The Triton Charge Radius to Next-to-next-to-leading order in Pionless Effective Field Theory
TThe Triton Charge Radius to Next-to-next-to-leading order inPionless Effective Field Theory
Jared Vanasse
1, 2, ∗ Department of Physics, Duke University, Durham, NC 27708, USA Department of Physics and Astronomy Ohio University, Athens OH 45701, USA (Dated: June 9, 2017)
Abstract
The triton point charge radius is calculated to next-to-next-to-leading order (NNLO) in pionlesseffective field theory (EFT( /π )), yielding a prediction of 1 . ± .
19 fm (leading order), 1 . ± .
08 fm(next-to leading order), and 1 . ± .
03 fm (NNLO) in agreement with the current experimentalextraction of 1 . ± .
040 fm [1]. The error at NNLO is due to cutoff variation ( ∼ /π ) error estimate ( ∼ /π ). ∗ Electronic address: [email protected]; Electronic address: [email protected] a r X i v : . [ nu c l - t h ] J un . INTRODUCTION If a system is probed at length scales, (cid:96) , larger than the range of the underlying interac-tion, r , then its interactions can be expanded in a series of contact interactions known asshort-range effective field theory (EFT) [2], and its applicability to any system for which (cid:96) > r is known as universality [3]. Short range EFT has been used in cold atom systems,halo nuclei using halo EFT, and for low-energy few-body nuclear systems using pionless EFT(EFT( /π )). For all of these systems the scattering length, a , is unnaturally large ( a (cid:29) r ). Thus at leading order (LO) the scattering length contribution is treated nonperturbatively,and higher order range corrections ([ r/a ] n ) are added perturbatively [4, 5].Nucleon-nucleon ( NN ) interactions are dominated by one pion exchange at large lengthscales. Thus for length scales (cid:96) > /m π (or energies E < m π /M N ) NN interactions canbe expanded in a short-range EFT known as EFT( /π ). The series of contact interactionsin EFT( /π ) can be written down as a Lagrangian of nucleon terms and possible externalcurrents. These terms are ordered by the power counting of EFT( /π ) [2, 4, 5] which hasthe expansion (1 / ( M N Q ))( Q/ Λ /π ) n , where ( Q/ Λ /π ) ∼ /
3, Λ /π ∼ m π , Q ∼ γ t , n ≥
0, and γ t ≈
45 MeV is the deuteron binding momentum. In addition to making EFT( /π ) tractable(one only needs a finite number of terms to a given order) the power counting also allowsfor an estimation of the error in calculations.LO EFT( /π ) has two low energy constants (LECs) in the two-body sector fit to the S and S bound and virtual bound state poles respectively, and one three-body LEC fit to athree-body datum. At next-to-leading order (NLO) there are two more LECs in the two-body sector fit to the effective ranges in the S and S channels. Next-to-next-to-leadingorder (NNLO) has a two-body LEC parametrizing the mixing between the NN S and D channels and an energy dependent three-body LEC [21]. Thus to NNLO in EFT( /π ) two-and three-body systems are characterized by seven LECs and predict observables to roughly6% accuracy. However, certain observables, such as the neutron-deuteron ( nd ) polarization Note, for nuclear systems the scattering length is fixed, but for cold atom systems the scattering lengthcan be made large by tuning a magnetic field near a Feshbach resonance. In the two-body sector the factor of 1 / ( M N Q ) only occurs for two-body resonant S -wave interactions,which are a leading contribution in the three-body sector. However, for higher two-body partial wavesthe factor of 1 / ( M N Q ) will not occur and n ≥ /π ). A y , are sensitive to higher order interactions and are three orders of magnitudesmaller than experiment at NNLO, which is the first order at which A y is non-zero. The A y observable is sensitive to two-body P -wave contact interactions that occur at N LO [6].EFT( /π ) (see e.g. Ref. [7] for a review) has been used with great success in the two-bodysector calculating deuteron electromagnetic form factors [8, 9], NN scattering [8, 10, 11],neutron-proton ( np ) capture [8, 9, 12] to ( < ∼ nd scattering [6, 18–23], pd scattering [24–27], nd capture [28, 29], andthe energy difference between H and He [25, 30, 31]. Previous three-body calculations of nd scattering in EFT( /π ) made use of the partial resummation technique [21]. This methodhas the advantage of being able to calculate diagrams that contain full off-shell scatteringamplitudes without needing to calculate the full off-shell scattering amplitude. However,this method suffers the drawback that it contains an infinite subset of higher order diagramsand although correct up to the order one is working is not strictly perturbative. This workwas improved upon in Ref. [23] where a new technique no more numerically complicatedthan the partial resummation technique but strictly perturbative was introduced. Thistechnique makes higher order strictly perturbative numerical calculations in nd scatteringmuch simpler [6]. However, this method initially suffered the drawback that it could notbe used to calculate perturbative corrections to three-body bound-state systems such as thetriton. This work corrects that drawback. Using the new perturbative method developedhere for bound states I will show that the triton charge radius has excellent agreement withexperiment at NNLO in EFT( /π ).Hagen et al. [32] calculated the point charge radius of halo nuclei to LO in halo EFTand introduced the concept of a trimer field to calculate vertex functions for bound-statecalculations. Building on that work a technique similar to Hagen et al. is introduced, but onethat can also calculate perturbative corrections to three-body bound states. This techniqueintroduces a triton auxiliary field and thus treats three-body forces in the doublet S -wavechannel differently, but analytically equivalent to previous approaches to all orders [23]. Inaddition it is shown how this technique improves the calculation of the LO three-body forceby removing the need for iterative numerical schemes. One can also now calculate the NNLOenergy dependent three-body force without the need for a numerical limiting procedure [33].The new technique also leads to slight numerical simplifications in the calculation of nd χ EFT) [35] potentials which give diffractionminima at the correct values of Q . From experimental data the triton point charge radiushas been extracted, most recently with a value of 1 . ± .
040 fm [1]. A NNLO EFT( /π )calculation of the triton point charge radius is accurate to roughly 1.5%. However, as I willshow cutoff variation gives an additional source of error leading to an overall error estimate of2%. This cutoff variation is either a signal of slow divergence or convergence. Either a carefulasymptotic analysis or a numerical calculation to higher cutoffs will be needed to answerthis unambiguously. However, reliable calculations to very large cutoffs (Λ > MeV) arecurrently unfeasible, due to numerical instabilities.This paper is organized as follows. In Sec. II properties of the two-body system in EFT( /π )necessary for three-body calculations are reviewed. Sec. III introduces new techniques for nd scattering, the connection between the auxiliary triton and non-auxiliary triton fieldapproach for three-body forces, and the calculation of perturbative corrections to the tritonvertex function. In Sec. IV it is shown how the triton auxiliary field is used to calculatethree-body forces in the doublet S -wave channel. Discussion of the calculation of the tritoncharge form factor to NNLO is given in Sec. V, results are shown in Sec. VI, and conclusionsgiven in Sec. VII. II. TWO-BODY SYSTEM
The two-body Lagrangian in EFT( /π ) is L = ˆ N † (cid:32) i∂ + (cid:126) ∇ M N (cid:33) ˆ N + ˆ t † i (cid:34) ∆ t − c t (cid:32) i∂ + (cid:126) ∇ M N + γ t M N (cid:33)(cid:35) ˆ t i (1)+ ˆ s † a (cid:34) ∆ s − c s (cid:32) i∂ + (cid:126) ∇ M N + γ s M N (cid:33)(cid:35) ˆ s a + y t (cid:104) ˆ t † i ˆ N T P i ˆ N + H . c . (cid:105) + y s (cid:104) ˆ s † a ˆ N T ¯ P a ˆ N + H . c . (cid:105) , where ˆ t i (ˆ s a ) is the spin-triplet iso-singlet (spin-singlet iso-triplet) dibaryon auxiliary field.The projector P i = √ σ σ i τ ( ¯ P a = √ τ τ a σ ) projects out the spin-triplet iso-singlet (spin-4inglet iso-triplet) combination of nucleons.At LO the bare deuteron propagator, i/ ∆ t , is dressed by the infinite sum of bubblediagrams in Fig. 1. The parameters are then fit to reproduce the deuteron pole at the physical (LO)(NLO) (NNLO) FIG. 1: The top equation shows the LO dressed spin-triplet dibaryon propagator, which can besolved analytically via a geometric series. The solid bar is the bare dibaryon propagator i/ ∆ t ,the single lines with arrows are nucleon propagators, the cross represents a NLO effective rangeinsertion from c (0)0 t , and the star a NNLO correction from c (1)0 t . position. At NLO the parameters are chosen to fix the deuteron pole at the same positionand give the correct residue about the deuteron pole. This parametrization is known as the Z -parametrization [22, 36] and is advantageous because it reproduces the correct residueabout the deuteron pole at NLO instead of being approached perturbatively order-by-orderas in the effective range expansion (ERE) parametrization. The same procedure is carriedout in the S channel except the virtual bound-state pole and its residue is fit to. Carryingout this procedure the coefficients are given by [22] y t = 4 πM N , ∆ t = γ t − µ, c ( n )0 t = ( − n ( Z t − n +1 M N γ t , (2) y s = 4 πM N , ∆ s = γ s − µ, c ( n )0 s = ( − n ( Z s − n +1 M N γ s , where γ t = 45 . Z t = 1 . γ s = − .
890 MeV is the S virtual bound-state momentum, and Z s = 0 . S pole [37]. The non-physical scale µ is introduced byusing dimensional regularization with the power divergence subtraction scheme [4, 5]. Allphysical observables are µ -independent.After fitting the coefficients, the spin-triplet and spin-singlet dibaryon propagators up to5nd including NNLO are given by iD NNLO { t,s } ( p , (cid:126) p ) = iγ { t,s } − (cid:113) (cid:126) p − M N p − i(cid:15) (3) × (cid:124)(cid:123)(cid:122)(cid:125) LO + Z { t,s } − γ { t,s } (cid:32) γ { t,s } + (cid:114) (cid:126) p − M N p − i(cid:15) (cid:33)(cid:124) (cid:123)(cid:122) (cid:125) NLO + (cid:18) Z { t,s } − γ { t,s } (cid:19) (cid:18) (cid:126) p − M N p − γ { t,s } (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) NNLO + · · · . The deuteron wavefunction renormalization is given by the residue about the deuteron poleof the spin-triplet dibaryon, which to NNLO yields Z D = 2 γ t M N (cid:124)(cid:123)(cid:122)(cid:125) LO + ( Z t − (cid:124) (cid:123)(cid:122) (cid:125) NLO + 0 (cid:124)(cid:123)(cid:122)(cid:125)
NNLO + · · · . (4)In the formalism used here higher-order corrections to the deuteron wavefunction renormal-ization will be built into the integral equation and do not need to be added separately. TheLO deuteron wavefunction renormalization is defined by Z LO = 2 γ t M N . (5) III. THREE-BODY SYSTEMA. Doublet Channel Scattering
The LO nd scattering amplitude in the doublet channel is given by an infinite sum ofdiagrams represented by the coupled-channel integral equations in Fig. 2. Single lines arenucleons and the double line (dashed double line) is the spin-triplet (spin-singlet) dibaryon.For the doublet S -wave channel there is also a contribution from a LO three-body force.However, in the approach used here three-body forces will be treated in separate diagramsdiscussed later. By projecting out the diagrams of Fig. 2 in the doublet channel and in apartial wave basis the integral equations can be written as an infinite set of matrix equations6 IG. 2: The coupled-channel integral equations for the LO doublet channel nd scattering amplitude.Single lines represent nucleons and double lines (dashed double lines) spin-triplet (spin-singlet)dibaryons. in cluster configuration (c.c.) space [22], which gives t (cid:96) ,d ( k, p ) = B (cid:96) ( k, p ) + K (cid:96) ( q, p, E ) ⊗ t (cid:96) ,d ( k, q ) , (6)where the subscript “ d ” refers to the doublet channel, and the superscript “ (cid:96) ” to the partialwave. The “ ⊗ ” notation is shorthand for the integration A ( q ) ⊗ B ( q ) = 12 π (cid:90) Λ0 dqq A ( q ) B ( q ) , (7)where Λ is a cutoff imposed to regulate divergences. Physical results should be Λ-independent for sufficiently large Λ. In the integral equation k is the magnitude of theincoming on-shell momentum in the nd center of mass (c.m.) frame and p is the magnitudeof the off-shell outgoing momentum. Since k is on-shell it is related to the total energy ofthe three-body system by E = k M N − γ t M N . t (cid:96)m,d ( k, p ) and the inhomogeneous term B (cid:96) ( k, p )are vectors in c.c. space, defined as t (cid:96)m,d ( k, p ) = t (cid:96)m,Nt → Nt ( k, p ) t (cid:96)m,Nt → Ns ( k, p ) , B (cid:96) ( k, p ) = πpk Q (cid:96) (cid:16) p + k − M N E − i(cid:15)pk (cid:17) − πpk Q (cid:96) (cid:16) p + k − M N E − i(cid:15)pk (cid:17) . (8)Here the subscript “ m ” refers to the order of the calculation ( m = 0 is LO, m = 1 is NLO,and etc.), t (cid:96)m,Nt → Nt ( k, p ) is the nd scattering amplitude, and t (cid:96)m,Nt → Ns ( k, p ) is the unphysicalamplitude of a neutron and deuteron going to a nucleon and spin-singlet dibaryon. In thisformalism B (cid:96) ( k, p ) = B (cid:96) ( k, p ) = 0, even for (cid:96) = 0, unlike in Ref. [23]. The function Q (cid:96) ( a ) isa Legendre function of the second kind and is related to standard Legendre polynomials by Q (cid:96) ( a ) = 12 (cid:90) − P (cid:96) ( x ) a + x dx. (9) This definition of the Legendre functions of the second kind differs from the normal convention by a phaseof ( − (cid:96) . K (cid:96) ( q, p, E ) is a matrix in c.c. space defined by K (cid:96) ( q, p, E ) = R ( q, p, E ) D (0) (cid:18) E − q M N , (cid:126) q (cid:19) , (10)where D ( n ) ( E, (cid:126) q ) = D ( n ) t ( E, (cid:126) q ) 00 D ( n ) s ( E, (cid:126) q ) (11)is a matrix of dibaryon propagators with n = 0 giving the LO dibaryon propagators, n = 1the NLO correction to the dibaryon propagators, and n = 2 the NNLO correction to thedibyaron propagators as in Eq. (3), and R ( q, p, E ) = − πqp Q (cid:96) (cid:18) q + p − M N E − i(cid:15)qp (cid:19) (cid:32) − − (cid:33) . (12)The half off-shell NLO correction to the doublet channel nd scattering amplitude is givenby the coupled-channel integral equations in Fig. 3, where the cross represents an effectiverange insertion. Iterating the inhomogeneous piece a single time in the kernel gives the
11 1 111
FIG. 3: The coupled-channel integral equations for the NLO correction to the doublet channel nd scattering amplitude. The cross refers to a single effective range insertion from c (0)0 t or c (0)0 s and thenumber “1” to the NLO correction to the nd scattering amplitude. integral equation for the NLO correction to nd scattering as in Ref. [23] along with anadditional diagram where an effective range insertion appears on an external dibaryon leg.In the on-shell limit the effective range insertion on the external dibaryon leg becomes theNLO wavefunction renormalization, which multiplies the LO nd scattering amplitude. Inother words, in the on-shell limit this integral equation gives the NLO correction to the nd scattering amplitude plus the LO nd scattering amplitude times the NLO deuteronwavefunction renormalization, or simply put all NLO contributions. The integral equationcan be written in c.c. space as t (cid:96) ,d ( k, p ) = t (cid:96) ,d ( k, p ) ◦ R (cid:18) E − (cid:126) p M N , (cid:126) p (cid:19) + K (cid:96) ( q, p, E ) ⊗ t (cid:96) ,d ( k, q ) , (13)8here “ ◦ ” is the Schur product (element wise matrix multiplication) and R ( p , (cid:126) p ) is avector in c.c. space defined by R ( p , (cid:126) p ) = Z t − γ t (cid:16) γ t + (cid:113) (cid:126) p − M N p − i(cid:15) (cid:17) Z s − γ s (cid:16) γ s + (cid:113) (cid:126) p − M N p − i(cid:15) (cid:17) . (14)Choosing the kinematics of the S ( S ) bound-state (virtual bound-state) pole for the upper(lower) component of R ( p , (cid:126) p ), R ( p , (cid:126) p ) reduces to c = Z t − Z s − , (15)which is the NLO correction to the wavefunction renormalization [22]. Similarly, the halfoff-shell NNLO correction to the nd scattering amplitude is given by the coupled-channelintegral equations in Fig. 4, where the star represents an insertion of c (1)0 t or c (1)0 s . In c.c. space
122 1 2 222
FIG. 4: The coupled-channel integral equations for the NNLO correction to the doublet channel nd scattering amplitude. The star refers to an insertion of c (1)0 t or c (1)0 s and the number “2” refersto the NNLO correction to the doublet channel nd scattering amplitude. the integral equation is given by t (cid:96) ,d ( k, p ) = (cid:2) t (cid:96) ,d ( k, p ) − c ◦ t (cid:96) ,d ( k, p ) (cid:3) ◦ R (cid:18) E − (cid:126) p M N , (cid:126) p (cid:19) (16)+ K (cid:96) ( q, p, E ) ⊗ t (cid:96) ,d ( k, q ) . In the ERE parametrization c = and the integral equations at NLO and NNLO look thesame. The presence of c ◦ t (cid:96) ,d ( k, p ) removes the ( Z t − t (cid:96) ,d ( k, k ) contribution that comesfrom t (cid:96) ,d ( k, p ) ◦ R (cid:16) E − (cid:126) p M N , (cid:126) p (cid:17) in the on-shell limit. Since the wavefunction renormalizationin the Z -parametrization is exact at NLO by construction, there is no ( Z t − correction. Since t (cid:96)m,Nt → Ns ( k, p ) is unphysical its normalization can be chosen arbitrarily without affecting physicalresults. . Three-Body Forces The above description for doublet channel nd scattering is incomplete since in the S -wavechannel a three-body force is required at LO [19]. The Lagrangian for the three-body forceup to NNLO is L = M N H (Λ)3Λ (cid:104) y t ˆ N † ( (cid:126)t · (cid:126) σ ) † − y s ˆ N † ( (cid:126)s · (cid:126) τ ) † (cid:105) (cid:104) y t ( (cid:126)t · (cid:126) σ ) ˆ N − y s ( (cid:126)s · (cid:126) τ ) ˆ N (cid:105) (17)+ M N H (Λ)3Λ (cid:104) y t ˆ N † ( (cid:126)t · (cid:126) σ ) † − y s ˆ N † ( (cid:126)s · (cid:126) τ ) † (cid:105) (cid:18) i(cid:126)∂ + γ t M N (cid:19) (cid:104) y t ( (cid:126)t · (cid:126) σ ) ˆ N − y s ( (cid:126)s · (cid:126) τ ) ˆ N (cid:105) .H (Λ) first occurs at LO and receives higher order corrections that can be written as H (Λ) = H , (Λ) (cid:124) (cid:123)(cid:122) (cid:125) LO + H , (Λ) (cid:124) (cid:123)(cid:122) (cid:125) NLO + H , (Λ) (cid:124) (cid:123)(cid:122) (cid:125) NNLO + · · · , (18)where the first subscript denotes that it is a contribution to H (Λ) and the second subscriptgives the order of the contribution. At NNLO a new energy-dependent three-body force H (Λ) appears [21]. The LO three-body force H , (Λ) does not renormalize an ultra-violetdivergence. Rather, the solution of the LO doublet S -wave nd scattering amplitude isnot unique in the limit where Λ → ∞ and this causes oscillations in the solution as Λ ischanged [21]. The physical explanation for H , (Λ) comes from the fact that in the doublet S -wave channel there is no Pauli blocking preventing the nucleons from falling to the center.Thus the doublet S -wave channel is sensitive to short range physics, which H , (Λ) encodes.The three-body force Lagrangian can be rewritten using a triton auxiliary field ˆ ψ , yielding L = ˆ ψ † (cid:34) Ω − h (Λ) (cid:32) i∂ + (cid:126) ∇ M N + γ t M N (cid:33)(cid:35) ˆ ψ + ∞ (cid:88) n =0 (cid:104) ω ( n ) t ˆ ψ † σ i ˆ N ˆ t i − ω ( n ) s ˆ ψ † τ a ˆ N ˆ s a (cid:105) (19)+ H . c .. A matching calculation shows that the parameters from each Lagrangian are related by H , (Λ)Λ = − ω (0) t ) π Ω = − ω (0) s ) π Ω = − ω (0) t ω (0) s π Ω , (20) H , (Λ)Λ = − ω (0) t ω (1) t π Ω = − ω (0) s ω (1) s π Ω = − ω (0) t ω (1) s π Ω = − ω (1) t ω (0) s π Ω , (21) H , (Λ)Λ = − ω (1) t ) + 2 ω (0) t ω (2) t )4 π Ω = − ω (1) s ) + 2 ω (0) s ω (2) s )4 π Ω (22)= − ω (1) s ω (1) t + 2 ω (0) t ω (2) s )4 π Ω = − ω (1) s ω (1) t + 2 ω (2) t ω (0) s )4 π Ω , H , (Λ)Λ = − ω (0) t ) π Ω M N h (Λ) = − ω (0) s ) π Ω M N h (Λ) = − ω (0) t ω (0) s π Ω M N h (Λ) . (23)It is convenient to make the definitions H LO = 4 H , (Λ)Λ , H NLO = 4 H , (Λ)Λ , H NNLO = 4 H , (Λ)Λ , (24)and (cid:98) H = 4 H , (Λ)Λ . (25)From these definitions follow the useful identities H NLO H LO = 2 ω (1) t ω (0) t , (26)and 2 ω (2) t ω (0) t = H NNLO H LO − ( H NLO ) ( H LO ) . (27) C. Triton Vertex Function
The LO triton vertex function is given by the coupled-channel integral equations in Fig. 5,where the triple line represents the triton propagator. These integral equations can be
FIG. 5: The coupled-channel integral equations for the LO triton vertex function, where the tripleline is the triton, and the filled circle is the LO triton vertex function. written in c.c. space as G ( E, p ) = (cid:101) B + K (cid:96) =00 ( q, p, E ) ⊗ G ( E, q ) , (28)where the “0” subscript indicates LO and (cid:101) B is a c.c space vector defined by (cid:101) B = (cid:32) (cid:33) . (29)11ote the kernel of these coupled-channel integral equations is the same as in LO nd scat-tering. The only difference between the integral equations for the LO triton vertex function G ( E, p ) and the LO nd scattering amplitude Eq. (6) is the inhomogeneous term. At theenergy of the bound state the matrix [ − K (cid:96) =00 ( q, p, E )] is invertible for all cutoffs for which H , (Λ) (cid:54) = 0. For cutoffs for which H , (Λ) = 0 the LO triton vertex is still well definedbecause the zero of H , (Λ) and the infinity of [ − K (cid:96) =00 ( q, p, E )] − have a well definedlimit. However, this is numerically tricky and therefore such cutoffs are avoided. G ( E, p )is defined in c.c. space by G ( E, p ) = G ,ψ → Nt ( E, p ) G ,ψ → Ns ( E, p ) , (30)where G ,ψ → Nt ( E, p ) ( G ,ψ → Ns ( E, p )) is the triton vertex function for an outgoing neutronand deuteron (nucleon and spin-singlet dibaryon) state. Note (cid:101) B is not the “physical”inhomogeneous term. The “physical” inhomogeneous term B is given by B = √ ω (0) t −√ ω (0) s . (31)Since an arbitrary normalization can be absorbed into both components of G ( E, p ) it isconvenient to use (cid:101) B instead of B . The “physical” triton vertex function Γ ( p ) is relatedto G ( E, p ) by Γ ( p ) = G ( E, p ) ◦ B (cid:112) Z ψ , (32)where the value of E is assumed fixed, and here Z ψ is the LO triton wavefunction renormal-ization to be defined below. Using G ( E, p ) instead of Γ ( p ) allows three-body forces to befactored out of expressions that would otherwise be absorbed into Γ ( p ).Adding a NLO effective range insertion to the triton vertex function can be achieved viathe coupled-channel integral equations in Fig. 6, which in c.c. space can be written as
11 11 11
FIG. 6: The coupled-channel integral equations for the NLO correction to the triton vertex function. ( E, p ) = G ( E, p ) ◦ R (cid:18) E − (cid:126) p M N , (cid:126) p (cid:19) + K (cid:96) =00 ( q, p, E ) ⊗ G ( E, q ) . (33)This equation is analogous to the NLO correction to the nd scattering amplitude Eq. (13).Two effective range insertions and c (1)0 t and c (1)0 s corrections to the triton vertex functionat NNLO can be added using the coupled-channel integral equations in Fig. 7, which in
122 1 22 22
FIG. 7: The coupled-channel integral equations for the NNLO correction to the triton vertexfunction. c.c. space are G ( E, p ) = (cid:104) G ( E, p ) − c ◦ G ( E, p ) (cid:105) ◦ R (cid:18) E − (cid:126) p M N , (cid:126) p (cid:19) + K (cid:96) =00 ( q, p, E ) ⊗ G ( E, q ) . (34)This equation is again entirely analogous to the integral equations for the NNLO correctionto nd scattering Eq. (16). In fact the only difference between the integral equations for thetriton vertex function and the nd scattering amplitude up to NNLO is the LO inhomogeneousterm.The function Σ P ( E ) is defined asΣ P ( E ) = (cid:90) d q (2 π ) iE − q − q M N + i(cid:15) (cid:2) i D (0) ( E − q , q ) i B (cid:3) · [ G ( E, q ) ◦ i B ] (35)and describes the sum of all triton-irreducible diagrams in Fig. 8. Note “ · ” represents theordinary dot product of two c.c space vectors. Subscript “0” denotes this is LO. Integrating Σ FIG. 8: Diagrammatic representation of the function Σ P ( E ). over the energy pole and angles, the expression for Σ P ( E ) becomes i Σ P ( E ) = − i ω (0) t ) π π (cid:90) Λ0 dqq D (0) t (cid:18) E − q M N , q (cid:19) G ,ψ → Nt ( E, q ) (36) − i ω (0) s ) π π (cid:90) Λ0 dqq D (0) s (cid:18) E − q M N , q (cid:19) G ,ψ → Ns ( E, q ) . n ( E ) = − π Tr (cid:20) D (0) (cid:18) E − q M N , q (cid:19) ⊗ G n ( E, q ) (cid:21) , (37)and using Eqs. (20) and (24) to rewrite ω (0) s and ω (0) t , Σ P ( E ) becomes i Σ P ( E ) = − i Ω H LO Σ ( E ) . (38)Using Σ P ( E ), the LO dressed triton propagator is given by the infinite sum of diagrams inFig. 9, which can be summed as a geometric series giving Σ Σ Σ FIG. 9: LO dressed triton propagator. The triangle is the dressed triton propagator, and the tripleline is the bare triton propagator i/ Ω. i ∆ (LO)3 ( E ) = i Ω + i Ω H LO Σ ( E ) + · · · = i Ω 11 − H LO Σ ( E ) . (39)This is the LO dressed triton propagator in the c.m. frame of the nd system. Thus thetriton propagator always has zero momentum. The formalism here can be straightforwardlygeneralized to include a triton propagator with non-zero momentum. At the bound-stateenergy B of the triton, the LO dressed triton propagator has a pole, giving the condition H LO = 1Σ ( B ) . (40)Setting B = E ( H) the three-body force can be fit to the triton binding energy E ( H) = − .
48 MeV [38]. Additionally, the LO triton binding energy can be calculated if a differentrenormalization condition is used for H LO . Considering higher orders beyond the workof Hagen et al. [32] the triton-irreducible functions Σ P ( E ) and Σ P ( E ) follow the Σ P ( E )definition and are given by the sum of diagrams in Fig. 10 and 11 respectively. Σ FIG. 10: Diagrammatic representation of the function Σ P ( E ). FIG. 11: Diagrammatic representation of the function Σ P ( E ). One finds that Σ P ( E ) and Σ P ( E ) are defined as i Σ P ( E ) = − i Ω H LO Σ ( E ) , i Σ P ( E ) = − i Ω H LO Σ ( E ) . (41)The NLO and NNLO corrections to the triton propagator are given by the diagrams inFig. 12. Summing the NLO diagrams gives ( NNLO )( NLO ) Σ H NLO Σ H NLO Σ Σ H NNLO H NLO Σ Σ Σ Σ h Σ Σ ( H NLO ) Σ FIG. 12: NLO and NNLO corrections to the triton propagator. The diagram with h comes fromthe kinetic term of the triton auxiliary field. i Ω 11 − H LO Σ ( E ) (cid:40) − i Ω H LO Σ ( E ) − i Ω (cid:32) ω (1) t ω (0) t (cid:33) H LO Σ ( E ) (cid:41) i Ω 11 − H LO Σ ( E ) (42)for the NLO correction to the triton propagator. The first (second) term comes from thefirst (second) diagram in the NLO box of Fig. 12. The second diagram in the NLO box isΣ P ( E ), but with a ω (0) t ( ω (0) s ) vertex replaced by ω (1) t ( ω (1) s ). A factor of two comes the factthe ω (1) t ( ω (1) s ) vertex can be on the left or the right of Fig. 8. Then using Eq. (26) the NLOcorrection to the triton propagator reduces to i Ω 11 − H LO Σ ( E ) {− i Ω H LO Σ ( E ) − i Ω H NLO Σ ( E ) } i Ω 11 − H LO Σ ( E ) . (43)15arrying out a similar procedure gives the triton propagator up to and including NNLO as i ∆ NNLO3 ( E ) = i Ω 11 − H LO Σ ( E ) (cid:20) H LO Σ ( E ) + H NLO Σ ( E )1 − H LO Σ ( E ) (44)+ H LO Σ ( E ) + H NLO Σ ( E ) + H NNLO Σ ( E ) + ( M N E + γ t ) (cid:98) H /H LO − H LO Σ ( E )+ [ H LO Σ ( E ) + H NLO Σ ( E )] [1 − H LO Σ ( E )] (cid:35) . The (cid:98) H /H LO term comes from the last NNLO diagram in Fig. 12. Fitting the LO three-bodyforce to the triton binding energy pole and ensuring that the pole is fixed at higher ordersimposes the conditions H LO Σ ( B ) + H NLO Σ ( B ) = 0 , (45)and H LO Σ ( B ) + H NLO Σ ( B ) + (cid:18) H NNLO + 43 ( M N B + γ t ) (cid:98) H (cid:19) Σ ( B ) = 0 . (46) H LO = 1 / Σ ( B ) has been used to rewrite the term with (cid:98) H . These two conditions fixtwo higher-order three-body forces, and H NNLO is fixed to the physical nd doublet S -wavescattering length. It will be shown later how this is done in the new formalism. The tritonwavefunction renormalization is the residue about the triton pole, which up to NNLO isgiven by Z ψ = −
1Ω 1 H LO Σ (cid:48) ( B ) (cid:20) − [ H LO Σ (cid:48) ( B ) + H NLO Σ (cid:48) ( B )] H LO Σ (cid:48) ( B ) (47) − [ H LO Σ (cid:48) ( B ) + H NLO Σ (cid:48) ( B ) + H NNLO Σ (cid:48) ( B )] + M N (cid:98) H /H LO H LO Σ (cid:48) ( B )+ [ H LO Σ (cid:48) ( B ) + H NLO Σ (cid:48) ( B )] [ H LO Σ (cid:48) ( B )] (cid:35) . Using Eqs. (40), (45), and (46) the dependence on H LO , H NLO , and H NNLO can be removed16ielding Z ψ = −
1Ω 1 H LO Σ (cid:48) ( B ) (cid:34) − (cid:18) Σ (cid:48) ( B )Σ (cid:48) ( B ) − Σ ( B )Σ ( B ) (cid:19) (48) − (cid:40) Σ (cid:48) ( B )Σ (cid:48) ( B ) − Σ ( B )Σ (cid:48) ( B )Σ ( B )Σ (cid:48) ( B ) + (cid:18) Σ ( B )Σ ( B ) (cid:19) − Σ ( B )Σ ( B )+ 43 M N (cid:98) H Σ ( B ) (cid:18) Σ ( B )Σ (cid:48) ( B ) − B − γ t M N (cid:19)(cid:41) + (cid:18) Σ (cid:48) ( B )Σ (cid:48) ( B ) − Σ ( B )Σ ( B ) (cid:19) (cid:35) . For the triton vertex function there is only one external triton propagator, and therefore thesquare root of Z ψ must be taken. Expanding the square root of Z ψ perturbatively to NNLOgives (cid:112) Z ψ = (cid:115) −
1Ω 1 H LO Σ (cid:48) (cid:124)(cid:123)(cid:122)(cid:125) LO − (cid:18) Σ (cid:48) Σ (cid:48) − Σ Σ (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) NLO (49) − (cid:34) Σ (cid:48) Σ (cid:48) + 12 Σ Σ (cid:48) Σ Σ (cid:48) − Σ Σ + 14 (cid:18) Σ Σ (cid:19) − (cid:18) Σ (cid:48) Σ (cid:48) (cid:19) + 43 M N (cid:98) H Σ (cid:18) Σ Σ (cid:48) − B − γ t M N (cid:19)(cid:35)(cid:124) (cid:123)(cid:122) (cid:125) NNLO + · · · . Here the explicit energy dependence for all Σ n functions has been dropped with the under-standing that all functions are evaluated at E = B . The “physical” triton vertex functionis calculated using Eq. (32). Using the definition of B and the triton wavefunction renor-malization, the LO renormalization for the triton vertex function G ( B, p ) is (cid:113) Z LO ψ = √ ω (0) t (cid:115) −
1Ω 1 H LO Σ (cid:48) ( B ) = (cid:115) − ω (0) t ) π Ω πH LO Σ (cid:48) ( B ) = (cid:114) π Σ (cid:48) ( B ) . (50)Eq. (20) has been used to simplify the expression. Thus the “physical” LO triton vertexfunction is given by Γ ( p ) = (cid:113) Z LO ψ G ( B, p ) . (51)This expression is equivalent to solving the homogeneous equation for the doublet S -wavechannel with a nonzero three-body force and then normalizing the result using techniquesin Refs. [25, 39]. The NLO triton vertex function is given by G ( B, p ), G ( B, p ) with the ω (0) t ( ω (0) s ) vertex replaced by ω (1) t ( ω (1) s ), and the LO triton vertex function times the NLOtriton wavefunction renormalization correction. The ω (1) t ( ω (1) s ) vertex can again be replaced17y a ratio of three-body forces as in the calculation of the triton propagator, and then theratio of three-body forces can be rewritten in terms of Σ n ( B ) using Eq. (45). With thesesimplifications the NLO triton vertex function is given by Γ ( p ) = (cid:113) Z LO ψ (cid:20) G ( B, p ) −
12 Σ (cid:48) Σ (cid:48) G ( B, p ) (cid:21) . (52)The calculation of the NNLO triton vertex function follows similarly and yields Γ ( p ) = (cid:113) Z LO ψ (cid:20) G ( B, p ) −
12 Σ (cid:48) Σ (cid:48) G ( B, p ) (53)+ (cid:40) (cid:18) Σ (cid:48) Σ (cid:48) (cid:19) −
12 Σ (cid:48) Σ (cid:48) − M N (cid:98) H Σ Σ (cid:48) (cid:41) G ( B, p ) (cid:35) . IV. DOUBLET S -WAVE SCATTERING In the formalism of this work the LO doublet S -wave on-shell nd scattering amplitudeis given by the sum of the two diagrams in Fig. 13. The first diagram is the solution ofEq. (6) for (cid:96) = 0. This diagram contains no three-body forces; all three-body force termsare contained in the second diagram. The sum of the two diagrams is given by FIG. 13: Diagrams for the LO doublet S -wave nd scattering amplitude. T LO ( k ) = Z LO t (cid:96) =00 ,Nt → Nt ( k, k ) + H LO − H LO Σ ( E ) πZ LO [ G ,ψ → Nt ( E, k )] . (54)In the new formalism the LO three-body force H LO is factored out of all numerically de-termined expressions. This is one advantage of this formalism. The LO three-body forcecan be found algebraically in terms of numerically determined quantities by fitting to thescattering length, a nd = 0 .
65 fm [40], which yields H LO = x x Σ (cid:16) − γ t M N (cid:17) , (55) The power of this formalism at LO lies in the fact that the triton pole contribution is contained solely inthe second diagram of Fig. 13. At higher orders contributions from poles are again clearly factored outin specific diagrams and can be easily read off. x = − (cid:16) πa nd M N + Z LO t (cid:96) =00 ,Nt → Nt (0 , (cid:17) πZ LO (cid:104) G ,ψ → Nt (cid:16) − γ t M N , (cid:17)(cid:105) . (56)The NLO nd scattering amplitude is given by the sum of diagrams in Fig. 14. The factorof two for the second diagram comes from including the time reversed diagram not explicitlyshown in Fig. 14. Summing these yields the NLO nd scattering amplitude Σ H NLO Σ { } FIG. 14: Diagrams for the NLO correction to the doublet S -wave nd scattering amplitude. Thefactor of two takes into account the diagram related by time reversal symmetry that is not shown. T NLO ( k ) = Z LO t (cid:96) =01 ,Nt → Nt ( k, k ) (57)+ πZ LO − H LO Σ ( E ) G ,ψ → Nt ( E, k ) [ H NLO G ,ψ → Nt ( E, k ) + 2 H LO G ,ψ → Nt ( E, k )]+ πH LO Z LO [ H LO Σ ( E ) + H NLO Σ ( E )][1 − H LO Σ ( E )] [ G ,ψ → Nt ( E, k )] . Again, the NLO three-body force is factored out of all numerically determined expressionsand therefore can be algebraically fit to the doublet S -wave nd scattering length. The NNLO nd scattering amplitude is given by the sum of diagrams in Fig. 15, which gives19 Σ Σ Σ Σ Σ Σ Σ } H NLO H NNLO Σ { } ( H NLO ) }{ Σ Σ Σ Σ { h FIG. 15: Diagrams for the NNLO correction to the doublet S -wave nd scattering amplitude. Thefactors of two take into account diagrams related by time reversal symmetry that are not shown. T NNLO ( k ) = Z LO t (cid:96) =02 ,Nt → Nt ( k, k ) (58)+ πZ LO − H LO Σ ( E ) G ,ψ → Nt ( E, k ) × [ H NNLO G ,ψ → Nt ( E, k ) + 2 H NLO G ,ψ → Nt ( E, k ) + 2 H LO G ,ψ → Nt ( E, k )]+ πH LO Z LO [ H LO Σ ( E ) + H NLO Σ ( E ) + H NNLO Σ ( E )][1 − H LO Σ ( E )] [ G ,ψ → Nt ( E, k )] + πH LO Z LO [ H LO Σ ( E ) + H NLO Σ ( E )] [1 − H LO Σ ( E )] [ G ,ψ → Nt ( E, k )] + πH NLO Z LO [ H LO Σ ( E ) + H NLO Σ ( E )][1 − H LO Σ ( E )] [ G ,ψ → Nt ( E, k )] + 2 πH LO Z LO [ H LO Σ ( E ) + H NLO Σ ( E )][1 − H LO Σ ( E )] G ,ψ → Nt ( E, k ) G ,ψ → Nt ( E, k )+ πH LO Z LO − H LO Σ ( E ) ( G ,ψ → Nt ( E, k )) + π ( M N E + γ t ) (cid:98) H Z LO [1 − H LO Σ ( E )] [ G ,ψ → Nt ( E, k )] , When k = 0 the term with (cid:98) H disappears and only one new three-body force H NNLO ispresent, which can again be solved algebraically and fit to the nd scattering length. (cid:98) H canthen be fit to the triton binding energy. In order to find the physical triton binding energythe scattering amplitude can be written in the form t ( k, p, E ) + t ( k, p, E ) + t ( k, p, E ) + · · · = Z ( k, p ) + Z ( k, p ) + Z ( k, p ) E − B − B − B + · · · (59)+ R ( k, p, E ) + R ( k, p, E ) + R ( k, p, E ) + · · · ,
20s an expansion about the bound-state pole [26, 39]. There is a pole at the physical tritonbinding energy E ( H) = B + B + B + · · · , with smooth residue c.c. space vector functions Z n ( k, p ) and smooth remainder c.c. space vector functions R n ( k, p, E ). Expanding thisexpression perturbatively gives at LO t ( k, p, E ) = Z ( k, p ) E − B + R ( k, p, E ) . (60)Now the power of this formalism becomes clear because from Eq. (54) it can clearly be seenthat the pole contribution comes from the second term. The location of the pole is given byEq. (40) and Z ( k, p ) is simply the residue about this pole, which is Z ( k, k ) = − πZ LO [ G ,ψ → Nt ( B , k )] Σ (cid:48) ( B ) . (61)At NLO the perturbative expansion of Eq. (59) gives t ( k, p, E ) = Z ( k, p ) E − B + B Z ( k, p )( E − B ) + R ( k, p, E ) . (62)Comparing to Eq. (57) and using the expression for Z ( k, k ), the contributions from the firstand second order pole can easily be extracted, giving the NLO correction to the bound-stateenergy B = − H LO Σ ( B ) + H NLO Σ ( B ) H LO Σ (cid:48) ( B ) , (63)and the NLO residue function Z ( k, k ) = − πZ LO G ,ψ → Nt ( B , k ) [ H NLO G ,ψ → Nt ( B , k ) + 2 H LO G ,ψ → Nt ( B , k )] H LO Σ (cid:48) ( B ) . (64)The NNLO perturbative expansion of Eq. (59) gives t ( k, p, E ) = Z ( k, p ) E − B + B Z ( k, p )( E − B ) + B Z ( k, p )( E − B ) + B Z ( k, p )( E − B ) + R ( k, p, E ) . (65)Since Z ( k, k ) and B are known, their second order pole contribution can be subtractedfrom Eq. (58) leaving the contribution from B , which is given by B = − H LO Σ ( B ) + H NLO Σ ( B ) + (cid:104) H NNLO + ( M N B + γ t ) (cid:98) H (cid:105) Σ ( B ) H LO Σ (cid:48) ( B ) (66) − B H LO Σ (cid:48) ( B ) + H NLO Σ (cid:48) ( B ) H LO Σ (cid:48) ( B ) − B Σ (cid:48)(cid:48) ( B )Σ (cid:48) ( B ) . To fit (cid:98) H to the bound-state energy one adjusts (cid:98) H such that E ( H) = B + B + B . Note thatif one sets B and B to zero then the constraints on the three-body forces are equivalent21o Eqs. (45) and (46) where the three-body forces were fit to the bound-state energy byfixing the pole position for the triton propagator. This formalism reproduces the resultsfor three-body forces and doublet S -wave scattering amplitudes found in Ref. [23] up tonumerical accuracy. But it is superior because it avoids iterative techniques for H LO andnumerical limiting procedures for (cid:98) H . V. TRITON CHARGE FORM FACTOR
The LO triton charge form factor is given by the sum of diagrams in Fig. 16, where thewavy blue lines are minimally coupled ˆ A photons. The form factor calculation is performedin the Breit frame in which the photon imparts no energy to the triton but only momentum.In the Breit frame one chooses the initial (final) momentum of the triton to be (cid:126) K ( (cid:126) P ). Themomentum imparted by the photon is (cid:126) Q = (cid:126) P − (cid:126) K , and the form factor only depends on thevalue (cid:126) Q . Summing all three diagrams in the Breit frame gives (a) (b) (c) FIG. 16: Diagrams for the LO triton charge form factor. The wavy blue lines represent minimallycoupled ˆ A photons. Z LO ψ (cid:88) j = a,b,c (cid:90) d k (2 π ) (cid:90) d p (2 π ) G T ( E, (cid:126) P , p , (cid:126) p ) χ j ( E, (cid:126) K , (cid:126) P , p , k , (cid:126) p , (cid:126) k ) G ( E, (cid:126) K , k , (cid:126) k ) , (67)where G ( E, (cid:126) K , k , (cid:126) k ) is the LO triton vertex function in a frame boosted by momentum (cid:126) K , and E = B + M N K , with B = E ( H) , is the total energy of the triton in this frame.The functional forms of χ j ( E, (cid:126) K , (cid:126) P , p , k , (cid:126) p , (cid:126) k ) are listed in App. A. Choosing the fourmomentum of the dibaryon (nucleon) to be [ E + k , (cid:126) k + (cid:126) K ] ([ E − k , − (cid:126) k + (cid:126) K ]) thetriton vertex function in the boosted frame is related to the triton vertex function in the22.m. frame via G ( E, (cid:126) K , k , (cid:126) k ) = (cid:101) B (68)+ (cid:34) R (cid:32) q, k, B + k − (cid:126) K · (cid:126) k M N + (cid:126) k M N (cid:33) D (0) (cid:18) B − (cid:126) q M N , (cid:126) q (cid:19)(cid:35) ⊗ G ( B , q ) . For diagram (a), χ a ( · · · ) gives delta functions over momentum and energy that remove theintegral over d p . Then integrating over the energy k and using Eq. (68) the LO contributionfrom diagram (a) can be written as F ( a )0 ( Q ) = Z LO ψ (cid:110) (cid:101) G T ( p ) ⊗ A ( p, k, Q ) ⊗ (cid:101) G ( k ) + 2 (cid:101) G T ( p ) ⊗ A ( p, Q ) + A ( Q ) (cid:111) . (69)The subscript “0” in the functions F n ( Q ), A n ( · · · ), A n ( Q ), and (cid:101) G n ( p ) refer to LO. NLO andNNLO contributions will be denoted by a “1” and “2” subscript respectively. The function A n ( p, k, Q ) is a matrix function in c.c. space, A n ( p, Q ) a vector function in c.c. space, and A n ( Q ) a scalar function. Further details of this calculation and the form of the functions A n ( · · · ) and A n ( Q ) are given in App. A. The vector function (cid:101) G n ( p ) in c.c. space is definedas (cid:101) G n ( p ) = D (0) (cid:18) B − (cid:126) p M N , (cid:126) p (cid:19) G n ( B , p ) . (70)Diagram (b) of Fig. 16 can be written as F ( b )0 ( Q ) = Z LO ψ (cid:101) G T ( p ) ⊗ B ( p, k, Q ) ⊗ (cid:101) G ( k ) , (71)where B ( p, k, Q ) is a matrix function in c.c. space given in the App. A. For diagram (c) F ( c )0 ( Q ) = Z LO ψ (cid:110) (cid:101) G T ( p ) ⊗ C ( p, k, Q ) ⊗ (cid:101) G ( k ) + C ( k, Q ) ⊗ (cid:101) G ( k ) (cid:111) , (72)where C ( p, k, Q ) is a matrix function in c.c. space and C ( k, Q ) a vector function inc.c. space . Summing the contribution from all diagrams the LO triton charge form fac-tor is given by F ( Q ) = F ( a )0 ( Q ) + F ( b )0 ( Q ) + F ( c )0 ( Q ) . (73)In the limit Q → F (0) = 1 up to numerical accuracy. It can be shown analytically thatin the limit Q → Note that in Ref. [32] only the first term for F ( c )0 ( Q ) exists. This is due to the difference in LO three-bodyforces between these two calculations. S -wave channel is recovered from F (0). This is shown in furtherdetail in App. C.The NLO correction to the triton charge form factor is given by the diagrams in Fig. 17.Diagrams (a) through (d) are added together while diagram (e) is subtracted to avoid double (a) (b) (c)1 1 1(d) (e) FIG. 17: Diagrams for the NLO correction to the triton charge form factor, where diagrams relatedby time reversal symmetry are not shown. The diagram in the dashed box is subtracted from theother diagrams to avoid double counting. The photon in diagram (d) is minimally coupled to thedibaryon. counting from diagram (a) and its time reversed version. The photon in diagram (d) is min-imally coupled via the dibaryon kinetic term. Diagrams related by time reversal symmetryare not shown in Fig. 17. The sum of diagrams (a)-(d) and subtraction of diagram (e) isgiven by Z LO ψ (cid:88) j = a,b,c (cid:90) d k (2 π ) (cid:90) d p (2 π ) (cid:110) G T ( E, (cid:126) P , p , (cid:126) p ) χ j ( E, (cid:126) K , (cid:126) P , p , k , (cid:126) p , (cid:126) k ) G ( E, (cid:126) K , k , (cid:126) k ) (74)+ G T ( E, (cid:126) P , p , (cid:126) p ) χ j ( E, (cid:126) K , (cid:126) P , p , k , (cid:126) p , (cid:126) k ) G ( E, (cid:126) K , k , (cid:126) k ) (cid:111) + Z LO ψ (cid:88) d, − e (cid:90) d k (2 π ) (cid:90) d p (2 π ) G T ( E, (cid:126) P , p , (cid:126) p ) χ j ( E, (cid:126) K , (cid:126) P , p , k , (cid:126) p , (cid:126) k ) G ( E, (cid:126) K , k , (cid:126) k ) . Functions χ j ( · · · ) for j = a, b, c are the same as in the LO case. At NLO there are newfunctions χ d ( · · · ) and χ e ( · · · ). To obtain Eq. (74) the LO expression Eq. (67) is replacedby NLO corrections wherever possible. The NLO correction to the triton vertex functionin a boosted frame is related to the NLO correction to the triton vertex function in the24.m. frame by G ( E, (cid:126) K , k , (cid:126) k ) = G ( E, (cid:126) K , k , (cid:126) k ) ◦ R (cid:18) E + k , (cid:126) k + 23 (cid:126) K (cid:19) (75)+ (cid:34) R (cid:32) q, k, B + k − (cid:126) K · (cid:126) k M N + (cid:126) k M N (cid:33) D (0) (cid:18) B − (cid:126) q M N , (cid:126) q (cid:19)(cid:35) ⊗ G ( B , q ) . Using Eq. (68) the NLO correction to the triton vertex function in a boosted frame can bewritten entirely in terms of c.m. quantities. The NLO contribution from diagram (a) minusdiagram (e) is given by F ( a )1 ( Q ) = Z LO ψ (cid:110) (cid:101) G T ( p ) ⊗ A ( p, k, Q ) ⊗ (cid:101) G ( k ) + 2 (cid:101) G T ( p ) ⊗ A ( p, k, Q ) ⊗ (cid:101) G ( k ) (76)+2 (cid:101) G T ( p ) ⊗ A ( p, Q ) + 2 (cid:101) G T ( p ) ⊗ A ( p, Q ) + A ( Q ) (cid:111) . To obtain this NLO expression one replaces all LO terms in Eq. (69) by their NLO coun-terparts. The functions A ( · · · ) and A ( Q ) only differ from A ( · · · ) and A ( Q ) by thereplacement of a LO dibaryon propagator by a NLO correction to the dibaryon propaga-tor. Again further details and their functional forms can be seen in App. A. The NLOcontribution from diagram (b) is given by F ( b )1 ( Q ) = Z LO ψ (cid:110) (cid:101) G T ( p ) ⊗ B ( p, k, Q ) ⊗ (cid:101) G ( k ) (cid:111) , (77)for diagram (c) by F ( c )1 ( Q ) = Z LO ψ (cid:110) (cid:101) G T ( p ) ⊗ C ( p, k, Q ) ⊗ (cid:101) G ( k ) + (cid:101) G T ( p ) ⊗ C ( p, k, Q ) ⊗ (cid:101) G ( k ) (78)+ (cid:101) G T ( p ) ⊗ C ( p, k, Q ) ⊗ (cid:101) G ( k ) + C ( k, Q ) ⊗ (cid:101) G ( k ) + C ( k, Q ) ⊗ (cid:101) G ( k ) (cid:111) . and finally diagram (d) by F ( d )1 ( Q ) = Z LO ψ (cid:110) (cid:101) G T ( p ) ⊗ D ( p, k, Q ) ⊗ (cid:101) G ( k ) + D ( k, Q ) ⊗ (cid:101) G ( k ) (cid:111) . (79)The function D n ( p, k, Q ) is a matrix function in c.c. space and D n ( k, Q ) a vector functionin c.c. space. For the functions D n ( · · · ) n = 0 does not occur; its first contribution isat NLO. The functions B ( p, k, Q ) and B ( p, k, Q ) also do not exist. Summing all of theNLO contributions, replacing ω (0) t and ω (0) s by ω (1) t and ω (1) s in the LO contributions, andmultiplying the LO contribution by the NLO triton wavefunction renormalization gives F ( Q ) = (cid:16) F ( a )1 ( Q ) + F ( b )1 ( Q ) + F ( c )1 ( Q ) + F ( d )1 ( Q ) (cid:17) − Σ (cid:48) Σ (cid:48) F ( Q ) , (80)25or the NLO correction to the triton charge form factor. In the limit Q → F (0) = 0 upto numerical accuracy.The NNLO correction to the triton charge form factor is given by the diagrams in Fig. 18.Diagrams of type (a) through (d) are added while diagrams (e) and (f) are subtracted to avoiddouble counting from (a) type diagrams and their time reversed versions. Again diagramsrelated by time reversal symmetry are not shown. Diagram (g) comes from gauging thekinetic term of the triton field. Analogously to the NLO case the sum of diagrams (a) h (d) (e) (f)1 1 (g) FIG. 18: Diagrams for the NNLO correction to the triton charge form factor, where diagramsrelated by time reversal symmetry are not shown. The diagrams in the dashed boxes are subtractedfrom the other diagrams to avoid double counting. Z LO ψ (cid:88) j = a,b,c (cid:90) d k (2 π ) (cid:90) d p (2 π ) (cid:110) G T ( E, (cid:126) P , p , (cid:126) p ) χ j ( E, (cid:126) K , (cid:126) P , p , k , (cid:126) p , (cid:126) k ) G ( E, (cid:126) K , k , (cid:126) k ) (81)+ G T ( E, (cid:126) P , p , (cid:126) p ) χ j ( E, (cid:126) K , (cid:126) P , p , k , (cid:126) p , (cid:126) k ) G ( E, (cid:126) K , k , (cid:126) k )+ G T ( E, (cid:126) P , p , (cid:126) p ) χ j ( E, (cid:126) K , (cid:126) P , p , k , (cid:126) p , (cid:126) k ) G ( E, (cid:126) K , k , (cid:126) k ) (cid:111) + Z LO ψ (cid:88) j = d, − e (cid:90) d k (2 π ) (cid:90) d p (2 π ) (cid:110) G T ( E, (cid:126) P , p , (cid:126) p ) χ j ( E, (cid:126) K , (cid:126) P , p , k , (cid:126) p , (cid:126) k ) G ( E, (cid:126) K , k , (cid:126) k )+ G T ( E, (cid:126) P , p , (cid:126) p ) χ j ( E, (cid:126) K , (cid:126) P , p , k , (cid:126) p , (cid:126) k ) G ( E, (cid:126) K , k , (cid:126) k ) (cid:111) + Z LO ψ (cid:88) j = − f (cid:90) d k (2 π ) (cid:90) d p (2 π ) G T ( E, (cid:126) P , p , (cid:126) p ) χ j ( E, (cid:126) K , (cid:126) P , p , k , (cid:126) p , (cid:126) k ) G ( E, (cid:126) K , k , (cid:126) k ) . The NNLO correction to the triton vertex function in a boosted frame is related to theNNLO correction to the triton vertex function in the c.m. frame via G ( E, (cid:126) K , k , (cid:126) k ) = (cid:104) G ( E, (cid:126) K , k , (cid:126) k ) − c ◦ G ( E, (cid:126) K , k , (cid:126) k ) (cid:105) ◦ R (cid:18) E + k , (cid:126) k + 23 (cid:126) K (cid:19) (82)+ (cid:34) R (cid:32) q, k, B + k − (cid:126) K · (cid:126) k M N + (cid:126) k M N (cid:33) D (0) (cid:18) B − (cid:126) q M N , (cid:126) q (cid:19)(cid:35) ⊗ G ( B , q ) . Using Eqs. (68) and (75) the NNLO correction to the triton vertex function in a boostedframe can be written in terms of c.m. quantities. The sum of type (a) diagrams minusdiagrams (e) and (f) gives F ( a )2 ( Q ) = Z LO ψ (cid:110) (cid:101) G T ( p ) ⊗ A ( p, k, Q ) ⊗ (cid:101) G ( k ) + 2 (cid:101) G T ( p ) ⊗ A ( p, k, Q ) ⊗ (cid:101) G ( k ) (83)+ 2 (cid:101) G T ( p ) ⊗ A ( p, k, Q ) ⊗ (cid:101) G ( k ) + (cid:101) G T ( p ) ⊗ A ( p, k, Q ) ⊗ (cid:101) G ( k )+2 (cid:101) G T ( p ) ⊗ A ( p, Q ) + 2 (cid:101) G T ( p ) ⊗ A ( p, Q ) + 2 (cid:101) G T ( p ) ⊗ A ( p, Q ) + A ( Q ) (cid:111) . As in the NLO case all functions in Eq. (69) are replaced by their NNLO counterparts.In addition terms where two expressions are replaced by their NLO counterparts are in-cluded. The functions A ( · · · ) and A ( Q ) are the same as A ( · · · ) and A ( Q ) respectivelyexcept with LO dibaryon propagators replaced by the NNLO correction to the dibaryonpropagators. Diagrams of type (b) give the contribution F ( b )2 ( Q ) = Z LO ψ (cid:110) (cid:101) G T ( p ) ⊗ B ( p, k, Q ) ⊗ (cid:101) G ( k ) + (cid:101) G T ( p ) ⊗ B ( p, k, Q ) ⊗ (cid:101) G ( k ) (cid:111) , (84)27iagrams of type (c) give F ( c )2 ( Q ) = Z LO ψ (cid:110) (cid:101) G T ( p ) ⊗ C ( p, k, Q ) ⊗ (cid:101) G ( k ) + 2 (cid:101) G T ( p ) ⊗ C ( p, k, Q ) ⊗ (cid:101) G ( k ) (85)+ 2 (cid:101) G T ( p ) ⊗ C ( p, k, Q ) ⊗ (cid:101) G ( k ) + (cid:101) G T ( p ) ⊗ C ( p, k, Q ) ⊗ (cid:101) G ( k )+ C ( k, Q ) ⊗ (cid:101) G ( k ) + C ( k, Q ) ⊗ (cid:101) G ( k ) + C ( k, Q ) ⊗ (cid:101) G ( k ) (cid:111) , and diagram (d) gives F ( d )2 ( Q ) = Z LO ψ (cid:110) (cid:101) G T ( p ) ⊗ D ( p, k, Q ) ⊗ (cid:101) G ( k ) + 2 (cid:101) G T ( p ) ⊗ D ( p, k, Q ) ⊗ (cid:101) G ( k ) (86)+ D ( k, Q ) ⊗ (cid:101) G ( k ) + D ( k, Q ) ⊗ (cid:101) G ( k ) (cid:111) . Finally, the contribution from diagram (g) is given by the constant term43 M N (cid:98) H Σ Σ (cid:48) (87)Summing all of the NNLO corrections to the triton charge form factor, replacing ω (0) t and ω (0) s by ω (2) t and ω (2) s and two factors of ω (1) t and ω (1) s in the LO contributions, replacing ω (0) t and ω (0) s by ω (1) t and ω (1) s in the NLO contributions, multiplying the NLO correction bythe NLO triton wavefunction renormalization, and multiplying the LO term by the NNLOtriton wavefunction renormalization yields the NNLO triton charge form factor F ( Q ) = (cid:16) F ( a )2 ( Q ) + F ( b )2 ( Q ) + F ( c )2 ( Q ) + F ( d )2 ( Q ) (cid:17) (88) − Σ (cid:48) Σ (cid:48) (cid:16) F ( a )1 ( Q ) + F ( b )1 ( Q ) + F ( c )1 ( Q ) + F ( d )1 ( Q ) (cid:17) + (cid:34)(cid:18) Σ (cid:48) Σ (cid:48) (cid:19) − Σ (cid:48) Σ (cid:48) − M N (cid:98) H Σ Σ (cid:48) (cid:35) F ( Q ) + 43 M N (cid:98) H Σ Σ (cid:48) In the limit Q → F (0) = 0. However, it is found that F (0) ∼ − ,which is only one order of magnitude smaller than the deviation of the LO value of the tritoncharge form factor from the value F (0) = 1 for Q ∼ . . This is due to the factthat this qauntity is very fine tuned with respect to the three-body force H NNLO : taking H NNLO fit to the triton binding energy and varying it by one part in 10 it is found that F (0) ∼ − . Despite the value of F (0) being highly fine tuned with respect to H NNLO nosuch level of fine tuning is seen for the NNLO correction to the triton point charge radius. Inother words the slope of the NNLO correction to the triton charge form factor with respectto Q is not fine tuned with respect to H NNLO , but the y-intercept is.28
I. TRITON POINT CHARGE RADIUS AND RESULTS
The triton charge form factor can be expanded in powers of Q yielding F ( Q ) = 1 − (cid:10) r H (cid:11) Q + · · · , (89)where δr C = (cid:113)(cid:10) r H (cid:11) is the triton point charge radius. At LO the triton charge form factoris given by F ( Q ) = 1 − (cid:10) r H (cid:11) Q + · · · , (90)where (cid:10) r H (cid:11) is the LO contribution to ( δr C ) . The NLO correction to the triton chargeform factor is given by F ( Q ) = − (cid:10) r H (cid:11) Q + · · · , (91)and the NNLO correction by F ( Q ) = − (cid:10) r H (cid:11) Q + · · · . (92) (cid:10) r H (cid:11) is the NLO correction to δr C and (cid:10) r H (cid:11) is the NNLO correction to δr C , and thesquare of the triton point charge radius to NNLO is simply given by (cid:10) δr C (cid:11) = (cid:10) r H (cid:11) + (cid:10) r H (cid:11) + (cid:10) r H (cid:11) + · · · . (93)Taking the square root of this expression and expanding perturbatively the triton pointcharge radius δr C up to NNLO is given by δr c = (cid:113)(cid:10) r H (cid:11) (cid:124)(cid:123)(cid:122)(cid:125) LO + 12 (cid:10) r H (cid:11) (cid:10) r H (cid:11) (cid:124) (cid:123)(cid:122) (cid:125) NLO + 12 (cid:10) r H (cid:11) (cid:10) r H (cid:11) − (cid:32) (cid:10) r H (cid:11) (cid:10) r H (cid:11) (cid:33) (cid:124) (cid:123)(cid:122) (cid:125) NNLO + · · · . (94)In order to calculate the point charge radius at each order the charge form factor can becalculated for low values of Q and a linear fit with respect to Q then performed to extractthe point charge radius. This procedure works well at LO, however, for higher cutoffs atNLO and NNLO this approach quickly runs into numerical issues and the point charge radiuscannot be reliably extracted. In order to circumvent this one expands the functions A n ( · · · ), A n ( Q ), B ( · · · ), C n ( · · · ), and D n ( · · · ) in powers of Q and extracts their Q pieces allowingfor a direct calculation of the point charge radius contributions. The Q parts of these29unctions can be simplified further by analytical integrations of angular integrals, therebyreducing potential numerical issues and speeding up calculations. The Q parts of thesefunctions are given in App. B.The triton charge radius r C is related to the triton point charge radius δr C by (cid:10) δr C (cid:11) = (cid:10) r C (cid:11) − (cid:10) r p (cid:11) − (cid:10) r n (cid:11) , (95)where r p = 0 . ± . r n = − . ± . [1]is the neutron charge radius squared, and r C = 1 . ± . δr c = 1 . ± .
040 fmis extracted.The cutoff dependence of the LO, NLO, and NNLO triton point charge radius is given inFig. 19. Small values of the cutoff should be ignored since they are sensitive to shifts in the C ha r ge R ad i u s δ r C [f m ] Cutoff [MeV] LONLONNLOExp-1.5978
FIG. 19: Cutoff dependence of the LO, NLO, and NNLO predictions for the triton point chargeradius. The pink band is a 15% error estimate for the LO triton point charge radius, the greenband is a 5% error estimate for the NLO triton point charge radius, and the blue band is a 1.5%error estimate for the NNLO triton point charge radius. The dotted line is the value extractedfrom experiment, 1 . ± .
040 fm [1], and the black lines its error. momentum in integrals from the finite cutoff regularization. However, for sufficiently largecutoffs all terms that go like 1 / Λ n are suppressed and all integrals are effectively invariantunder a shift in momentum. In Fig. 19 the LO pink band corresponds to a 15% error aboutthe LO point charge radius prediction, the NLO green band corresponds to a 5% error aboutthe NLO point charge radius prediction, and the NNLO blue band to a 1.5% error about the30NLO point charge radius prediction. The LO and NLO bands converge as a function ofcutoff, while the NNLO band has a very slight cutoff variation. The LO triton point chargeradius converges to a value of 1.14 fm and the NLO value to 1.59 fm. In the region of cutoffsfrom 1000 to 10 MeV the NNLO point charge radius varies from 1.62 fm to 1.63 fm. TheNLO (NNLO) value is within 5% (1.5%) of the experimental number for the triton pointcharge radius of 1 . ± .
040 fm [1]. From LO to NLO a large change is seen in the pointcharge radius. This large change from LO to NLO is typical in the Z -parametrization wherefixing the residue about the poles of the deuteron and S virtual bound state makes a largecorrection from LO to NLO. Further examples of this behavior can be seen in Ref. [36] forthe np phase shift in the S channel.The LO prediction for the triton point charge radius is more than 15% away from theexperimental error bars. However, calculating the LO triton point charge radius in theunitary limit yields the numerical result M N E H (cid:10) r H (cid:11) = (1 + s ) / ≈ . This gives further confidence thatthe LO result, despite perhaps seeming too small, is indeed correct. At NNLO a pointcharge radius of 1 . ± .
03 fm is predicted, which agrees with the experimental extractionwithin errors, where the error comes from a 1.5% error estimate from EFT( /π ) and also a 1%error from cutoff variation. It is still an open question whether the NNLO result is strictlyconverging as Λ → ∞ . In order to address this issue either a detailed asymptotic analysismust be carried out or a calculation to cutoffs large enough where signs of convergence orlack thereof can be clearly seen. However, the NNLO calculation suffers from numerical noiseat large cutoffs (Λ > MeV) and new numerical techniques would be needed to deal withthe fine tuning of three-body forces at large cutoffs. Dealing with this fine tuning could alsoallow reliable calculations of the triton charge form factor and not just the triton point chargeradius to higher cutoffs at NNLO. Finally, a previous EFT( /π ) calculation using wavefunctionmethods obtained a LO prediction of 2 . ± . /π ) prediction of 1 . ± . The usual EFT( /π ) error is 30%, 10%, and 3% for LO, NLO, and NNLO respectively. Taking the squareroot to get the charge radius divides this percent error in half. The number s = 1 . ... is a universal number coming from the solution of the asymptotic form of thetriton vertex function[41 ? , 42]. Ref. [3] actually calculates the point matter radius in the unitary andequal mass limit, but this is equivalent to the point charge radius in this limit. M N E H (cid:10) r H (cid:11) = (1 + s ) / ≈ .
224 in the unitaryand equal mass limit.The point charge radius of the triton was obtained using Eq. (95) and the charge radiusof the proton from electron scattering. However, spectroscopy from muonic hydrogen finds aproton charge radius of 0.84087(39) fm [45], which is about seven standard deviations awayfrom the averaged results of electron scattering and electronic hydrogen spectroscopy [46].This discrepancy is known as the “proton radius puzzle”. An extensive review can be foundin Ref. [47] and an overview of certain current and ongoing experimental efforts in Ref. [48].Possible solutions lie in the way that functions are fit to electron scattering data to extractthe charge radius [49]. However, this would not explain the discrepancy between muonichydrogen and electronic hydrogen spectroscopy data. Both experimental [50–52] and theo-retical [53] efforts are being carried out to reexamine the electronic hydrogen spectroscopyresults. Other possible theoretical explanations include using new muonic forces [54–56] andnew proton structures [57–62]. Using the value for the proton charge radius from muonichydrogen gives a triton point charge radius of 1 . ± .
040 fm. The approximate 1%difference between the experimental triton point charge radius from muonic hydrogen andelectron scattering would require a N LO calculation in EFT( /π ) to distinguish them. Notea N LO calculation does not give direct information about the fundamental interactionsgiving rise to the proton structure in the triton, but only to correlations within and betweenthe triton and deuteron structures.A comparison of various calculations of the triton point charge radius is shown in Table I.The results of Ref. [63] use the Lanzcos sum rule and the effective interaction hypersphericalharmonics method with the two-body Argonne-v18 (AV18) [64] and three-body Urbana IX(UIX) [65] (AV18/UIX) potential to obtain a triton point charge radius of 1.593 fm andusing a two- [66] and three-body [67] χ EFT potential they find a triton point charge radiusof 1.617 fm. Ref. [68] uses the AV18/UIX potential with the hyperspherical harmonics (HH)method to get a triton point charge radius of 1.582 fm. Using Green’s function MonteCarlo (GFMC) with the AV18 and three-body Illinois 7 (IL7) [69] potential (AV18/IL7) atriton point charge radius of 1.58 fm is found [70]. χ EFT predicts a triton point chargeradius of 1.594(8), where the error comes from looking at the cutoff dependence of the tritonpoint charge radius [35]. The NNLO results of this work and other lower-order EFT( /π )calculations are displayed as well. Also shown in Table I are predictions for the triton32 ethod B H [MeV] δr C [fm]AV18/UIX [63] 8.473 1.593 χ EFT [63] 8.478 1.617AV18/UIX HH [68] 8.479 1.582AV18/IL7 GFMC[70] 8.50(1) 1.58 χ EFT N3LO/N2LO [35] – 1.594(8)EFT( /π ) (LO) [43] – 2.1(6)EFT( /π ) (NLO) [44] – 1.6(2)EFT( /π ) (NNLO) – 1.62(3)Experiment: 8.4818Experiment: e − µ − χ EFT calculation of Ref. [63]. The error for the triton binding energy for the GFMC resultscomes from statistical errors in Monte Carlo calculations. All other errors are estimates from EFTor experimental errors. The error for the χ EFT value of δr C comes from varying the cutoff of thecalculation [35]. Experimental numbers for the triton point charge radius are given using both theproton charge radius from electron scattering data and muonic hydrogen data. binding energy. For EFT predictions the triton binding energy is fit to and therefore notshown . Most techniques predict the triton binding energy reasonably well, but the GFMCseems to slightly overpredict it, and its error comes from Monte Carlo statistics. All PMCsseem to predict roughly the same triton point charge radius, with the exception of the χ EFTresult from Ref. [63], which favors the triton point charge radius using the proton chargeradius from muonic hydrogen. None of the PMC values have any error estimates. TheEFT( /π ) predictions agree with the triton point charge radius within their respective errors. χ EFT seems to agree quite well with experiment and also has a small error. However,estimating the error with cutoff variation should be done with caution [71]. The three-body terms using the χ EFT potential in Ref. [63] are clearly not fit exactly to the triton bindingenergy. For further details of how their three-body parameters are chosen consult Ref. [67] II. CONCLUSIONS
Building upon the work of Hagen et al. [32] I have introduced a technique to treatperturbative corrections to bound-state calculations for EFTs of short range interactions.This work focused on the use of these techniques in EFT( /π ), but they are equally usefulfor halo EFT or cold atom systems. In addition, this new technique leads to numericalsimplifications in calculating nd scattering amplitudes and the LO three-body force in thedoublet S -wave channel. It also allows the NNLO energy dependent three-body force to befixed to the triton bound-state energy without the need for a limiting procedure [33].Using this new technique the triton point charge radius was calculated to NNLO inEFT( /π ), giving a LO value of 1 . ± .
19 fm, a NLO value of 1 . ± .
08 fm, and aNNLO value of 1 . ± .
03 fm. The LO value disagrees with the experimental extrac-tion of 1 . ± .
040 fm [1] by about 40%, which is more than the LO estimated EFT( /π )error of 15%. However, it was found at LO that it agrees with analytical calculations inthe unitary and equal mass limit [3]. At NLO the value of 1 . ± .
08 fm agrees with theexperimental extraction within the expected 5% error. The error for the NNLO value comesfrom the expected 1.5% error at NNLO in EFT( /π ) and from the slight cutoff variation ofthe calculation. Within these errors the NNLO prediction of 1 . ± .
03 fm agrees with theexperimental extraction. Future work should address the cutoff variation at NNLO, and seeif the results actually converge as a function of cutoff. In addition future work should carryout a more rigorous error analysis by means of Bayesian statistics [72].Fitting the three-body force to the triton binding energy in the unitary limit the tritonpoint charge radius is 1 .
05 fm. Including the proper NN scattering lengths gives the LOvalue 1 .
14 fm, and including range corrections up to NNLO gives the value 1 . ± .
03 fm.Thus range corrections give significant contributions to the triton point charge radius withrespect to the unitary limit. Despite this, a controlled expansion in terms of a finite numberof parameters from the unitary limit is observed, and therefore the triton can be thought ofas being in the so called “Efimov window” [73].Future work will also consider the He point charge radius, which in the absence ofCoulomb is the isospin mirror of the current calculation presented here. Coulomb effects canbe included in this formalism straightforwardly either perturbatively or nonperturbatively.For a description of He it should be sufficient to treat Coulomb fully perturbatively [31].34n addition future work will consider the magnetic moments of the triton and He as wellas their magnetic radii. The magnetic radii are of interest because they will be measured togreater precision in upcoming experiments using spectroscopy of µ He + [74]. EFT( /π ) offersa way to make precision calculations for these observables in a controlled expansion matchedon to low energy nuclear observables. Acknowledgments
I would like to thank Roxanne Springer, Thomas Mehen, Daniel Phillips, Hans-WernerHammer, Bijaya Acharya, and Chen Ji for useful discussions during the course of this work.In addition I would also like to thank the ExtreMe Matter Institute EMMI at the GSIHelmholtz Centre for Heavy Ion Research and the Institute for Nuclear Theory INT program16-01: “Nuclear Physics from Lattice QCD” for support during the completion of this work.This material is based upon work supported by the U.S. Department of Energy, Office ofScience, Office of Nuclear Physics, under Award Number DE-FG02-05ER41368 and AwardNumber DE-FG02-93ER40756
Appendix A:
The function ( χ jia ( · · · )) µανβ is given by (cid:16) χ jia ( E, (cid:126) K , (cid:126) P , p , k , (cid:126) p , (cid:126) k ) (cid:17) µανβ = ie (2 π ) δ ( k − p ) δ (3) (cid:18) (cid:126) k − (cid:126) p − (cid:126) Q (cid:19) (A1) × i D (0) (cid:18) E + k , (cid:126) k + 23 (cid:126) K (cid:19) i E − k − ( (cid:126) k − (cid:126) K ) M N + i(cid:15) × i E − k − ( (cid:126) k − (cid:126) Q − (cid:126) P ) M N + i(cid:15) (cid:18) τ (cid:19) µν δ αβ δ ij , where α ( β ) is the initial (final) nucleon spin, µ ( ν ) the initial (final) nucleon isospin, and i ( j ) the initial (final) dibaryon polarization. Using the projection operators as defined inRef. [22] to project the c.c. space spin-isospin operator into the doublet S -wave channelyields 13 (cid:32) σ j τ B (cid:33) (cid:0) τ (cid:1) δ ij (cid:0) τ (cid:1) δ AB (cid:32) σ i τ A (cid:33) = (cid:32) (cid:33) . (A2)35hus the function χ a ( · · · ) is a matrix in c.c. space given by χ a ( E, (cid:126) K , (cid:126) P , p , k , (cid:126) p , (cid:126) k ) = ie (2 π ) δ ( k − p ) δ (3) (cid:18) (cid:126) k − (cid:126) p − (cid:126) Q (cid:19) (A3) × i D (0) (cid:18) E + k , (cid:126) k + 23 (cid:126) K (cid:19) i E − k − ( (cid:126) k − (cid:126) K ) M N + i(cid:15) × i E − k − ( (cid:126) k − (cid:126) Q − (cid:126) P ) M N + i(cid:15) (cid:32) (cid:33) . Plugging χ a ( · · · ) into Eq. (67) the integration over d p is removed by the delta functions.Integrating over the energy pole the integration over dk leaves only a d k integration. NextEq. (68) is used to rewrite the triton vertex function in the boosted frame in terms of thetriton vertex function in the c.m. frame. The momentum (cid:126) k from Eq. (67) and momentum (cid:126) q from Eq. (68) are interchanged, and then (cid:126) q → (cid:126) q + (cid:126) Q . This shift makes the time reversalsymmetry of the expressions manifest. Finally, integrating over the azimuthal angle of (cid:126) q leaves a double integral for the analytical forms of the functions A n ( · · · ) and A n ( Q ) whichare given by A n ( p, k, Q ) = M N (cid:12)(cid:12)(cid:12) (cid:90) Λ0 dqq (cid:90) − dx qQx kp (cid:113) q + qQx + Q (cid:113) q − qQx + Q (A4) × Q k + q + Q + ( y − ) qQx − M N B k (cid:113) q + qQx + Q Q p + q + Q + ( y − ) qQx − M N B p (cid:113) q − qQx + Q × D ( n ) s (cid:18) B − q M N − Q M N + (cid:18) − y (cid:19) qQxM N , (cid:126) q (cid:19) (cid:32) − − (cid:33) , A n ( p, Q ) = − M N π (cid:12)(cid:12)(cid:12) (cid:90) Λ0 dqq (cid:90) − dx qQx p (cid:113) q − qQx + Q (A5) × Q p + q + Q + ( y − ) qQx − M N B p (cid:113) q − qQx + Q × D ( n ) s (cid:18) B − q M N − Q M N + (cid:18) − y (cid:19) qQxM N , (cid:126) q (cid:19) (cid:32) − (cid:33) , Note all of the functions here should be similar to those found in Hagen et al. [32], in the limit wherethe core mass equals the neutron mass. However, where I find the term Q / (12 M N ) in the dibaryonpropagator for the functions A n ( · · · ) and A n ( Q ) they find Q / (8 M N ). A n ( Q ) = M N π (cid:12)(cid:12)(cid:12) (cid:90) Λ0 dqq (cid:90) − dx qQx D ( n ) s (cid:18) B − q M N − Q M N + (cid:18) − y (cid:19) qQxM N , (cid:126) q (cid:19) , (A6)where (cid:12)(cid:12)(cid:12) f ( y ) = f (1) − f (0) . (A7)The matrix (vector) of the function A n ( p, k, Q ) ( A n ( p, Q )) is defined in c.c. space. Toobtain the c.c. space matrix for A n ( p, k, Q ) the c.c. space matrix from χ a ( · · · ) is multipliedon either side by a c.c. space matrix from the LO kernel leading to (cid:32) − − (cid:33) (cid:32) (cid:33) (cid:32) − − (cid:33) = (cid:32) − − (cid:33) , (A8)giving the c.c. space matrix as defined in Eq. (A4).The function ( χ jib ( · · · )) µανβ is given by (cid:16) χ jib ( E, (cid:126) K , (cid:126) P , p , k , (cid:126) p , (cid:126) k ) (cid:17) µανβ = i πeM N iD (0) x (cid:18) E + k , (cid:126) k + 23 (cid:126) K (cid:19) i E − k − ( (cid:126) k − (cid:126) K ) M N + i(cid:15) (A9) × i E − p − ( (cid:126) p − (cid:126) P ) M N + i(cid:15) i E + k + p − ( (cid:126) k + (cid:126) p − (cid:126) Q + (cid:126) K ) M N + i(cid:15) × i E + k + p − ( (cid:126) k + (cid:126) p + (cid:126) Q + (cid:126) P ) M N + i(cid:15) iD (0) w (cid:18) E + p , (cid:126) p + 23 (cid:126) P (cid:19) (cid:20) P ( w ) i † (cid:18) τ (cid:19) P ( x ) j (cid:21) αµβν , where P ( x ) j = √ P j ( P ( x ) j = √ P j ) for x = t ( x = s ) in the spin-triplet iso-singlet (spin-singlet iso-triplet) channel. Here the indices “ i ” and “ j ” are either spinor or isospinorindices depending on the values of ( x ) and ( w ). The values of ( x ) and ( w ) pick out thematrix element of ( χ jib ( · · · )) µανβ in c.c. space. Projecting ( χ jib ( · · · )) µανβ onto the doublet S -wave channel gives χ b ( E, (cid:126) K , (cid:126) P , p , k , (cid:126) p , (cid:126) k ) = i πeM N i D (0) (cid:18) E + k , (cid:126) k + 23 (cid:126) K (cid:19) i E − k − ( (cid:126) k − (cid:126) K ) M N + i(cid:15) (A10) × i E − p − ( (cid:126) p − (cid:126) P ) M N + i(cid:15) i E + k + p − ( (cid:126) k + (cid:126) p − (cid:126) Q + (cid:126) K ) M N + i(cid:15) × i E + k + p − ( (cid:126) k + (cid:126) p + (cid:126) Q + (cid:126) P ) M N + i(cid:15) (cid:32) − (cid:33) i D (0) (cid:18) E + p , (cid:126) p + 23 (cid:126) P (cid:19) χ b ( · · · ) into Eq. (67) and then integrating over the energy poles removes the dp and dk integrals. After performing these integrations the LO triton vertex functions arealready in the c.m. frame, leaving only six integrations to be performed. Integrating over oneof the azimuthal angles and noting that Eq. (71) already has two integrations, the function B ( p, k, Q ) has three remaining integrals and is defined by B ( p, k, Q ) = − M N (cid:90) − dx (cid:90) − dy (cid:90) π dφ (A11) × k + p + kp (cid:16) xy + √ − x (cid:112) − y cos φ (cid:17) − Q ( kx + 2 py ) + Q − M n B × k + p + kp (cid:16) xy + √ − x (cid:112) − y cos φ (cid:17) + Q (2 kx + py ) + Q − M n B × (cid:32) − (cid:33) . Time reversal symmetry in this expression is immediately apparent as it is invariant underthe transformation k ←→ p , and Q → − Q .The function ( χ jic ( · · · )) µανβ is given by (cid:16) χ jic ( E, (cid:126) K , (cid:126) P , p , k , (cid:126) p , (cid:126) k ) (cid:17) µανβ = (A12) i eM N Q (2 π ) δ ( k − p ) δ (3) (cid:18) (cid:126) p − (cid:126) k − (cid:126) Q (cid:19) i E − k − ( (cid:126) k − (cid:126) K ) M N + i(cid:15) × arctan Q (cid:113) ( (cid:126) k + (cid:126) K ) − M N E − M N k + 2 (cid:113) ( (cid:126) k + (cid:126) Q + (cid:126) K ) − M N E − M N k × iD (0) w (cid:18) E + k , (cid:126) k + 23 (cid:126) K (cid:19) iD (0) x (cid:18) E + k , (cid:126) k + (cid:126) Q + 23 (cid:126) K (cid:19) Tr (cid:20) P ( x ) j (cid:18) τ (cid:19) P ( w ) i † (cid:21) δ αβ δ µν , which projected onto the doublet S -wave channel gives χ c ( E, (cid:126) K , (cid:126) P , p , k , (cid:126) p , (cid:126) k ) = (A13) i eM N Q (2 π ) δ ( k − p ) δ (3) (cid:18) (cid:126) p − (cid:126) k − (cid:126) Q (cid:19) i E − k − ( (cid:126) k − (cid:126) K ) M N + i(cid:15) × arctan Q (cid:113) ( (cid:126) k + (cid:126) K ) − M N E − M N k + 2 (cid:113) ( (cid:126) k + (cid:126) Q + (cid:126) K ) − M N E − M N k × i D (0) (cid:18) E + k , (cid:126) k + 23 (cid:126) K (cid:19) (cid:32) (cid:33) i D (0) (cid:18) E + k , (cid:126) k + (cid:126) Q + 23 (cid:126) K (cid:19) , dk leaves only the integration d k .After this, one LO triton vertex function is in the c.m. frame and the other is not and mustbe rewritten using Eq. (68). Integrating over the azimuthal angle the functions C n ( · · · ) aregiven by C n ( p, k, Q ) = − M N πQ (cid:90) − dx (A14) × arctan Q (cid:113) k − M N B + 2 (cid:113) k + Qkx + Q − M N B × p (cid:113) k + kQx + Q Q p + k + kQx + Q − M N B p (cid:113) k + kQx + Q × (cid:32) − − (cid:33) D ( n ) (cid:18) B − k M N − Qkx M N − Q M N , k (cid:19) , and C n ( k, Q ) = M N Q (cid:90) − dx (A15) × arctan Q (cid:113) k − M N B + 2 (cid:113) k + Qkx + Q − M N B × (cid:32) − (cid:33) T D ( n ) (cid:18) B − k M N − Qkx M N − Q M N , k (cid:19) . In the current form of the functions C n ( · · · ) time reversal invariance is not immediatelyapparent. Recasting these expressions into an immediately apparent time reversal invari-ant form requires shifting momentum before integrating out angles. However, the gain inanalytical insight is outweighed by the loss in numerical efficiency and the form above iskept.Diagram (d) is essentially diagram (c) without the two-body sub-diagram and therefore39 χ jid ( · · · )) µανβ is similar to ( χ jic ( · · · )) µανβ and is given by (cid:16) χ jid ( E, (cid:126) K , (cid:126) P , p , k , (cid:126) p , (cid:126) k ) (cid:17) µανβ = (A16) ie (2 π ) δ ( k − p ) δ (3) (cid:18) (cid:126) p − (cid:126) k − (cid:126) Q (cid:19) i E − k − ( (cid:126) k − (cid:126) K ) M N + i(cid:15) × iD (0) w (cid:18) E + k , (cid:126) k + 23 (cid:126) K (cid:19) iD (0) x (cid:18) E + k , (cid:126) k + (cid:126) Q + 23 (cid:126) K (cid:19) T ijwx δ αβ δ µν , where T ijwx = δ wx ( c (0)0 t δ wt δ ij + c (0)0 s δ ws δ i δ j ). The function δ wt picks out the contributionfrom the spin-triplet dibaryon and δ ws from the spin-singlet dibaryon. The indices i and j in δ i δ j are isospin indices and correspond to the fact that only the the np spin-singletdibaryon is charged and not the nn spin-singlet dibaryon. Projecting ( χ jid ( · · · )) µανβ onto thedoublet S -wave channel yields χ d ( E, (cid:126) K , (cid:126) P , p , k , (cid:126) p , (cid:126) k ) = (A17) ie (2 π ) δ ( k − p ) δ (3) (cid:18) (cid:126) p − (cid:126) k − (cid:126) Q (cid:19) i E − k − ( (cid:126) k − (cid:126) K ) M N + i(cid:15)i D (0) (cid:18) E + k , (cid:126) k + 23 (cid:126) K (cid:19) c (0)0 t c (0)0 s i D (0) (cid:18) E + k , (cid:126) k + (cid:126) Q + 23 (cid:126) K (cid:19) The calculation of the functions D n ( · · · ) is analogous to the calculation of C n ( · · · ) andyields D n ( p, k, Q ) = π (cid:90) − dx (A18) × p (cid:113) k + kQx + Q Q p + k + kQx + Q − M N B p (cid:113) k + kQx + Q × n (cid:88) j =1 c ( j − t − c ( j − s − c ( j − t c ( j − s D ( n − j ) (cid:18) B − k M N − Qkx M N − Q M N , k (cid:19) , and D n ( k, Q ) = − (cid:90) − dx (A19) × n (cid:88) j =1 c ( j − t − c ( j − s T D ( n − j ) (cid:18) B − k M N − Qkx M N − Q M N , k (cid:19) . χ e ( · · · ) and χ f ( · · · ) are the same as χ a ( · · · ), but with the LO dibaryonpropagator replaced by its corresponding NLO and NNLO correction. The NLO and NNLOresults for type (a) diagrams Eqs. (76), (83), (A4), (A5), and (A6) already contain thesubtraction of diagrams (e) and (f) and therefore χ e ( · · · ) and χ f ( · · · ) are not shown. Appendix B:
Expanding the scalar function A n ( Q ) as a function of Q and picking out the Q contri-bution gives 12 ∂ ∂Q A n ( Q ) (cid:12)(cid:12)(cid:12) Q =0 = 23 (cid:90) Λ0 dqq f n ( q ) , (B1)where f ( q ) = M N π (cid:101) D D (cid:110) q ( D − D (cid:101) D + 2 (cid:101) D ) + 4 D (cid:101) D (3 (cid:101) D − γ s ) (cid:111) , (B2) f ( q ) = ( Z s − f ( q ) , (B3)and f ( q ) = (cid:18) Z s − γ s (cid:19) (cid:20)(cid:16) (cid:101) D − γ s (cid:17) f ( q ) + M N π (cid:101) D D (cid:110) (cid:101) D D − q ( γ s − (cid:101) D ) (cid:111)(cid:21) . (B4)The variables D and (cid:101) D are given by (cid:101) D = (cid:114) q − M N E , D = γ s − (cid:101) D. (B5)Extracting the Q part of the c.c. space vector functions A n ( p, Q ) gives12 ∂ ∂Q A n ( p, Q ) (cid:12)(cid:12)(cid:12) Q =0 = (cid:90) Λ0 dqq f n ( p, q ) (cid:32) − (cid:33) , (B6)where f ( p, q ) = − πf ( q ) 1 pq Q ( a ) (B7) − M N π D pq ) (cid:26) a (1 − a ) + (cid:20)(cid:18) qp + pq (cid:19) (1 + 3 a ) − a (3 + a ) (cid:21) − a ) (cid:27) − M N π (cid:101) D D pq ) (cid:26) (cid:101) D D (cid:20) − a + (cid:18)(cid:18) qp + 8 pq (cid:19) a − a ) (cid:19) − a ) (cid:21) − ( γ s − (cid:101) D ) 92 q − a (cid:27) , ( p, q ) = (cid:18) Z s − γ s (cid:19) (cid:20) ( γ s + (cid:101) D ) f ( p, q ) − πDf ( q ) 1 pq Q ( a ) (B8) − M N π (cid:101) D D pq ) (cid:26)(cid:20) (cid:101) D D − q ( γ s − (cid:101) D ) (cid:21) − a − (cid:101) D D (cid:20) a ) − (cid:18) qp + 8 pq (cid:19) a (cid:21) − a ) (cid:27)(cid:21) , and f ( p, q ) = (cid:18) Z s − γ s (cid:19) (cid:104) ( (cid:101) D − γ s ) f ( p, q ) (B9) − M N π (cid:101) D D (cid:110) (cid:101) D D − q ( γ s − (cid:101) D ) (cid:111) pq Q ( a ) − M N π (cid:101) DD pq ) (cid:26)(cid:104) (cid:101) DD + 9 q (cid:105) − a − (cid:101) DD (cid:20) a ) − (cid:18) qp + 8 pq (cid:19) a (cid:21) − a ) (cid:27)(cid:21) . The variable a is defined by a = q + p − M N Eqp . (B10)Pulling out the Q part of the c.c. space matrix functions A n ( p, k, Q ) gives12 ∂ ∂Q A n ( p, k, Q ) (cid:12)(cid:12)(cid:12) Q =0 = (cid:90) Λ0 dqq f n ( p, k, q ) (cid:32) − − (cid:33) , (B11)where f ( p, k, q ) = − π (cid:26) f ( k, q ) 1 pq Q ( a ) + f ( p, q ) 1 kq Q ( b ) (cid:27) − π f ( q ) 1 kq Q ( b ) 1 pq Q ( a ) (B12)+ M N
54 1 (cid:101) DD q k p (cid:26) (cid:101) DD (cid:18)(cid:20) − b )(1 − a ) + 4 qp a (1 − b ) + 4 qk b (1 − a ) (cid:21) + 2 ab (cid:20) kp (1 − b ) + pk (1 − a ) (cid:21) + 2 b kq (cid:2) b − (1 + a ) (cid:3) + 2 a pq (cid:2) a − (1 + b ) (cid:3) +2 kq (cid:18) qp a − (cid:19) (1 − b ) Q ( b ) + 2 pq (cid:16) qk b − (cid:17) (1 − a ) Q ( a ) (cid:19) − b ) (1 − a ) + q (cid:18)(cid:20) kq b + pq a − kq pq ab (cid:21) + kq (1 − b ) (cid:18) − a pq (cid:19) Q ( b )+ pq (1 − a ) (cid:18) − b kq (cid:19) Q ( a ) − kq pq (1 − b )(1 − a ) Q ( b ) Q ( a ) (cid:19) − b ) (1 − a ) (cid:27) , ( p, k, q ) = (cid:18) Z s − γ s (cid:19) ( γ s + (cid:101) D ) f ( p, k, q ) − πf ( k, q ) 1 pq Q ( a ) − πf ( p, q ) 1 kq Q ( b ) (B13)+ (cid:18) Z s − γ s (cid:19) M N
54 1 (cid:101) DD q k p (cid:26)(cid:20) kq b + pq a − kq pq ab (cid:21) + kq (1 − b ) (cid:18) − a pq (cid:19) Q ( b ) + pq (1 − a ) (cid:18) − b kq (cid:19) Q ( a ) − kq pq (1 − b )(1 − a ) Q ( b ) Q ( a ) (cid:27) − b )(1 − a )+ 2 π (cid:18) Z s − γ s (cid:19) ( γ s + (cid:101) D ) (cid:20) f ( k, q ) 1 pq Q ( a ) + f ( p, q ) 1 kq Q ( b ) (cid:21) − π (cid:18) f ( q ) − (cid:18) Z s − γ s (cid:19) ( γ s + (cid:101) D ) f ( q ) (cid:19) pq Q ( a ) 1 kq Q ( b ) , and f ( p, k, q ) = (cid:18) Z s − γ s (cid:19) ( (cid:101) D − γ s ) f ( p, k, q ) − πf ( k, q ) 1 pq Q ( a ) − πf ( p, q ) 1 kq Q ( b )(B14)+ (cid:18) Z s − γ s (cid:19) M N
27 1 D q k p (cid:26)(cid:20) kq b + pq a − kq pq ab (cid:21) + kq (1 − b ) (cid:18) − a pq (cid:19) Q ( b ) + pq (1 − a ) (cid:18) − b kq (cid:19) Q ( a ) − kq pq (1 − b )(1 − a ) Q ( b ) Q ( a ) (cid:27) − b )(1 − a )+ 2 π (cid:18) Z s − γ s (cid:19) ( (cid:101) D − γ s ) (cid:20) f ( k, q ) 1 pq Q ( a ) + f ( p, q ) 1 kq Q ( b ) (cid:21) − π (cid:32) f ( q ) − (cid:18) Z s − γ s (cid:19) ( (cid:101) D − γ s ) f ( q ) (cid:33) pq Q ( a ) 1 kq Q ( b ) . The variable b is defined as b = q + k − M N Eqk . (B15)Extracting the Q part of the c.c. space matrix function B ( p, k, Q ) gives12 ∂ ∂Q B ( p, k, Q ) (cid:12)(cid:12)(cid:12) Q =0 = − M N π p k − a ) (B16) × (cid:26) a − a − a − p + k pk a − a (cid:27) (cid:32) − (cid:33) , where a = p + k − M N Epk . (B17)43he Q part of the c.c. space vector function C n ( k, Q ) is12 ∂ ∂Q C n ( k, Q ) (cid:12)(cid:12)(cid:12) Q =0 = g ( n ) t ( k ) − g ( n ) s ( k ) T , (B18)where g (0) { t,s } ( k ) = M N (cid:101) D D { t,s } (cid:110) (cid:101) D D { t,s } (2 (cid:101) D − γ { t,s } ) + k ( γ { t,s } − (cid:101) D ) D { t,s } + 2 k (cid:101) D (cid:111) , (B19) g (1) { t,s } ( k ) = (cid:18) Z { t,s } − γ { t,s } (cid:19) (cid:34) ( γ { t,s } + (cid:101) D ) g (0) { t,s } ( k ) (B20)+ M N (cid:101) D D { t,s } (cid:110) (cid:101) D D { t,s } + k ( (cid:101) D − D { t,s } ) (cid:111)(cid:35) , and g (2) { t,s } ( k ) = (cid:18) Z { t,s } − γ { t,s } (cid:19) (cid:34) ( (cid:101) D − γ { t,s } ) g (0) { t,s } ( k ) (B21)+ M N (cid:101) D D { t,s } (cid:26) (cid:101) D D { t,s } + k (cid:18) (cid:101) D − D { t,s } (cid:19)(cid:27)(cid:35) . For these functions and all functions below in this appendix, a is given by Eq. B17 and thevariables (cid:101) D , D t , and D s are defined as (cid:101) D = (cid:114) k − M N E , D t = γ t − (cid:101) D , D s = γ s − (cid:101) D. (B22)Note the notation { t, s } is a shorthand for two different functions one with subscript t andthe other with subscript s . The Q dependence of the c.c. space matrix function C n ( p, k, Q )is given by 12 ∂ ∂Q C n ( p, k, Q ) (cid:12)(cid:12)(cid:12) Q =0 = g ( n ) t ( p, k ) − g ( n ) s ( p, k ) − g ( n ) t ( p, k ) g ( n ) s ( p, k ) , (B23)where g (0) { t,s } ( p, k ) = − πg (0) { t,s } ( k ) 1 pk Q ( a ) (B24) − M N π (cid:101) DD { t,s } pk (cid:26) pk − a + 1 p (cid:18) a + a (cid:16) pk (cid:17) − pk (1 + a ) (cid:19) − a ) (cid:27) − M N π kp (cid:101) D D { t,s } (cid:26) k Q ( a ) − pk − pk a − a (cid:27) (cid:104) γ { t,s } − (cid:101) D (cid:105) , (1) { t,s } ( p, k ) = (cid:18) Z { t,s } − γ { t,s } (cid:19) (cid:34) ( γ { t,s } + (cid:101) D ) g (0) { t,s } ( p, k ) (B25) − M N π (cid:101) D D { t,s } pk Q ( a ) (cid:110) (cid:101) D D { t,s } + k ( (cid:101) D − D { t,s } ) (cid:111) − kp M N π (cid:101) D D { t,s } (cid:26) pk − a − k a − a − k Q ( a ) (cid:27)(cid:35) , and g (2) { t,s } ( p, k ) = (cid:18) Z { t,s } − γ { t,s } (cid:19) (cid:104) ( (cid:101) D − γ { t,s } ) g (0) { t,s } ( p, k ) (B26) − M N π (cid:101) D D { t,s } pk Q ( a ) (cid:26) (cid:101) D D { t,s } + k (cid:18) (cid:101) D − D { t,s } (cid:19)(cid:27) − kp M N π (cid:101) DD { t,s } (cid:26) pk − a − k a − a − k Q ( a ) (cid:27)(cid:35) . Extracting the Q term of the c.c. space vector function D n ( k, Q ) gives12 ∂ ∂Q D n ( k, Q ) (cid:12)(cid:12)(cid:12) Q =0 = h ( n ) t ( k ) c (0)0 t − h ( n ) s ( k ) c (0)0 s T , (B27)where h (1) { t,s } ( k ) = − (cid:101) D D { t,s } (cid:110) (cid:101) D D { t,s } + k (3 (cid:101) D − γ { t,s } ) (cid:111) (B28)and h (2) { t,s } ( k ) = 0 (B29)Note there is no n = 0 value for the D n ( · · · ) functions. Finally, the Q piece of the c.c. spacematrix function D n ( p, kQ ) is given by12 ∂ ∂Q D n ( p, k, Q ) (cid:12)(cid:12)(cid:12) Q =0 = h ( n ) t ( p, k ) c (0)0 t − h ( n ) s ( p, k ) c (0)0 s − h ( n ) t ( p, k ) c (0)0 t h ( n ) s ( p, k ) c (0)0 s , (B30)where h (1) { t,s } ( p, k ) = − πh (1) { t,s } ( k ) 1 pk Q ( a ) (B31)+ 2 π D { t,s } pk ) (cid:20)(cid:18) kp + pk (cid:19) a − a − (cid:21) − a ) − π (cid:101) DD { t,s } pk (cid:40) Q ( a ) + a − kp − a (cid:41) , h (2) { t,s } ( p, k ) = − (cid:18) Z { t,s } − γ { t,s } (cid:19) (cid:20) D { t,s } h (1) { t,s } ( p, k ) + 2 πD { t,s } h (1) { t,s } ( k ) 1 pk Q ( a ) (B32) − π (cid:101) DD { t,s } pk (cid:26)(cid:20) kp − a (cid:21) − a − Q ( a ) (cid:27)(cid:35) Appendix C:
Taking the limit Q → − ieF ( a )0 (0) = − ieπ M N (cid:16)(cid:101) Γ ( q ) (cid:17) T ⊗ q δ ( q − (cid:96) ) (cid:113) q − M N B (cid:32) (cid:33) ⊗ (cid:101) Γ ( (cid:96) ) (C1)+ i πeM N (cid:16)(cid:101) Γ ( q ) (cid:17) T ⊗ q (cid:96) − ( q + (cid:96) − M N B ) (cid:32) − − (cid:33) ⊗ (cid:101) Γ ( (cid:96) ) , where (cid:101) Γ ( q ) = D (0) (cid:18) B − q M N , q (cid:19) Γ ( q ) . (C2)In order to obtain the expression for F ( a )0 (0) it is easiest to take the limit Q → Q → − ieF ( b )0 (0) = − i πeM N (cid:16)(cid:101) Γ ( q ) (cid:17) T ⊗ q (cid:96) − ( q + (cid:96) − M N B ) (cid:32) − (cid:33) ⊗ (cid:101) Γ ( (cid:96) ) , (C3)and for the LO diagram (c) − ieF ( c )0 (0) = − ieπ M N (cid:16)(cid:101) Γ ( q ) (cid:17) T ⊗ q δ ( q − (cid:96) ) (cid:113) q − M N B (cid:32) (cid:33) ⊗ (cid:101) Γ ( (cid:96) ) . (C4)Combining all these terms the total LO triton charge form factor in the limit Q → F (0) = 2 πM N (cid:16)(cid:101) Γ ( q ) (cid:17) T ⊗ π q δ ( q − (cid:96) ) (cid:113) q − M N B (cid:32) (cid:33) (C5) − q (cid:96) − ( q + (cid:96) − M N B ) (cid:32) − − (cid:33)(cid:41) ⊗ (cid:101) Γ ( (cid:96) ) . F (0) derived here. Therefore, it automatically follows that F (0) = 1 if thetriton vertex function is properly renormalized. Appendix D
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