˜ G λ + ( E i ) (cid:3) − V. (375)Using Eq. (370) we can rewrite the effective potential into Q ( E i ) = (cid:2) − V λ ( E i − H ) − (cid:3) − V = ( E i − H − V λ + V λ ) 1 E i − H − V λ V = V + V E i − λHλ V. (376)We project Eq. (371) to the model space and get ηT ( E i ) η = Q η ( E i ) + Q η ( E i ) ˜ G η + ( E i ) ηT ( E i ) η , (377)with Q η ( E i ) = ηQ ( E i ) η. (378)From now on we will work only with the model projectedpotential Q η ( E i ) rather than with full Q ( E i ). To abbre-viate the notation we make a replacement in the notation Q η → Q using from now on Q ( E i ) = ηV η + ηV E i − λHλ V η. (379)The effective potential Q ( E i ) is energy-dependent and isdifficult to deal with in practical calculation with A >
2. For this reason, we can define an energy-independentpotential by using the folded-diagram technique. For thispurpose, we define higher Q -boxes by Q n ( E , . . . , E n +1 ) = n +1 (cid:88) k =1 C k ( E , . . . , E n +1 ) Q ( E k ) , (380)with C k ( E , . . . , E n +1 ) = k − (cid:89) i =1 E k − E i n +1 (cid:89) j = k +1 E k − E j . (381)The higher Q -boxed can be rewritten into a simpler formof Eq. (360) by using the partial fraction decomposition n +1 (cid:88) k =1 C k ( E , . . . , E n +1 ) 1 E k − λHλ = ( − n n +1 (cid:89) k =1 E k − λHλ , (382)as well as n +1 (cid:88) k =1 C k ( E , . . . , E n +1 ) = 0 . (383)To define a folded-diagram we follow [147] and applythe T-matrix equivalence approach. We start with the firstiteration of the energy-dependent potential and replace ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 45 the energy E i in the Q -box by the energy of the state onwhich the Q -box operates. To compensate this change weneed to add to the changed expression a folded-diagram (cid:104) f | Q ( E i ) ˜ G η + ( E i ) Q ( E i ) | i (cid:105) = (384) (cid:88) α (cid:104) f | Q ( E α ) | α (cid:105)(cid:104) α | ˜ G η + ( E i ) Q ( E i ) | i (cid:105) − (cid:104) f | Q (cid:48) (cid:90) Q | i (cid:105) . Eq. (384) defines a folded-diagram by a difference betweenthe original and the modified iterations. Having defined afolded-diagram we can express it in terms of the higher Q -boxes: (cid:104) f | Q (cid:48) (cid:90) Q | i (cid:105) = (cid:88) α (cid:104) f | (cid:0) Q ( E α ) − Q ( E i ) (cid:1) | α (cid:105)× (cid:104) α | ˜ G η + ( E i ) Q ( E i ) | i (cid:105) = (cid:88) α (cid:104) f | (cid:0) Q ( E α ) − Q ( E i ) (cid:1) | α (cid:105) E i − E α (cid:104) α | Q ( E i ) | i (cid:105) = − (cid:88) α (cid:104) f | Q ( E i , E α ) | α (cid:105)(cid:104) α | Q ( E i ) | i (cid:105) , (385)where in the last step we used the definition of the higher Q -boxes given in Eqs. (380) and (381). As a direct con-sequence of Eq. (360) there are no pure nucleon cuts ina folded-diagram. If we would define an effective energy-independent potential by (cid:104) α | V eff | β (cid:105) = (cid:104) α | Q ( E β ) | β (cid:105) − (cid:104) α | Q (cid:48) (cid:90) Q | β (cid:105) , (386)the sum of the zeroth and the first iteration of this poten-tial would reproduce the original result for the T-matrix(up to the two Q -boxes approximation). This idea can begeneralized to any number of iteration. Let us demonstratethis for two iterations. We define a twice-folded-diagram (cid:104) f | Q (cid:48) (cid:82) Q (cid:82) Q | i (cid:105) by (cid:104) f | Q ( E i ) ˜ G η + ( E i ) Q ( E i ) ˜ G η + ( E i ) Q ( E i ) | i (cid:105) = (cid:88) α,β (cid:104) f | Q ( E α ) | α (cid:105)(cid:104) α | ˜ G η + ( E i ) Q ( E β ) | β (cid:105)(cid:104) β | ˜ G η + ( E i ) Q ( E i ) | i (cid:105)− (cid:88) β (cid:104) f | Q (cid:48) (cid:90) Q | β (cid:105)(cid:104) β | ˜ G η + ( E i ) Q ( E i ) | i (cid:105)− (cid:88) α,β (cid:104) f | Q ( E α ) | α (cid:105)(cid:104) α | ˜ G η + ( E i ) | β (cid:105)(cid:104) β | Q (cid:48) (cid:90) Q | i (cid:105) + (cid:104) f | Q (cid:48) (cid:90) Q (cid:90) Q | i (cid:105) = (cid:88) α,β (cid:104) f | Q ( E α ) | α (cid:105) (cid:104) α | Q ( E β ) | β (cid:105) E i − E α (cid:104) β | Q ( E i ) | i (cid:105) E i − E β + (cid:88) α,β (cid:104) f | Q ( E β , E α ) | α (cid:105)(cid:104) α | Q ( E β ) | β (cid:105) (cid:104) β | Q ( E i ) | i (cid:105) E i − E β + (cid:88) α,β (cid:104) f | Q ( E α ) | α (cid:105) E i − E α (cid:104) α | Q ( E i , E β ) | β (cid:105)(cid:104) β | Q ( E i ) | i (cid:105) + (cid:104) f | Q (cid:48) (cid:90) Q (cid:90) Q | i (cid:105) . (387) For twice-folded-diagram we get (cid:104) f | Q (cid:48) (cid:90) Q (cid:90) Q | i (cid:105) = (cid:88) α,β (cid:20) (cid:104) f | Q ( E i ) | α (cid:105) E i − E α (cid:104) α | Q ( E i ) | β (cid:105) E i − E β − (cid:104) f | Q ( E α ) | α (cid:105) E i − E α (cid:104) α | Q ( E β ) | β (cid:105) E i − E β − (cid:104) f | Q ( E β , E α ) | α (cid:105)× (cid:104) α | Q ( E β ) | β (cid:105) E i − E β − (cid:104) f | Q ( E α ) | α (cid:105) E i − E α (cid:104) α | Q ( E i , E β ) | β (cid:105) (cid:21) × (cid:104) β | Q ( E i ) | i (cid:105) = (cid:88) α,β (cid:20)(cid:18) (cid:104) f | Q ( E i ) | α (cid:105) E i − E α − (cid:104) f | Q ( E α ) | α (cid:105) E i − E α − (cid:104) f | Q ( E β , E α ) | α (cid:105) (cid:19) (cid:104) α | Q ( E β ) | β (cid:105) E i − E β + (cid:104) f | Q ( E i , E α ) | α (cid:105)× (cid:104) α | Q ( E i , E β ) | β (cid:105) (cid:21) (cid:104) β | Q ( E i ) | i (cid:105) = (cid:88) α,β (cid:20)(cid:18) (cid:104) f | Q ( E i ) | α (cid:105) ( E i − E α )( E i − E β ) + (cid:104) f | Q ( E β ) | α (cid:105) ( E β − E i )( E β − E α )+ (cid:104) f | Q ( E α ) | α (cid:105) ( E α − E i )( E α − E β ) (cid:19) (cid:104) α | Q ( E β ) | β (cid:105) + (cid:104) f | Q ( E i , E α ) | α (cid:105)(cid:104) α | Q ( E i , E β ) | β (cid:105) (cid:21) (cid:104) β | Q ( E i ) | i (cid:105) = (cid:88) α,β (cid:20) (cid:104) f | Q ( E i , E α , E β ) | α (cid:105)(cid:104) α | Q ( E β ) | β (cid:105) + (cid:104) f | Q ( E i , E α ) | α (cid:105)(cid:104) α | Q ( E i , E β ) | β (cid:105) (cid:21) (cid:104) β | Q ( E i ) | i (cid:105) . (388)For the effective potential we get (cid:104) α | V eff | β (cid:105) = (cid:104) α | Q ( E β ) | β (cid:105) − (cid:104) α | Q (cid:48) (cid:90) Q | β (cid:105) + (cid:104) f | Q (cid:48) (cid:90) Q (cid:90) Q | i (cid:105) . (389)To reproduce higher number of iterations one should in-clude more and more foldings into the effective potential.Up to the three Q -boxes one gets (cid:104) α | V eff | β (cid:105) = (cid:104) α | Q ( E β ) | β (cid:105) + (cid:104) α | Q ( E β , E γ ) | γ (cid:105)(cid:104) γ | Q ( E β ) | β (cid:105) + (cid:104) α | Q ( E β , E δ , E γ ) | γ (cid:105)(cid:104) γ | Q ( E δ ) | δ (cid:105)(cid:104) δ | Q ( E β ) | β (cid:105) + (cid:104) α | Q ( E β , E γ ) | γ (cid:105)(cid:104) γ | Q ( E β , E δ ) | δ (cid:105)(cid:104) δ | Q ( E β ) | β (cid:105) . (390)From three and a higher number of iterations one canderive in a similar way further corrections to the energy-independent potential. H Unitary equivalence of S-matrix in this appendix we want to clarify under which conditionthe scattering matrix for original and transformed time-dependent interactions remains the same? Assume that we have a time-dependent Hamiltonian H ( t ) = H + V ( t )and a state | Ψ ( t ) (cid:105) which satisfies the Schr¨odinger equation i ∂∂t | Ψ ( t ) (cid:105) = H ( t ) | Ψ ( t ) (cid:105) . (391)Here H denotes a free Hamiltonian . Let | i ( t ) (cid:105) and | f ( t ) (cid:105) be stationary eigenstates of the free Hamiltonian i ∂∂t | α ( t ) (cid:105) = H | α ( t ) (cid:105) , α = i, f, (392)which can be written in the form | i ( t ) (cid:105) = e − i E i t | i (cid:105) , | f ( t ) (cid:105) = e − i E f t | i (cid:105) (393)Let U ( t ) = 1 + δU ( t ) be a time-dependent unitary trans-formation with existing Fourier transform for δU ( t ). Thismeans, in particular, that δU ( t ) decreases fast enoughwith increasing | t | . Then a transformed state | Ψ U ( t ) (cid:105) = U † ( t ) | Ψ ( t ) (cid:105) (394)satisfies the Schr¨odinger equation i ∂∂t | Ψ U ( t ) (cid:105) = H U ( t ) | Ψ U ( t ) (cid:105) , (395)with H U ( t ) = U † ( t ) H ( t ) U ( t ) + i ˙ U † ( t ) U ( t ) , (396)where dot on top of the letter denotes a time-derivative˙ U † = ∂∂t U † ( t ) . (397)We show now that the scattering matrices for both Hamil-tonians H ( t ) and H U ( t ) are the same. To give the scat-tering matrix in terms of Hamiltonians and keep notationshort we change the notation for a free Green-function(compared to Appendix G, see Eq. (370)) which we de-note now by G (+)0 ( t ). The free Green-function satisfies (cid:18) i ∂∂t − H (cid:19) G (+)0 ( t ) = δ ( t ) . (398)The formal solution of this equation is given by G (+)0 ( t ) = − i θ ( t ) e − iH t , (399)where θ ( t ) is a step function θ ( t >
0) = 1 , θ ( t <
0) = 0 . (400)The state | Ψ (+) ( t ) (cid:105) which satisfies | Ψ (+) ( t ) (cid:105) = | i ( t ) (cid:105) + (cid:90) ∞−∞ dt (cid:48) G (+)0 ( t − t (cid:48) ) V ( t (cid:48) ) | Ψ (+) ( t (cid:48) ) (cid:105) , (401) It could be also any time-independent Hamiltonian whiche.g. can have bound states as eigenstates. In our case this willbe kinetic energy of nucleons and pions. satisfies also the original Schr¨odinger equation (391). The i(cid:15) -prescription chosen in Eq. (399) is chosen to make theintegrand of Eq. (401) vanish for t (cid:48) approaching minusinfinity. The step function in Eq. (401) causes the state | Ψ (+) ( t ) (cid:105) for t → −∞ to become a free incoming asymp-totic state | Ψ (+) ( −∞ ) (cid:105) = | i ( −∞ ) (cid:105) . (402)The scattering matrix for the Hamiltonian H ( t ) is givenby [145] (cid:104) f | S | i (cid:105) = (cid:104) f | i (cid:105) − i (cid:104) f | T | i (cid:105) , (403)where the on-shell T-matrix is defined by (cid:104) f | T | i (cid:105) = lim t →∞ i (cid:90) ∞−∞ dt (cid:48) (cid:104) f ( t ) | G (+)0 ( t − t (cid:48) ) V ( t (cid:48) ) | Ψ (+) ( t (cid:48) ) (cid:105) . (404)Note that in this formulation energy-conservation deltafunction is not yet extracted out of the T-matrix. A fullGreen-function is defined via a differential equation (cid:18) i ∂∂t − H ( t ) (cid:19) G (+) ( t, t (cid:48) ) = δ ( t − t (cid:48) ) ,G (+) ( t, t (cid:48) ) (cid:18) − i ←− ∂∂t (cid:48) − H ( t (cid:48) ) (cid:19) = δ ( t − t (cid:48) ) . (405)which is equivalent to an iterative solution of the integralequation G (+) ( t, t (cid:48) ) = G (+)0 ( t − t (cid:48) )+ (cid:90) dt G (+)0 ( t − t ) V ( t ) G (+) ( t , t (cid:48) ) . (406)Indeed applying i∂/∂t − H ( t ) on both sides of Eq. (406)we get (cid:18) i ∂∂t − H ( t ) (cid:19) G (+) ( t, t (cid:48) ) = δ ( t − t (cid:48) ) − V ( t ) G (+)0 ( t − t (cid:48) )+ (cid:90) dt (cid:18) i ∂∂t − H ( t ) (cid:19) G (+)0 ( t − t ) V ( t ) G (+) ( t , t (cid:48) )= δ ( t − t (cid:48) ) + V ( t ) (cid:18) G (+) ( t, t (cid:48) ) − G (+)0 ( t − t (cid:48) ) − (cid:90) dt G (+)0 ( t − t ) V ( t ) G (+) ( t , t (cid:48) ) (cid:19) = δ ( t − t (cid:48) ) . (407)If we know the full Green function we know the solutionof the Schr¨odinger equation | Ψ (+) ( t ) (cid:105) = | i ( t ) (cid:105) + (cid:90) ∞−∞ dt (cid:48) G (+) ( t, t (cid:48) ) V ( t (cid:48) ) | i ( t (cid:48) ) (cid:105) , (408)Using Eq. (408) and (cid:104) f ( t ) | G (+)0 ( t − t (cid:48) ) = (cid:104) f ( t (cid:48) ) | (409) ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 47 we can rewrite the on-shell T-matrix to (cid:104) f | T | i (cid:105) = i (cid:90) ∞−∞ dt (cid:104) f ( t ) | V ( t ) | i ( t ) (cid:105) + i (cid:90) ∞−∞ dt dt (cid:104) f ( t ) | V ( t ) G (+) ( t , t ) V ( t ) | i ( t ) (cid:105) . (410)We write now the step function as θ ( t ) = lim (cid:15) → (cid:90) dω πi e iωt ω − i(cid:15) . (411)0+ means here that (cid:15) > (cid:15) > (cid:15) > ∂∂t θ ( t ) = (cid:90) dω πi i ω − i(cid:15) + i(cid:15)ω − i(cid:15) e iωt = δ ( t ) − (cid:15)θ ( t ) . (412)In the case of a free Green function, this leads to (cid:18) i ∂∂t − H + i(cid:15) (cid:19) G (+)0 ( t ) = δ ( t ) . (413)For the full Green function, we get a similar relation (cid:18) i ∂∂t − H ( t ) + i(cid:15) (cid:19) G (+) ( t, t (cid:48) ) = δ ( t − t (cid:48) ) . (414)We can explicitly see this if we apply the operator i∂/∂t − H ( t ) + i(cid:15) to Eq. (406) (cid:18) i ∂∂t − H ( t ) + i(cid:15) (cid:19) G (+) ( t, t (cid:48) ) = δ ( t − t (cid:48) )+ V ( t ) (cid:18) G (+) ( t, t (cid:48) ) − G (+)0 ( t − t (cid:48) ) − (cid:90) dt G (+)0 ( t − t ) V ( t ) G (+) ( t , t (cid:48) ) (cid:19) = δ ( t − t (cid:48) ) . (415)We use now V ( t ) = H ( t ) − H and rewrite (cid:90) dt G (+) ( t , t ) V ( t ) | i ( t ) (cid:105) = (cid:90) dt G (+) ( t , t ) (cid:0) H ( t ) − H (cid:1) | i ( t ) (cid:105) = (cid:90) dt G (+) ( t , t ) (cid:18) H ( t ) − i ∂∂t (cid:19) | i ( t ) (cid:105) = (cid:90) dt G (+) ( t , t ) (cid:18) H ( t ) + i ←− ∂∂t − i(cid:15) + i(cid:15) (cid:19) | i ( t ) (cid:105) = − (cid:90) dt δ ( t − t ) | i ( t ) (cid:105) + i(cid:15) (cid:90) dt G (+) ( t , t ) | i ( t ) (cid:105) = −| i ( t ) (cid:105) + i(cid:15) (cid:90) dt G (+) ( t , t ) | i ( t ) (cid:105) . (416) Using Eq. (416) we can rewrite T-matrix of Eq. (410) into (cid:104) f | T | i (cid:105) = i (cid:90) ∞−∞ dt dt (cid:104) f ( t ) | V ( t ) i(cid:15)G (+) ( t , t ) | i ( t ) (cid:105) . (417)With the same steps as in Eq. (416) we get (cid:90) dt (cid:104) f ( t ) | V ( t ) G (+) ( t , t )= −(cid:104) f ( t ) | + i(cid:15) (cid:90) dt (cid:104) f ( t ) | G (+) ( t , t ) . (418)Using Eq. (418) we can rewrite the T-matrix into (cid:104) f | T | i (cid:105) = (cid:15) (cid:90) ∞−∞ dt (cid:104) f ( t ) | i ( t ) (cid:105)− i(cid:15) (cid:90) ∞−∞ dt dt (cid:104) f ( t ) | G (+) ( t , t ) | i ( t ) (cid:105) = − i(cid:15) (cid:90) ∞−∞ dt dt (cid:104) f ( t ) | G (+) ( t , t ) | i ( t ) (cid:105) . (419)In the last step of Eq. (419) we used the fact that (cid:15) (cid:90) ∞−∞ dt (cid:104) f ( t ) | i ( t ) (cid:105) = (cid:15) πδ ( E f − E i ) (cid:104) f | i (cid:105) , (420)which vanishes in the limit (cid:15) → (cid:104) f | T | i (cid:105) = − i(cid:15) (cid:90) ∞−∞ dt dt (cid:104) f ( t ) | U ( t ) × U † ( t ) G (+) ( t , t ) U ( t ) U † ( t ) | i ( t ) (cid:105) . (421)We denote the transformed Green function by G (+) U ( t, t (cid:48) ) = U † ( t ) G (+) ( t, t (cid:48) ) U ( t (cid:48) ) , (422)which satisfies (cid:18) i ∂∂t − H U ( t ) (cid:19) G (+) U ( t, t (cid:48) ) = δ ( t − t (cid:48) ) ,G (+) U ( t, t (cid:48) ) (cid:18) − i ←− ∂∂t (cid:48) − H U ( t (cid:48) ) (cid:19) = δ ( t − t (cid:48) ) . (423)Eq. (423) follows directly from multiplying Eq. (407) with U † ( t ) and U ( t (cid:48) ) from left and right, respectively. Due toEq. (423) we also have G (+) U ( t, t (cid:48) ) = G (+)0 ( t − t (cid:48) ) + (cid:90) dt G (+)0 ( t − t ) V U ( t ) G (+) U ( t , t (cid:48) ) = G (+)0 ( t − t (cid:48) ) + (cid:90) dt G (+) U ( t, t ) V U ( t ) G (+)0 ( t − t (cid:48) ) , (424)where V U ( t ) = H U ( t ) − H . (425) We see that the difference between G (+) and G (+) U is in theHamiltonian. G (+) and G (+) U are full Green-functions withHamiltonians H ( t ) and H U ( t ), respectively. The originalT-matrix expressed in terms of G (+) U is given by (cid:104) f | T | i (cid:105) = − i(cid:15) (cid:90) ∞−∞ dt dt (cid:104) f ( t ) | U ( t ) G (+) U ( t , t ) × U † ( t ) | i ( t ) (cid:105) . (426)On the other hand the T-matrix with the transformedHamiltonian H U is given by (cid:104) f | T U | i (cid:105) = − i(cid:15) (cid:90) ∞−∞ dt dt (cid:104) f ( t ) | G (+) U ( t , t ) | i ( t ) (cid:105) . (427)We want to show that in the limit (cid:15) →
0+ the two T-matrices are equal:lim (cid:15) → (cid:104) f | T | i (cid:105) = lim (cid:15) → (cid:104) f | T U | i (cid:105) , (428)if the unitary transformation satisfieslim (cid:15) → (cid:15) (cid:90) dt G (+)0 ( t − t ) δU † ( t ) | i ( t ) (cid:105) = 0 , lim (cid:15) → (cid:15) (cid:90) dt (cid:104) f ( t ) | δU ( t ) G (+)0 ( t − t ) = 0 . (429)To do this we use Eq. (424) to rewrite the T-matrix into (cid:104) f | T | i (cid:105) = − i(cid:15) (cid:90) ∞−∞ dt dt (cid:104) f ( t ) | U ( t ) (cid:20) G (+)0 ( t − t )+ (cid:90) dt G (+)0 ( t − t ) V U ( t ) G (+)0 ( t − t )+ (cid:90) dt dt G (+)0 ( t − t ) V U ( t ) G (+) U ( t , t ) V U ( t ) G (+)0 ( t − t ) (cid:21) U † ( t ) | i ( t ) (cid:105) . (430)A time integration over the time t gives (cid:90) dt G (+)0 ( t , t ) | i ( t ) (cid:105) = (cid:90) dt θ ( t − t ) × e − i ( H − i(cid:15) )( t − t ) e − iE i t | i (cid:105) = | i (cid:105) e − iE i t × (cid:90) t −∞ dt e − (cid:15) ( t − t ) = | i ( t ) (cid:105) (cid:15) , (431)Similarly, for the integration over t we get (cid:90) dt (cid:104) f ( t ) | G (+)0 ( t − t ) = (cid:104) f ( t ) | (cid:15) . (432)Due to condition in Eq. (429) all contributions propor-tional to δU ( t ) do not generate poles in small (cid:15) and forthis reason vanish in the limit (cid:15) → More elegant proof for time-independent unitary transfor-mations can be found in [15]. Note that we did not considerhere nonperturbative effects but concentrated only on pertur-bation theory. More rigorous proof for time-independent uni-tary transformations can be found in [31].
I Transfer matrix with time-dependentinteraction
In this appendix we derive a transfer matrix for time-dependent interaction. We start with the Schr¨odinger equa-tion (cid:18) i ∂∂t − H (cid:19) | Ψ ( t ) (cid:105) = V ( t ) | Ψ ( t ) (cid:105) . (433)To keep the notation short we again change the notationfor a free retarded Green function which satisfies (cid:18) i ∂∂t − H (cid:19) G + ( t − t (cid:48) ) = δ ( t − t (cid:48) ) (434)with a constraint G + ( t − t (cid:48) ) = 0 , for t < t (cid:48) . (435)The solution is given by G + ( t − t (cid:48) ) = − i θ ( t − t (cid:48) ) e − i ( H − i (cid:15) ) ( t − t (cid:48) ) . (436)The solution of the Schr¨oding equation can be written as | Ψ + ( t ) (cid:105) = | φ ( t ) (cid:105) + (cid:90) ∞−∞ dt (cid:48) G + ( t − t (cid:48) ) V ( t (cid:48) ) | Ψ + ( t (cid:48) ) (cid:105) , (437)with the state | φ ( t ) (cid:105) satisfying a free Schr¨odinger equation (cid:18) i ∂∂t − H (cid:19) | φ ( t ) (cid:105) = 0 . (438)We take now a Fourier transform of Eq. (437) by multi-plying both sides by e iEt and integrating over time: | ˜ Ψ + ( E ) (cid:105) = | ˜ φ ( E ) (cid:105) + (cid:90) ∞−∞ dt dt (cid:48) e i Et G + ( t − t (cid:48) ) V ( t (cid:48) ) | Ψ + ( t (cid:48) ) (cid:105) . (439)For the Fourier transform we use | ˜ Ψ + ( E ) (cid:105) = (cid:90) ∞−∞ dt e iEt | Ψ + ( t ) (cid:105) , | ˜ φ ( E ) (cid:105) = (cid:90) ∞−∞ dt e iEt | φ ( t ) (cid:105) . (440)To simplify Eq. (439) we Fourier-transform the free Green-function G + ˜ G + ( E ) = (cid:90) ∞−∞ dt e iEt G + ( t ) = − i (cid:90) ∞ dt e i ( E − H + i (cid:15) ) t = 1 E − H + i (cid:15) . (441)The backward Fourier transformations are given by G + ( t ) = (cid:90) dE π e − i Et ˜ G + ( E ) , | Ψ + ( t ) (cid:105) = (cid:90) dE π e − i Et | ˜ Ψ + ( E ) (cid:105) . (442) ermann Krebs: Nuclear Currents in Chiral Effective Field Theory 49 Using Eq. (442) we can rewrite Eq. (439) into | ˜ Ψ + ( E ) (cid:105) = | ˜ φ ( E ) (cid:105) + ˜ G + ( E ) (cid:90) dE (cid:48) π ˜ V ( E − E (cid:48) ) | ˜ Ψ + ( E (cid:48) ) (cid:105) (443)= | ˜ φ ( E ) (cid:105) + ˜ G + ( E ) (cid:90) dE (cid:48) π T ( E, E (cid:48) ) | φ ( E (cid:48) ) (cid:105) . (444)Eq. (444) defines the transition matrix in the presence ofan external source. It satisfies an integral equation T ( E, E (cid:48) ) = ˜ V ( E − E (cid:48) )+ (cid:90) dE (cid:48)(cid:48) π ˜ V ( E − E (cid:48)(cid:48) ) ˜ G + ( E (cid:48)(cid:48) ) T ( E (cid:48)(cid:48) , E (cid:48) ) . (445)which is also equivalent to T ( E, E (cid:48) ) = ˜ V ( E − E (cid:48) )+ (cid:90) dE (cid:48)(cid:48) π T ( E, E (cid:48)(cid:48) ) ˜ G + ( E (cid:48)(cid:48) ) ˜ V ( E (cid:48)(cid:48) − E (cid:48) ) . (446)Rewriting Eq. (446) into˜ V ( E − E (cid:48) ) = (cid:90) dE (cid:48)(cid:48) π T ( E, E (cid:48)(cid:48) ) (cid:18) πδ ( E (cid:48)(cid:48) − E (cid:48) ) − ˜ G + ( E (cid:48)(cid:48) ) ˜ V ( E (cid:48)(cid:48) − E (cid:48) ) (cid:19) , (447)and replacing ˜ V ( E − E (cid:48) ) in Eq. (443) by the left-handside of Eq. (447) we get | ˜ Ψ + ( E ) (cid:105) = | ˜ φ ( E ) (cid:105) + ˜ G + ( E ) (cid:90) dE (cid:48) π (cid:90) dE (cid:48)(cid:48) π T ( E, E (cid:48)(cid:48) ) × (cid:18) πδ ( E (cid:48)(cid:48) − E (cid:48) ) − ˜ G + ( E (cid:48)(cid:48) ) ˜ V ( E (cid:48)(cid:48) − E (cid:48) ) (cid:19) | ˜ Ψ + ( E (cid:48) ) (cid:105) . Using | ˜ φ ( E (cid:48)(cid:48) ) (cid:105) = (cid:90) dE (cid:48) π (cid:18) πδ ( E (cid:48)(cid:48) − E (cid:48) ) − ˜ G + ( E (cid:48)(cid:48) ) ˜ V ( E (cid:48)(cid:48) − E (cid:48) ) (cid:19) | ˜ Ψ + ( E (cid:48) ) (cid:105) , (448)we indeed get Eq. (444).We decompose now the interaction into the time-dependentand the time-independent parts V ( E − E (cid:48) ) = 2 πδ ( E − E (cid:48) ) v + v ( E − E (cid:48) ) , (449)where v denotes the time-independent nuclear force and v an axial vector source dependent interaction which van-ishes when the axial-vector source is switched off. Simi-larly, we can decompose the transition matrix T ( E, E (cid:48) ) = 2 πδ ( E − E (cid:48) ) t ( E ) + t ( E, E (cid:48) ) . (450)The off-shell transfer matrix t ( E ) satisfies the Lippmann-Schwinger equation t ( E ) = v + v ˜ G + ( E ) t ( E ) . (451) t transfer matrix satisfies t ( E, E (cid:48) ) = v ( E − E (cid:48) ) + v ˜ G + ( E ) t ( E, E (cid:48) )+ v ( E − E (cid:48) ) ˜ G + ( E (cid:48) ) t ( E (cid:48) )+ (cid:90) dE (cid:48)(cid:48) π v ( E − E (cid:48)(cid:48) ) ˜ G + ( E (cid:48)(cid:48) ) t ( E (cid:48)(cid:48) , E (cid:48) ) . (452)The last term contributes only to the processes with atleast two external sources, so we neglect this term here.If we only concentrate on one external source coupling weget (cid:0) − v ˜ G + ( E ) (cid:1) t ( E, E (cid:48) ) = v ( E − E (cid:48) ) × (cid:0) G + ( E (cid:48) ) t ( E (cid:48) ) (cid:1) = v ( E − E (cid:48) ) × (cid:0) G + ( E (cid:48) ) v (cid:0) − ˜ G + ( E (cid:48) ) v (cid:1) − (cid:1) = v ( E − E (cid:48) ) × (cid:18) − ˜ G + ( E (cid:48) ) v + ˜ G + ( E (cid:48) ) v (cid:19)(cid:0) − ˜ G + ( E (cid:48) ) v (cid:1) − = v ( E − E (cid:48) ) (cid:0) − ˜ G + ( E (cid:48) ) v (cid:1) − (453)So we get t ( E, E (cid:48) ) = (cid:0) − v ˜ G + ( E ) (cid:1) − v ( E − E (cid:48) ) (cid:0) − ˜ G + ( E (cid:48) ) v (cid:1) − . (454)We see that all energies which appear on the left-hand sideof v are the final state energies and all energies whichappear on the right-hand side of v are the initial stateenergies. References
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