Effective field theory for dilute Fermi systems at fourth order
EEffective field theory for dilute Fermi systems at fourth order
C. Wellenhofer,
1, 2, ∗ C. Drischler,
3, 4, 5, † and A. Schwenk
1, 2, 6, ‡ Technische Universit¨at Darmstadt, Department of Physics, 64289 Darmstadt, Germany ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, 64291 Darmstadt, Germany Facility for Rare Isotope Beams, Michigan State University, MI 48824, United States of America Department of Physics, University of California, Berkeley, CA 94720, United States of America Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, United States of America Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany
We discuss high-order calculations in perturbative effective field theory for fermions at low energyscales. The Fermi-momentum or k F a s expansion for the ground-state energy of the dilute Fermi gasis calculated to fourth order, both in cutoff regularization and in dimensional regularization. For thecase of spin one-half fermions we find from a Bayesian analysis that the expansion is well-convergedat this order for | k F a s | (cid:46) .
5. Further, we show that Pad´e-Borel resummations can improve the con-vergence for | k F a s | (cid:46)
1. Our results provide important constraints for nonperturbative calculationsof ultracold atoms and dilute neutron matter.
I. INTRODUCTION
Over the last two decades, striking progress inquantum many-body physics has been achieved espe-cially through well-controlled experiments with ultracoldatoms and the development of efficient computationalmethods. Parallel to this, the conception of effectivefield theory (EFT) has equipped advanced many-bodycalculations with a firm theoretical basis. Here, we makea new contribution to these advances by providing an-alytic EFT results at high orders for a central problemof many-body theory and experiment: the ground-stateenergy of the dilute Fermi gas.Effective field theory is deeply connected with the no-tion of universality, for which the dilute Fermi gas is aclassic example. This universal many-body system de-scribes both the physics of cold atomic gases as well asthat of the dilute nuclear matter present in the crust ofneutron stars. In ultracold-atom experiments, Feshbachresonances allow one to tune the interaction strength viathe application of external fields. This makes it possibleto probe low-density Fermi systems over a wide rangeof many-body dynamics, in particular at the unitarylimit of infinite scattering length and through the BCS-BEC crossover [1–5]. Moreover, continuous progress withquantum Monte Carlo (QMC) methods [4, 6, 7] has en-abled computations of strongly interacting dilute Fermigases with a high precision comparable to that of ex-perimental measurements. High-order analytic calcula-tions that provide precision benchmarks for QMC andexperiment represent an important tool for making fur-ther progress in this field. This is the focus of the presentwork.Effective field theory provides the basis for such ana-lytic benchmark calculations. In this context, the prob-lem of renormalization, which historically has presented a ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] notable barrier for many-body calculations at high ordersin perturbation theory, has been cleared up completely(in the perturbative case) [8–10]. While perturbativeEFT calculations are generally restricted to low densitiesand weak interactions, respectively, they are still usefulin many ways. Regarding the nuclear many-body prob-lem [11–15], they provide viable input for constrainingnuclear matter computations and neutron star model-ing. Via resummation methods, they also give accessto approximate analytic results of large-scattering lengthphysics.Here, we present in detail the calculation and re-sults to fourth order in the perturbative EFT for zero-temperature many-fermion systems at very low energies,i.e., the renowned Fermi-momentum or k F a s expansionfor the ground-state energy of the dilute Fermi gas [16–26]. In that, we follow up on our recent Letter [27] wherethe first fourth-order results have been presented. In thepresent paper, we expand substantially on the results andpresentation of Ref. [27]. First, in Sec. II we discuss inmore detail the contact EFT formalism for fermions atvery low energy scales. In Sec. III we then present thedetails of the calculation of the Fermi-momentum expan-sion to fourth order for the case of spin one-half fermions.The case of spins greater than one-half is examined in de-tail in Sec. IV using two different regularization schemes:cutoff regularization and dimensional regularization. Ourfourth-order results for the ground-state energy of thegeneral dilute Fermi gas are then summarized in Sec. V.Using Bayesian methods, in Sec. VI we investigate theconvergence of the Fermi-momentum expansion. There,we also study various Pad´e and Borel approximants con-structed from the expansion. Finally, Sec. VII providesa short summary. We note the following typos in Ref. [27]: in Eq. (21) and (25) afactor M is missing, and below Eq. (24) it should read II6(ii)instead of III6(ii). a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b II. EFFECTIVE FIELD THEORY FORNONRELATIVISTIC FERMIONS
The effective field theory (EFT) Lagrangian L EFT fordilute Fermi systems is composed of the most generaltwo- and multi-fermion contact interactions consistentwith Galilean invariance, parity, and time-reversal invari-ance. Up to field redefinitions, its leading terms are givenby (see, e.g., Refs. [10, 28–32]) L EFT = ψ † (cid:20) i∂ t + −→ ∇ M (cid:21) ψ − C ψ † ψ ) + C (cid:104) ( ψψ ) † ( ψ ←→ ∇ ψ ) + h.c. (cid:105) + C (cid:48) ψ ←→ ∇ ψ ) † · ( ψ ←→ ∇ ψ ) − D ψ † ψ ) + . . . , (1)where ψ are nonrelativistic fermion fields, ←→ ∇ = ←− ∇ − −→ ∇ is the Galilean invariant derivative, h.c. the Hermitianconjugate, and M the fermion mass. The couplings of thecontact interactions C , C , C (cid:48) , D , . . . , called low-energyconstants (LECs), have to be fitted to experimental dataor (if possible) matched to an underlying theory. (Forrecent work aimed at rooting contact EFT for nucleonsin lattice QCD calculations, see Refs. [33, 34].)A truncation scheme, known as power counting, is re-quired to organize the (infinite number of) EFT operatorsin a systematic way. In perturbative EFT calculations,the power counting must conform to the requirement thatultraviolet (UV) divergences can be renormalized at eachorder. For the contact EFT given by Eq. (1), this isachieved by ordering contributions in perturbation the-ory according to the (naive) mass dimension σ of theLECs, i.e., σ ( C ( (cid:48) )2 n ) = 2 n + 1 , (2) σ ( D ( (cid:48) )2 n ) = 2 n + 4 , (3) σ ( E ( (cid:48) )2 n ) = 2 n + 7 , (4)etc., where the LECs E n correspond to four-fermion in-teractions.In the following, we first discuss in Sec. II A the rela-tion between N -body scattering diagrams and the MBPTseries for dilute Fermi systems. This is following by theanalysis of UV power divergences and two-body scatter-ing diagrams in Sec. II B. In Sec. II C we then exam-ine the ladder diagrams of MBPT. Next, in Sec. II D westudy the renormalization of logarithmic UV divergencesand the associated nonanalytic terms in the perturbativeEFT expansion. Finally, Sec. II E briefly discusses differ-ent partial resummations for systems with a large S -wavescattering length. A. Renormalization from few-body tomany-body systems
The nonrelativistic field theory specified by the La-grangian L EFT is equivalent to a Hamiltonian approachwith N -body potentials. The regularized two- and three-body potentials are given by (cid:104) p (cid:48) | V (2)EFT | p (cid:105) = (cid:104) C (Λ) + C (Λ)( p (cid:48) + p ) / C (cid:48) (Λ) p (cid:48) · p + . . . (cid:105) f ( p/ Λ) f ( p (cid:48) / Λ) , (5) (cid:104) p (cid:48) q (cid:48) | V (3)EFT | pq (cid:105) = (cid:104) D (Λ) + . . . (cid:105) f ( p/ Λ) f ( q/ Λ) × f ( p (cid:48) / Λ) f ( q (cid:48) / Λ) . (6)Here, p ( (cid:48) ) and q ( (cid:48) ) are relative and Jacobi momenta, re-spectively, and f ( p/ Λ) is a regulator function that sup-presses high-momentum modes. Later we will also con-sider dimensional regularization (DR), but for now weuse a (Galilean invariant) momentum regulator.The superficial degree of divergence d of an N -bodyscattering diagram is given by d = 5 L − I + V (cid:88) j =1 [ σ ( g j ) − , (7)where L is the loop number, I the number of internallines, V the number of vertices, and g j ∈ { C n , D n , . . . } ;see, e.g., Refs. [35, 36] for details. [If there are subdiver-gences the actual degree of divergence can be larger than d .] The MBPT diagrams are obtained from scatteringdiagrams by closing the external lines (and excluding oc-cupied states in loop integrals) of a single scattering dia-gram, or by closing and connecting the external lines ofseveral diagrams. Since the hole propagators associatedwith closed external lines are bounded (or exponentiallydecaying at finite temperature), the renormalization ofMBPT follows from the renormalization of scattering di-agrams. For nonrelativistic contact interactions, N -bodyscattering diagrams can have only up to N intermediatelines between adjacent vertices, so only N (cid:48) -body inter-actions with N (cid:48) (cid:54) N appear in a given the N -body sec-tor. This implies that the renormalization of the EFTinteractions can be set up hierarchically, starting fromthe renormalization of two-body interactions in the two-body sector, then three-body, and so on, up to a giventruncation order in the power counting. B. Two-body scattering
In the nonrelativistic EFT, the only two-body scatter-ing diagrams are ladder diagrams (corresponding to iter-ations of the Lippmann-Schwinger equation), see Fig. 1.This makes the two-body sector very simple: all loop + + + + . . .
Figure 1. The two-body scattering diagrams. By closing the external lines one obtains the particle-particle ladder diagramsof MBPT. The momentum integration associated with the closed lines has the effect that the (MBPT) ladder series has zeroradius of convergence (renormalon divergence), in contrast to the series of two-body scattering diagrams (a geometric series).See the text for details. integrals factorize, with factors J n ( k, Λ) given by J n ( k, Λ) = ∞ (cid:90) dq q n k − q + i(cid:15) f ( q/ Λ) . (8)To extract the power divergence we rescale the loop mo-mentum as q → q / Λ, leading to J n ( k, Λ) = I UV n ( k, Λ) + J Rn ( k ) , (9)where J Rn ( k ) = iπ k n +1 (10)and I UV n ( k, Λ) = I UV , ∞ n ( k, Λ) + I UV , n ( k, Λ), with I UV , ∞ n ( k, Λ) = − n (cid:88) m =0 α m Λ m +1 k n − m ) , (11) I UV , n ( k, Λ) Λ →∞ −−−−→ , (12)where α m are regulator-dependent constants. Theeffective-range expansion (ERE) for the on-shell T ma-trix reads [10] T ( k, cos ϑ ) = 4 πM (cid:26) ∞ (cid:88) n =0 τ ( s ) n k n (cid:124) (cid:123)(cid:122) (cid:125) T ( s ) ( k ) + ∞ (cid:88) n =2 τ ( p ) n [ k cos ϑ )] n (cid:124) (cid:123)(cid:122) (cid:125) T ( p ) ( k, cos ϑ ) + . . . (cid:27) , (13)where k and ϑ are the scattering momentum and angle,and τ ( s ) n = { a s , − ia s , − a s + a s r s , i ( a s − a s r s ) ,a s − a s r s + a s r s + a s v s , . . . } , (14) τ ( p ) n = { a p , . . . } , (15)with a s and a p the S - and P -wave scattering length, re-spectively, r s the S -wave effective range, and v s the S -wave shape parameter. Matching the regularized EFTperturbation series to Eq. (13) leads (in the infinite-cutofflimit Λ → ∞ ) to C (Λ) = C + C (cid:88) ν =1 (cid:18) α C M π Λ (cid:19) ν + α C C M π Λ + . . . , (16) C (Λ) = C + α C C Mπ Λ + . . . , (17) C (cid:48) (Λ) = C (cid:48) + . . . , (18) where the cutoff-dependent parts are counterterms thatcancel UV divergences and the omitted terms correspondto counterterms beyond fourth order. Note that all thecounterterms required to renormalize C -only contribu-tions to the T matrix are included in C (Λ); i.e., the C term corresponds to a perturbatively renormalizable in-teraction. For spin multiplicities g >
2, this feature ishowever restricted to the two-body sector (see Sec. II D).The (renormalized) LECs are given by C = 4 πa s M , C = C a s r s , C (cid:48) = 4 πa p M , (19)etc. The perturbative EFT expansion is viable through-out the energy range appropriate to the EFT only if thesize of the LECs conforms to the power counting; i.e., C ∼ M Λ b , C ∼ C (cid:48) ∼ M Λ b , (20)etc., corresponding to a s ∼ r s ∼ a p ∼ / Λ b . Here, Λ b is the “hard scale” beyond which the EFT descriptionbreaks down. The scaling given by Eq. (20) is commonlyreferred to as the “natural” case. The EFT perturba-tion series then corresponds to an expansion in powers of Q/ Λ b . C. Many-body ladder diagrams and renormalons
Closing the external lines of two-body scattering dia-grams, one obtains the particle-particle ( pp ) ladder dia-grams of MBPT. For these diagrams, the factors corre-sponding to the pp bubbles are given by J n ( P, k,
Λ) = (cid:90) d q π q n k − q ¯ n | P − q | / ¯ n | P + q | / f ( q/ Λ) , (21)where ¯ n k = θ ( k − k F ), k F is the Fermi momentum, and q is the relative momentum of the two particle lines in agiven pp bubble. The hole lines correspond to integratingover P and k . The pp bubble can be separated as J n ( P, k ) = I UV n ( k, Λ) + I Rn ( P, k ) , (22)where the cutoff-independent part is given by J Rn ( P, k ) = (cid:90) d q π q n k − q (cid:2) ¯ n | P − q | / ¯ n | P + q | / − (cid:3) = k F P k (cid:12)(cid:12)(cid:12)(cid:12) k F + P − kk F + P + k (cid:12)(cid:12)(cid:12)(cid:12) + k − P − k P ln (cid:12)(cid:12)(cid:12)(cid:12) ( k F + P ) − k k − P − k (cid:12)(cid:12)(cid:12)(cid:12) . (23)Notably, the series of pp ladder diagrams is a di-vergent asymptotic series with zero radius of conver-gence [24, 37, 38]. The physical context of this so-called “renormalon divergence” is the Cooper pairingphenomenon [37]. Mathematically, the divergence is dueto the singularities of J Rn ( P, k ) at the boundaries of thehole-line integrals (i.e., the Lebesgue dominated conver-gence theorem is not satisfied). D. Multi-fermion scattering and logarithms
While the two-body scattering diagrams involve onlyUV power divergences [see Sec. II B], multi-fermion scat-tering involves also logarithmic divergences ∼ ln(Λ /Q ),where Q is an invariant kinematical variable. For scatter-ing diagrams, Q is an external momentum, and in MBPTat zero temperature Q is the Fermi momentum k F . Thatis, logarithmic UV divergences appear with a ratio ofscales, which implies that their coefficients must be reg-ulator independent (in contrast to the coefficients of UVpower divergences), see also Ref. [10]. Renormalizationremoves the dependence on the UV cutoff Λ such thatthe logarithms become ln(Λ /Q ), where Λ is an arbi-trary auxiliary scale [see Sec. IV for details]. The depen-dence on Λ is canceled by the “running” with Λ of themulti-fermion coupling g j associated with the respectivecounterterm. Note that this cancelation requires that theinvolved terms are kept together, i.e., independent par-tial resummations are inhibited by the requirement of Λ independence.For g >
2, the first logarithms in perturbative N -bodyscattering (for N (cid:62)
3) appear from the C interaction atorder 3 N −
5, i.e., at fourth order in the three-body sec-tor. (The first momentum-dependent logarithmic diver-gence appears at order 3 N − D ,etc., conforming to the perturbative EFT power count-ing.) The fourth-order three-body scattering diagramswith logarithmic divergences, Γ and Γ , are shown inFig. 2; the associated many-body diagrams are listed inFig. 3. They are renormalized by the contributions from Note that (in contrast to, e.g., relativistic φ theory [39]) therenormalon divergence occurs for both the renormalized and theregularized perturbation series. The MBPT series has still zeroradius convergence if the ladders are resummed [37, 40]; thelarge-order behavior is however (expected to be) dominated byrenormalons [37]. the leading three-body contact interaction with coupling D (corresponding to the last diagram in Fig. 2). Thisrequires that the cutoff dependence of the D coupling isthen given by D (Λ) = D (Λ ) + ηM C ln(Λ / Λ ) . (24)The regulator-independent coefficient η is obtained fromthe evaluation of the diagrams Γ and Γ (plus thebubble-counterterm diagram Γ ct2 ) of Fig. 2, or equiva-lently, from the evaluation of the corresponding many-body diagrams; see Sec. IV for details. The dependenceof the first term D (Λ ) is such that D (Λ) is indepen-dent of the auxiliary scale Λ . The value of D (Λ ) hasto be fixed (for a given choice of Λ ) by matching to few-or many-body data.For g = 2, all logarithmic divergences from S -wave in-teractions cancel, as required by the Pauli principle (theleading three-body contact interactions are Pauli blockedfor g = 2). That is, for g = 2 the S -wave part of theMBPT series is completely determined by two-body scat-tering (i.e., by the ERE). For P -wave interactions or S -wave interactions in g > N -body couplings is needed for per-turbative renormalization beyond the two-body sector.Finally, we note that the contact interactions betweenfermions can be rewritten such that they involve thepropagation of (so-called) dimer fields, and carrying outthe partial diagrammatic resummations that renormal-ize the dimer propagator makes the C part of the per-turbation series for three-body scattering UV finite alsofor g > D (but no higher-orderthree-body interactions) [41, 42]. (Perturbatively ex-panding the nonperturbative three-body scattering am-plitude then allows one to determine the perturbative D from nonperturbative three-body data [43].) A generalunderstanding of the problem of nonperturbatively renor-malizing irrelevant operators and its relation to the stan-dard (order-by-order) perturbative EFT renormalizationis however still missing [44]. E. Resummations for large scattering length
If there is a two-body bound-state at threshold the S -wave scattering length a s is unnaturally large, and in thiscase the perturbative EFT expansion is of limited use. Inthe two-body sector, this case can be straightforwardlydealt with by resumming the C contributions and adding C , . . . perturbatively, which leads to [28, 45] T ( s ) ( k ) = 1 a s + ik + r s k (cid:16) a s + ik (cid:17) + . . . . (25)Such a simple analytic resummation of the C con-tributions is however not possible for the much more Γ Γ Γ ct2 D Figure 2. The first three-body scattering diagrams with logarithmic divergences Γ and Γ . Also shown is the countertermdiagram Γ ct2 for the pp bubble of Γ (the counterterm is depicted as a shaded blob). The fourth diagram is the leading three-body contact contribution in three-fermion scattering, which includes the counterterm for the logarithmic UV divergences ofΓ and Γ . Closing the external lines one obtains from Γ and Γ the MBPT diagrams with logarithmic divergences II5, II6,IIA1, and III1 shown in Fig. 3 below. See Sec. IV for details on the evaluation of these diagrams. complicated MBPT series. A notable benchmark fornonperturbative many-body treatments of the C termis given by the a s → ∞ limit, corresponding to theunitary Fermi gas. From dimensional analysis it fol-lows that the ground-state energy density of the uni-tary Fermi gas of spin one-half fermions is given by E ( k F ) = ξE ( k F ), where E ( k F ) = k / (10 π M ) is thenoninteracting ground-state energy density and ξ is theBertsch parameter. From experiments with ultracoldatoms [1], the value ξ ≈ . pp ) ladders gives ξ pp ≈ .
237 [32, 46],and resumming also hole-hole ( hh ) and mixed pp - hh lad-ders gives ξ ladders ≈ . In addition, a value for ξ (cid:15) ≈ .
475 was deduced in Ref. [48] by expanding in termsof (cid:15) = 4 − d , where d is the number of space dimensions,and subsequently interpolating between the (cid:15) = 2 and (cid:15) = 0 results for E ( k F ). Even more close comes the value ξ LW ≈ .
36 obtained from a self-consistent Luttinger-Ward type approach with resummed ladders [49] (see alsoRef. [50]). (See also Ref. [40] for finite-temperature cal-culations based on Borel-resummed diagrammatic MonteCarlo calculations.) The most accurate value has beenobtained from QMC computations, ξ QMC = 0 . ξ may also be obtained by applyingresummation methods such as Pad´e approximants to the a s part of the Fermi-momentum expansion [51, 52]. Wewill present results from this approach in Sec. VI B. III. FOURTH-ORDER TERM FORSPIN ONE-HALF FERMIONS
We now start with the discussion of the perturba-tive EFT expansion for dilute many-fermion systems atfourth order. Logarithms and multi-fermion interactionsarise only for spin multiplicities g >
2, the intricacies ofthis case are postponed until Sec. IV. Here, we discuss The resummation of particle-hole ladders (“ring diagrams”) be-comes relevant for large values of g , in particular regarding theexpansion about the large- g limit [47]. the spin one-half case, g = 2, but we leave the notationgeneral such that the results not affected by logarithmicterms can be carried over to g >
2. There are two differ-ent types of contributions at fourth order: (i) the second-order MBPT diagram with one C and one C vertex, and(ii) fourth-order MBPT diagrams with four C vertices(for g > D vertex). In each case, (in cutoff regularization) one hasalso two-body counterm contributions from lower-orderMBPT diagrams.For the calculation of the contribution (i), see Refs. [27,53]. The calculation of the contribution (ii) is much moreinvolved. Among the possible MBPT diagrams with four C vertices, only those without single-vertex loops haveto be considered at zero temperature. This is becauseall diagrams with single-vertex loops are removed byfirst-order mean-field (i.e., Hartree-Fock) insertions [54],and for a momentum-independent interaction, first-ordermean-field renormalization at zero temperature has noeffect for a uniform system. Therefore, as can easily beverified explicitly, these diagrams cancel each other ateach order. The 39 remaining fourth-order many-bodydiagrams can be divided into four topological species: • I(1-6): ladder diagrams, • IA(1-3): ring diagrams, • II(1-12), IIA(1-6): other two-particle irreducible di-agrams, • III(1-12): two-particle reducible diagrams,where we have labeled diagrams according to groupsthat are closed under permutations of the vertices:I(1–6), IA(1–3), II(1–12), IIA(1–6), III(1–12). DiagramsIII(3,6,11,12) are anomalous and thus give no contribu-tion in zero-temperature MBPT [54, 55]. The 33 remain-ing diagrams are shown in Fig. 3.Diagrams I1, I6, and IA1 are the fourth-order versionsof the third-order pp , hh , and ph diagrams; see, e.g.,Ref. [56]. Diagrams I(2-5) are mixed pp - hh ladder di-agrams. The diagrams in the pairs I(3,4), III(7,8) andIII(9,10) can be combined to get simplified energy de-nominators; I(2,5), II(1,2), II(3,4), II(7,8), II(11,12) andIIA(2,4) give identical results for a spin-independent po-tential; and for a momentum-independent potential thecontribution from I(3+4) is half of that from I(2+5). I1 I2 I3 I4 I5 I6IA1 IA2 IA3II1 II2 II3 II4 II5 II6II7 II8 II9 II10 II11 II12IIA1 IIA2 IIA3 IIA4 IIA5 IIA6III1 III2 III7 III8 III9 III10
Figure 3. The 33 fourth-order Hugenholtz diagrams I(1-6), IA(1-3), II(1-12), IIA(1-6), and III(1,2,7-10. Diagrams II5 and IIA1(corresponding to Γ ) as well as II6 and III1 (corresponding to Γ ) have logarithmic UV divergences. The ladder diagrams I(1-6) are most conveniently com-puted by expanding the semianalytic formula for the lad-der resummation derived by Kaiser [46]. The expressionsobtained in this way can be derived from the usual many-body expressions by introducing relative momentum co-ordinates and applying various partial-fraction decom-positions as well as the Poincar´e-Bertrand transforma-tion formula [57]. For the numerical evaluation of theIA diagrams, it is more convenient to use single-particlemomenta instead of relative momenta, because then thephase space is less complicated. The II, IIA and III dia-grams without divergences can be evaluated in the sameway as the IA diagrams. The following diagrams involvedivergences: • I(1,2,4,5), II(1,2,6), III(1,8): UV power diver-gences, • II(5,6), IIA1, III1: logarithmic UV divergences, • III(1,2,8,10): energy-denominator divergences.The UV power divergences, corresponding to pp bub-bles, are renormalized in terms of (low-order) diagramswith two-body counterterm vertices. For g = 2, the log-arithmic UV divergences cancel in the sums II5+IIA1and II6+III1. Finally, the energy-denominator diver-gences correspond to higher-order poles at the integra-tion boundary; they cancel in the sums III(1+8) andIII(2+10).The counterterms for power divergences can be imple-mented by performing subtractions in the bubble partsof the integrands. For example, using a sharp cutoff, f ( p/ Λ) = θ (Λ − p ), and scaling all momenta by a factor k F , the regularized expression for II(1+2) is given by E (Λ) = − ζ ( g − (cid:88) i , j , kd θ cd θ kc θ je θ de n ijk ¯ n cde D cd,ij D de,ik × (cid:88) a θ ab ¯ n ab D ab,ij (cid:12)(cid:12)(cid:12)(cid:12) dummy b = i + j − ac = i + j − de = i + k − d . (26)Here, (cid:80) i ≡ (cid:82) d i/ (2 π ) , the distribution functions are n ij... ≡ n i n j · · · and ¯ n ab... ≡ ¯ n a ¯ n b · · · , with n i ≡ θ (1 − i )and ¯ n a ≡ θ ( a − D ab,ij ≡ ( a + b − i − j ) / (2 M ). Moreover, ζ = k g ( g − C , and θ ab ≡ θ (Λ /k F − | a − b | / g is obtainedby inserting a factor δ σ ,σ (cid:48) δ σ ,σ (cid:48) − δ σ ,σ (cid:48) δ σ ,σ (cid:48) for eachvertex and summing over the spins σ ( (cid:48) )1 , σ ( (cid:48) )2 of the in-and outgoing lines. (For P -wave interactions the factor is δ σ ,σ (cid:48) δ σ ,σ (cid:48) + δ σ ,σ (cid:48) δ σ ,σ (cid:48) .) For details on the diagram-matic rules, see, e.g., Ref. [10, 58]. The renormalizedexpression is given by E R4,II(1+2) = − ζ ( g − (cid:88) i , j , ka , d n ijk ¯ n cde D cd,ij D de,ik × (cid:20) ¯ n ab D ab,ij − D aa, (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) dummy c = i + j − de = i + k − db = i + j − a . (27)This expression can be further simplified such that onlyone unbounded integral appears, i.e., using (cid:88) a (cid:20) ¯ n ab D ab,ij − D aa, (cid:21) = − (cid:88) a n a + n b − n ab D ab,ij (28)we find E R4,II(1+2) = 2 ζ ( g − (cid:88) i , j , k , ad n ijka ¯ n cde D cd,ij D de,ik PD ab,ij (cid:12)(cid:12)(cid:12)(cid:12) dummy c = i + j − de = i + k − db = i + j − a , (29)where P denote the Cauchy principal value. The directapplication of Eq. (28) is prohibited for II6 and III1, be-cause in that case the pertinent energy denominators in-volve additional particle momenta. The regularized ex-pression for II6 is given by E (Λ) = − ζ ( g − (cid:88) i , j , ka , c θ ab θ ka θ cd θ je θ be × n ijk ¯ n abcde D ab,ij D be,ik D bcd,ijk (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dummy b = i + j − ad = k + a − ce = k + a − j . (30) Throughout the paper, we label the cutoff-independent renormal-ized expressions corresponding to UV divergent diagrams with asubscript “R”.
Substituting K = ( i + j ) / p = ( i − j ) / z = k , A =( a − b ) /
2, and Y = ( c − d ) /
2, and omitting redundantregulator functions, we have E (Λ) = − M ζ ( g − (cid:88) K , p , zA , Y n ijk ¯ n abcde θ A θ Y A − p × A + p ) · ( A − K + z ) 1 Y + R , (31)where R = (3 A + K − z ) · ( A − K + z ) / − p . The UVpower subdivergence can now be separated via1 Y − p + R = 1 Y (cid:124)(cid:123)(cid:122)(cid:125) (cid:59) E − R ( Y + R ) Y (cid:124) (cid:123)(cid:122) (cid:125) (cid:59) E . (32)For the UV power divergence of II6(i), the countertermcan be implemented analogous to Eq. (28). The sec-ond part II6(ii) is only logarithmically UV divergent.For g = 2, the logarithmic divergence is canceled if weadd the III1 term, which requires (due to the energy-denominator divergence) to add also III(7+8). The reg-ularized expression for III(1+7+8) is given by E (Λ) = − ζ ( g − (cid:88) i , j , ka , c θ ab θ ab n ijk ¯ n abc D ab,ij × (cid:18) θ ka θ cd ¯ n d D bcd,ijk − θ cd (cid:48) ¯ n d (cid:48) D cd (cid:48) ,ik (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dummy b = i + j − ad = k + a − cd (cid:48) = i + k − c . (33)The energy-denominator divergence corresponds to D ab,ij = 0, and in that case the two terms in the largeparentheses cancel each other. For III(1+8) also the lin-ear UV divergences are removed. For g = 2, the con-tribution from the sum II6(ii)+III(1+7+8) is then givenby E R4,II6(ii)+II(1+7+8) (cid:12)(cid:12) g =2 = − M ζ (cid:88) K , p , zA , Y n ijk ¯ n abc A − p × G , (34)with G = 1( A + p ) · ( A − K + z ) R ( Y + R ) Y + 1 A − p (cid:20) Y + R − Y + R (cid:48) (cid:21) , (35) The counterterms for the power divergences of III1 and III8would come from diagrams with single-vertex loops. where R (cid:48) = − ( K + p − Z ) /
4. Finally, the regularizedexpressions for II5 and IIA1 are given by E (Λ) = − ζ ( g − (cid:88) i , j , ka , c θ ab θ kb θ cd θ ad θ ke θ ce × n ijk ¯ n abcde D ab,ij D ce,ij D acd,ijk (cid:12)(cid:12)(cid:12)(cid:12) dummy b = i + j − ad = i + j + k − a − ce = i + j − c , (36) E (Λ) = − ζ (3 g − (cid:88) i , j , ka , c θ ab θ kb θ cd θ ad θ je θ ce × n ijk ¯ n abcde D ab,ij D ce,ik D acd,ijk (cid:12)(cid:12)(cid:12)(cid:12) dummy b = i + j − ad = i + j + k − a − ce = i + k − c . (37)For g = 2, the sum of these contribution is UV finite, andis given by E R4,II5+IIA1 (cid:12)(cid:12) g =2 = ζ (cid:88) i , j , ka , c n ijk ¯ n abcd D ab,ij D acd,ijk (cid:12)(cid:12)(cid:12)(cid:12) dummydummy b = i + j − ad = i + j + k − a − c × (cid:18) ¯ n e D ce,ij (cid:12)(cid:12)(cid:12)(cid:12) dummydummy e = i + j − c − ¯ n e (cid:48) D ce (cid:48) ,ik (cid:12)(cid:12)(cid:12)(cid:12) dummydummy e (cid:48) = i + k − c (cid:19) . (38)The contributions from II5+IIA1 as well asII6(ii)+II(1+7+8) can of course also be evaluatedby subtracting the individual logarithmic divergences,i.e., by adding the respective (counterterm) parts of D (Λ) (only the sum of these parts vanishes for g = 2),see Sec. IV. We have however found that evaluating thesums II5+IIA1 and II6(ii)+II(1+7+8) provides betternumerical precision (see Table I). IV. FOURTH-ORDER TERM FORHIGHER SPINS
For g >
2, the logarithmic divergences of II6, IIA1, II5and III(1+7+8) are canceled by the contribution fromthe first-order MBPT diagram with D vertex. In cutoffregularization, this cancelation is tantamount to Λ ∂∂ Λ D (Λ) = ηM C , (39) As discussed in Sec. II D, the logarithmic divergences ∼ ln Λare accompanied by logarithms of an invariant kinematical vari-able Q , ∼ ln Q , such that only logarithms of dimensionless argu-ments ∼ ln(Λ /Q ) appear. After renormalization, this becomes ∼ ln(Λ /Q ). In the many-body case (i.e., after the hole-line in-tegration) it is Q = k F , while in the three-body scattering case Q is an external momentum. For convenience and to treat thethree- and many-body cases on the same footing, we omit the ∼ ln Q terms in the analysis of logarithmic divergences below. where the coefficient η is determined by the logarithmicUV divergence. This can be integrated as D (Λ) = D (Λ ) + ηM C ln(Λ / Λ ) , (40)where Λ is an arbitrary auxiliary scale, and D (Λ) isindependent of Λ , as evident from the running with Λ according to Eq. (39) of the integration constant D (Λ ): D (Λ (cid:48) ) = D (Λ ) + ηM C ln(Λ (cid:48) / Λ ). The value of D (Λ ) has to be fixed (for a given choice of Λ ) bymatching to few- or many-body data (see, e.g., Ref. [43]).For further details we refer to the general discussion oflogarithmic divergences in EFT provided in Sec. II D.Below, we first show how the fourth-order term for g > A. Cutoff regularization
The coefficient η = η + η is determined by the log-arithmic divergence of II6+IIA1+II5+III(1+7+8), orequivalently, by the logarithmic divergence of the three-body scattering diagrams Γ and Γ (see Fig. 2). Using asharp cutoff, f ( p/ Λ) = θ (Λ − p ), the regularized expres-sion for diagram Γ is given byΓ (Λ) = − C J (Λ) , (41)with J (Λ) = (cid:88) x , x , x , l , l θ x l θ x k θ x l θ x l θ x l θ x k (cid:48) × D ∗ x l ,k ,k D ∗ x l l ,k k k D ∗ x l ,k (cid:48) k (cid:48) × δ x l , k k δ x l l , k k k δ x l , k (cid:48) k (cid:48) , (42)where k , , and k (cid:48) , , are the three-momenta ofthe in- and outgoing particles, respectively, with k + k + k = k (cid:48) + k (cid:48) + k (cid:48) , and x , , and l , are theloop momenta, and D ∗ = D − i(cid:15) . The factor 3 comesfrom cyclic permutations of the initial and final lines,the factor 2 is due to the number of equivalent contrac-tions for a given choice of final and initial lines, and thefactor 1 /
3! is due to final-state antisymmetrization. Sim-ilarly, the regularized expression for the sum of diagramsΓ and Γ ct2 of Fig. 2 is given by[Γ + Γ ct2 ](Λ) = − C J (Λ) , (43)with J (Λ) = (cid:88) x , x , x , l , l θ x l θ x k θ x l θ x l θ x k (cid:48) × D ∗ x l ,k ,k D ∗ x l ,k (cid:48) ,k (cid:48) (cid:34) D ∗ x l l ,k k k − D ∗ l l (cid:35) × δ x l , k k δ x l l , k k k δ x l , k (cid:48) k (cid:48) , (44)where the term in squared brackets involves the countert-erm for the two-particle bubble. Overall, the logarithmicdivergence is given byΓ (Λ) Λ →∞ −−−−→ − η M C ln(Λ) , (45)[Γ + Γ ct2 ](Λ) Λ →∞ −−−−→ − η M C ln(Λ) . (46)To determine η , we can set all external momenta tozero, i.e., J , (Λ) k ( (cid:48) )1 , , → −−−−−−→ M I , (Λ) . (47)Introducing relative momenta q and q such that { l , , x , , } = { q , ( q + 2 q ) / , q , ( q − q ) / , − q } ,the integral I (Λ) is given by I (Λ) = Λ (cid:90) ℵ d q (2 π ) (cid:90) ℵ d q (2 π ) q (cid:20) q / q − q (cid:21) , (48)where the integral boundaries are with respect to theradial coordinates. To have an infrared finite expres-sion we have, as a formal intermediate step, intro-duced an arbitrary infrared cutoff ℵ . This integral canbe expressed in terms of the inverse tangent integralTi ( x ) = [Li ( ix ) − Li ( ix )] / (2 i ), with Li ( z ) the com-plex dilogarithm, i.e., − I (Λ) = − √ π Ti (cid:18) q √ q (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) Λ , Λ ℵ , ℵ . (49)Using Ti ( x ) = Ti (1 /x ) + ( π/ x ) ln x as well asTi (0) = 0 we find (for ℵ → η = − √ π . (50)For Γ , this method to extract logarithms is prohibitedby the θ x l factor and a nontrivial angular integral. Thisproblem can be avoided in DR where loop integrals re-main invariant under translations of the integration vari-ables. As shown in Sec. IV B, one obtains η = 1 π . (51)From this, the renormalized fourth-order contribution tothe ground-state energy is E ( k F ) = χ (cid:2) D (Λ ) + ηM C ln( k F / Λ ) (cid:3) + (cid:88) i E R4,i + . . . , (52)where i ∈ { II5,IIA1,II6,III(1+7+8) } , and the ellipses re-fer to contributions from other fourth-order diagrams.The factor χ corresponding to the first-order three-bodydiagram is χ = g ( g − g − (cid:88) ijk n ijk = α ( g − M π a s , (53) with α = nε F ( k F a s ) ( g − n = g k / (6 π ) is thefermion number density and ε F = k / (2 M ) the nonin-teracting Fermi energy. Finally, the terms E R4,i are givenby E R4,II5 = lim Λ →∞ [ E (Λ) + ( g − L (Λ)] , (54) E R4,IIA1 = lim Λ →∞ [ E (Λ) + (3 g − L (Λ)] , (55) E R4,II6 = lim Λ →∞ (cid:104) E R4,II6(i) + E (Λ) (56)+( g − L (Λ)] ,E R4,III(1+7+8) = lim Λ →∞ (cid:2) E (Λ) + ( g − L (Λ) (cid:3) . (57)Here, the terms L (Λ) and L (Λ) cancel the logarithmicparts of the respective many-body diagrams, ∼ ln(Λ /k F ),with 4( g − L (Λ)+2( g − L (Λ) = χηM C ln(Λ /k F )matching the form of the logarithm in Eq. (52). They aregiven by L (Λ) = α π ln(Λ /k F ) , (58) L (Λ) = − α √ π ln(Λ /k F ) , (59)which matches (with different phase-space prefactors) thelogarithmic parts of the three-body scattering integrals J (Λ) and J (Λ), respectively. One finds that E R4,II5 = α ( g − × . , (60) E R4,IIA1 = α (3 g − × . , (61) E R4,II6 = − α ( g − × . , (62) E R4,III(1+7+8) = − α ( g − × . . (63)The sum of the first and second two contributions is givenby E R4,II5+IIA1 = α (cid:2) . g − × . (cid:3) , (64) E R4,II6+III(1+7+8) = α (cid:2) − . − ( g − × . (cid:3) , (65)where in each case the leading term corresponds to theresult obtained in the g = 2 calculation of Sec. III. B. Dimensional regularization
The DR calculation of the logarithmic terms is similarto the calculation of the corresponding terms for bosonic See Eq. (32) for the splitting of II6 into a power-divergent partII6(i) and a logarithmically divergent part II6(ii). is determined by the integral I D = µ − D ) (cid:90) d D l (2 π ) D (cid:90) d D l (2 π ) D l + ℵ ) × l + l + l · l + ℵ ) 1( l + ℵ ) . (66)Here, µ is a momentum scale introduced to maintain thecorrect mass dimension, and the scale ℵ serves to admitthe use of Eq. (72) below. [Note that this is a differentscale from the ℵ used in Sec. IV A within cutoff regular-ization.] Introducing Feynman parameters we obtain I D = µ − D ) (cid:90) d D l (2 π ) D (cid:90) d D l (2 π ) D (cid:90) dx − x (cid:90) dy × l ( x + y ) + l (1 − x ) + l · l y + ℵ ] . (67)Shifting l → − l y/ (2 x + 2 y ) and rescaling the inte-gration variables leads to I D = µ − D ) F D (cid:90) d D l (2 π ) D (cid:90) d D l (2 π ) D l + l + ℵ ) , (68)with F D = (cid:90) dx − x (cid:90) dy (cid:20) ( x + y )(1 − x ) − y (cid:21) − D/ . (69)We can expand this in (cid:15) = D − F D =3+ (cid:15) = F + (cid:15) F (cid:48) + O ( (cid:15) ), with F = (cid:90) dx − x (cid:90) dy (cid:20) ( x + y )(1 − x ) − y (cid:21) − / = 4 π , (70) F (cid:48) = − (cid:90) dx − x (cid:90) dy ln (cid:16) x + y )((1 − x ) − y (cid:17) (cid:104) ( x + y )(1 − x ) − y (cid:105) / ≈ . . (71)Applying the relation [35] (cid:90) d D q q + ℵ ) n = 1 ℵ n − D Γ( n − D/ n ) , (72)and analytically continuing to D = 3 + (cid:15) , we then find I D =3+ (cid:15) = 148 π (cid:20) − (cid:15) − ℵ /µ ) + ζ + O ( (cid:15) ) (cid:21) , (73)where ζ = ln(4 π ) − γ E − F (cid:48) / (4 π ) ≈ . γ E ≈ . D = 3the left side of Eq. (72) is UV divergent for n (cid:62) − / n = 3 /
2. This isthe well-known feature that power divergences are auto-matically set to zero in DR.Efimov [19] and Bishop [25] extracted the leading log-arithms by introducing a cutoff Λ on one of the loop mo-menta l , only. DR makes clear why this method givesthe correct result: the analytic continuation D → (cid:15) can be performed for individual subintegrals individually.For diagrams with subdivergences this procedure wouldin fact be required to obtain finite results. This is thecase for diagram Γ , where the divergent integral is I D ,a = µ − D (cid:90) d D l (2 π ) D l + ℵ ) 1( l + ℵ ) × µ − D (cid:90) d D l (2 π ) D l + l + l · l + ℵ ) . (74)Shifting l → l − l /
2, performing the l integration,and analytic continuation to D = 3 + (cid:15) leads to I D ,a = − µ − D π (cid:90) d D l (2 π ) D l + ℵ ) 1( l + ℵ ) (cid:114) l + ℵ , (75)which, for D = 3, indeed diverges only logarithmically inthe UV. Note that also the IR divergence (for ℵ = 0, D = 3) of the l integrals has been eliminated. However,to get to a form where we can apply Eq. (72) we wouldproceed instead as I D ,a = µ − D ) (cid:90) d D l (2 π ) D (cid:90) d D l (2 π ) D (cid:90) dx (1 − x ) × l + l x + l · l x + ℵ ] . (76)Shifting l → l − l x/ I D ,a = µ − D ) F D ,a (cid:90) d D l (2 π ) D (cid:90) d D l (2 π ) D l + l + ℵ ) , (77)with F D ,a = (cid:90) dx (1 − x ) (cid:20) x − x (cid:21) − D/ , (78) In particular, setting D = 3 and ℵ = 0, and introducing an UVcutoff on l is equivalent to the calculations by Efimov [19] andBishop [25]. D (cid:62)
2, obviously a manifestation ofthe subdivergence. This singularity can be removed byadding the term I D ,b = − µ − D ) (cid:90) d D l (2 π ) D l + ℵ ) 1( l + ℵ ) × (cid:90) d D l (2 π ) D l + ℵ ) , (79)i.e., I D ,b = µ − D ) F D ,b (cid:90) d D l (2 π ) D (cid:90) d D l (2 π ) D l + l + ℵ ) , (80)where F D ,b = − (cid:90) dx (1 − x ) (cid:2) x − x (cid:3) − D/ (81)is also singular for D (cid:62)
2, but F D = F D ,a + F D ,b is finite.Expanding F D =3+ (cid:15) = F + (cid:15) F (cid:48) + O ( (cid:15) ) , where F = − √ , (82) F (cid:48) = − π − √ (cid:0) ln (4 / − (cid:1) ≈ − . , (83)we find for I D = I D ,a + I D ,b : I D = − √ π (cid:20) − (cid:15) − ℵ /µ ) + ζ + O ( (cid:15) ) (cid:21) , (84)with ζ = ln(4 π ) − γ E + F (cid:48) / (2 √ ≈ . µ matches the one of ln Λ in the cutoff calcu-lation, see Eq. (50).Subtracting in Eqs. (73) and (84) only the di-vergent parts ∼ /(cid:15) corresponds to minimal sub-traction (MS). The coupling D is then fixed as D = D (cid:63) ( µ ) + ηM C / (2 (cid:15) ), where the scaling of D (cid:63) ( µ )with µ is identical to the scaling of D (Λ ) with Λ, i.e.,instead of Eq. (39) we have µ ∂∂µ D (cid:63) ( µ ) = ηM C . (85)The couplings D (cid:63) ( µ ) and D (Λ ) are not identical for µ = Λ , i.e., they differ in terms of a subtraction constantspecific to the respective regularization and subtractionprocedure. Instead of Eq. (52) we have E ( k F ) = χ (cid:2) D (cid:63) ( µ ) + ηM C ln( k F /µ ) (cid:3) + (cid:88) i E MS4,i + . . . , (86) with E MS4,II5 = E ( ℵ ) + ( g − L (cid:63) ( ℵ ) , (87) E MS4,IIA1 = E ( ℵ ) + (3 g − L (cid:63) ( ℵ ) , (88) E MS4,II6 = E R4,II6(i) + E ( ℵ ) + ( g − L (cid:63) ( ℵ ) , (89) E MS4,III(1+7+8) = E ( ℵ ) + ( g − L (cid:63) ( ℵ ) , (90)where 4 L (cid:63) ( ℵ ) + 2 L (cid:63) ( ℵ ) = χηM C ln( ℵ /k F ), with L (cid:63) ( ℵ ) = α π (cid:20) ζ ℵ /k F ) (cid:21) , (91) L (cid:63) ( ℵ ) = − α √ π (cid:20) ζ ℵ /k F ) (cid:21) , (92)and the terms E ( ℵ ) are given by subtracting from therespective integrands their values with the denominatorsreplaced by the ones corresponding to I D , . For example,the term E ( ℵ ) is given by E ( ℵ ) = − ζ ( g − (cid:88) i , j , ka , c n ijk × (cid:34) ¯ n abcde D ab,ij D ce,ij D acd,ijk (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dummy b = i + j − ad = i + j + k − a − ce = i + j − c − D ℵ ab D ℵ ce D ℵ acd (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dummy b = − ad = − a − ce = − c (cid:35) , (93)with D ℵ ab = D ab + ℵ / ( k M ). One finds E MS4,II5 = − α ( g − × . , (94) E MS4,IIA1 = − α (3 g − × . , (95) E MS4,II6 = α ( g − × . , (96) E MS4,III(1+7+8) = α ( g − × . . (97)The sums of the first two and the last two contributionsare given by E MS4,II5+IIA1 = α (cid:2) . − ( g − × . (cid:3) , (98) E MS4,II6+III(1+7+8) = α (cid:2) − . g − × . (cid:3) . (99)From Eqs. (64) and (65), the relation between the “MS”values and the “R” ones is given by E MS4,II5+IIA1 = E R4,II5+IIA1 − α ( g − × . , (100) E MS4,II6+III(1+7+8) = E R4,II6+III(1+7+8) + α ( g − × . . (101)As required, the difference between the “MS” values andthe “R” values vanishes for g = 2, see Sec. III. For II6 we separate again the power-divergent part II6(i), seeEq. (32). V. GROUND-STATE ENERGY ATFOURTH ORDER
Here, we summarize the results for the low-density ex-pansion for the ground-state energy density E ( k F ) of thedilute Fermi gas. The expansion reads E ( k F ) = n ε F (cid:20)
35 + ( g − ∞ (cid:88) ν =1 C ν ( k F ) (cid:21) , (102)with n = g k / (6 π ) the fermion number density, ε F = k / (2 M ) the noninteracting Fermi energy, and g the spin multiplicity. The expansion coefficients up tofourth order are given by C ( k F ) = 23 π k F a s , (103) C ( k F ) = 435 π (11 − k F a s ) , (104) C ( k F ) = (cid:104) . . g − (cid:105) ( k F a s ) + 110 π ( k F a s ) k F r s + 15 π g + 1 g − k F a p ) , (105) C ( k F ) = − . k F a s ) + 0 . k F a s ) k F r s + γ ( k F ) ( g −
2) ( k F a s ) . (106)The first two terms are the only ones for which closed-form expressions are known; these where first derivedby Lenz [16] in 1929 and Lee and Yang [17] as well asde Dominicis and Martin [18] in 1957, respectively. Thethird-order term was first computed by de Dominicis andMartin [18] in 1957 for hard spheres with two isospinstates, by Amusia and Efimov [20] in 1965 for a singlespecies of hard spheres, and then by Efimov [22] in 1966for the general dilute Fermi gas. It was also computedsubsequently by various authors [10, 23–25, 46, 53, 56].Initial studies of the fourth-order term for g = 2 wereperformed by Baker in Refs. [21, 24, 51, 52], see alsoRef. [27] for a discussion of these.Up to third order, only two-body (i.e., ERE) param-eters appear and the expansion is a polynomial in theFermi momentum k F . At higher orders N (cid:62)
4, logarith-mic terms ∼ k n F ln( k F / Λ ) enter, starting at N = 4 for g >
2; for g = 2, no logarithms emerge from S -wave in-teractions (as a consequence of the Pauli exclusion princi-ple). The logarithms are accompanied by multi-fermioncouplings [at fourth order, the coupling D (Λ )] whosedependence on the auxiliary scale Λ is such that theFermi-momentum expansion is independent of Λ . Themulti-fermion couplings are renormalization scheme de-pendent and have to be matched to few-body (or many-body) observables calculated in the same scheme. Usinga Galilean invariant regulator function and subtractingonly divergent terms (“R” scheme), the g > γ ( k F ) Table I. Results for the contributions to the regular (i.e., non-logarithmic) a s part of C ( k F ). Diagrams with ∗ ( ∗∗ ) have UVpower (logarithmic) divergences, which are subtracted by therespective counterterm contributions. Diagrams with ∗∗∗ haveenergy-denominator singularities. For the diagrams with loga-rithmic divergences, “(R)” denotes the result obtained using aregulator function and subtracting only divergent terms, and“(MS)” the result corresponding to DR with minimal sub-traction. The uncertainty estimates take into account boththe statistical Monte Carlo uncertainties and variations of thecutoff. The g factors are listed without the generic factor g ( g − g factor valueI1 ∗ . ∗ +I3+I4 ∗ +I5 ∗ . − . g ( g −
3) + 4 − . g ( g −
3) + 4 − . g ( g −
3) + 4 − . ∗ +II2 ∗ g − . g − − . ∗∗ (R) g − . ∗∗ (MS) g − − . ∗∗ , ∗ (R) g − − . ∗∗ , ∗ (MS) g − . g − . g − . g − − . g − − . ∗∗ (R) 3 g − . ∗∗ (MS) 3 g − − . g − . g − − . g − . g − . ∗∗∗ , ∗∗ , ∗ (R)+III7+III8 ∗∗∗ , ∗ g − − . ∗∗∗ , ∗∗ , ∗ (MS)+III7+III8 ∗∗∗ , ∗ g − . ∗∗∗ +III9+III10 ∗∗∗ g − . g =2 . ∗ g =2 − . (cid:80) diagrams ,g =2 − . of the fourth-order term takes the form γ R4 ( k F ) = M D (Λ )108 π a s + 0 . − . g − π (cid:16) π − √ (cid:17) ln( k F / Λ ) . (107)On the other hand, using DR with minimal subtraction The logarithmic part of Eq. (107) was first derived by Efimov [19,22] and subsequently in Refs. [9, 23, 25, 59]. Note that in theliterature [9, 10, 19, 22, 23, 25, 43, 59] the arbitrary scale Λ isusually set to Λ = 1 /a s . γ MS4 ( k F ) = M D (cid:63) (Λ )108 π a s − . − . g − π (cid:16) π − √ (cid:17) ln( k F / Λ ) . (108)The scaling of D (Λ ) and D (cid:63) (Λ ) with Λ is identical,and determined by the Λ independence of γ ( k F ). Thevalues of D (Λ ) and D (cid:63) (Λ ) differ by a subtraction con-stant, i.e., D (cid:63) (Λ ) = D (Λ ) − π a s M × . . (109)Although the subtraction constant is arbitrary, it is nev-ertheless pertinent to specify its value (i.e., to specify therenormalization scheme) in order to predict many-bodyresults from few-body data, or vice versa.The individual diagrammatic contributions to the C part of the fourth-order term are listed in Table I. Thecomputations have been carried out using the MonteCarlo framework introduced in Ref. [60] to evaluate high-order many-body diagrams, see also Ref. [27]. The re-sults for the contributions that involve logarithmic di-vergences, II5, II6, IIA1, and III(1+7+8), have thelargest numerical uncertainties. For g = 2, slightlymore precise results can be given for II5+IIA1 andII6+III(1+7+8), because then no logarithmic diver-gences occur (see Sec. III). VI. CONVERGENCE ANALYSIS ANDRESUMMATIONS
As discussed above, for spin one-half fermions ( g = 2)the logarithmic terms from S -wave interactions cancel(by virtue of the Pauli principle). Logarithms still arisefrom P -wave interactions at higher orders, i.e., at a cer-tain order N log . The Fermi-momentum expansion for E = E/E , truncated at an order N < N log , is thus apolynomial in δ = k F a s : E N ( δ ) = 1 + N (cid:88) ν =1 ε ν δ ν , (110)where E = 3 nk / (10 M ) is the energy density ofthe free Fermi gas, and the expansion coefficients ε ν ≡ ε ν ( a s , r s , a p , . . . ) are completely determined by theERE. In the following, we analyze the convergence be-havior of Eq. (110) for two different cases. First, weexamine the case where all ERE parameters beyond a s are zero, which we denote by LO. Here, the coefficientsin the k F a s expansion are given by { ε ν } = (cid:26) π , − π , . , − . , . . . (cid:27) . (111) Second, we consider the hard-sphere gas (HS) where a s =3 r s / a p , leading to { ε ν } = (cid:26) π , − π , . , . , . . . (cid:27) . (112)In Sec. VI A we examine the convergence behavior ofthe LO and HS expansions and analyze the uncertaintiesof the predictions for E/E . We will find that in bothcases the Fermi-momentum expansion is well-convergedat fourth-order for | δ | (cid:46) .
5. In Sec. VI B we then showthat Pad´e and Borel resummations allow us to extend thedomain of convergence to | δ | (cid:46)
1. Finally, in Sec. VI Cwe discuss the challenges regarding the calculation of theFermi-momentum expansion beyond fourth order.
A. Perturbative convergence anduncertainty estimates
In Ref. [27] we assessed the convergence pattern of the k F a s expansion at a given order N (cid:54) ε N +1 = ± max [ ε ν (cid:54) N ]. This spansan uncertainty band of width ∆ E N = 2 | ε N +1 | δ N +1 .Here, we use the pointwise Bayesian model with conju-gate distributions developed in Refs. [61, 62] to estimate ε N +1 given the computed coefficients. This model al-lows one to evaluate posterior distributions analytically(given the conjugate prior) rather than through MonteCarlo sampling. Specifically, we treat the coefficients ε ν as random numbers drawn from a single normal distri-bution, pr (cid:0) ε ν | ¯ c (cid:1) i.i.d. ∼ √ π ¯ c exp (cid:20) − ε ν c (cid:21) , (113)with mean zero and variance ¯ c . The computed coeffi-cients ε ν (cid:54) are assumed to be known draws from thisa priori unknown distribution function, while ε ν> areunknown. We also assume a scaled inverse- χ prior on ¯ c ,pr (cid:0) ¯ c (cid:1) ∼ ( τ η ) η Γ( η ) exp (cid:104) − η τ c (cid:105) ¯ c η ) , (114)with η degrees of freedom and scale parameter τ . Byadjusting the hyperparameters we can incorporate ourprior estimate of the (not computed) higher-order co-efficients. We fix η = 3 and determine τ by the re-quirement that the mean value η τ / ( η −
2) equals | max [ ε ν (cid:54) N ] | . This prior choice disfavors high values for z ∼ · · · is a common notation in statistics that reads “the vari-able z is distributed as · · · ”. The “i.i.d.” above the ∼ indicatesa set of independent and identically distributed (i.i.d.) randomvariables. c and thus ε ν>N . Using Bayes’ theorem and marginal-izing over ¯ c , one then finds that the posterior for a co-efficient at order n > N is given by the Student- t distri-bution [61], i.e.,pr (cid:0) ε ν>N | { ε ν } Nν =1 , τ (cid:1) ∼ t η (cid:0) ε ν ; 0 , τ (cid:1) , (115)with t η ( x ; µ, τ ) = 1 (cid:112) πητ Γ( η +12 )Γ( η ) (cid:18) x − µ ) ητ (cid:19) − η +12 . (116)Here, the scale parameter τ satisfies ητ = η τ + N (cid:88) ν =1 ε ν . (117)Further, η = η + n c , where n c is the number of coeffi-cients in the set { ε ν } n c ν =1 used to inform the probabilitydistribution. We consider all available coefficients, i.e., n c = 4, so that all four known coefficients are used foreach N ∈ { , , , } in Eq. (115). Finally, from Bayes’theorem one then finds that the posterior distributionrepresenting the uncertainty of E N ( δ ) is given by [61]pr (cid:0) E N ( δ ) | { ε ν } Nν =1 , τ (cid:1) ∼ t η (cid:16) E ( δ ); E N ( δ ) , δ N +1) τ (cid:17) , (118)where the variable E ( δ ) corresponds to the presumed ex-act results.The convergence behavior of the Fermi-momentum ex-pansion for the LO and the HS case is examined in Fig. 4.There, we show the perturbative results for E = E/E obtained for truncation orders N = 2 , , | δ | (cid:46) . | δ | (cid:38) δ = − . E QMC ≈ . .
5% ( E ≈ . .
8% ( E ≈ . .
4% ( E ≈ . .
1% ( E ≈ . δ = +0 . E QMC ≈ . .
2% ( E ≈ . .
5% ( E ≈ . .
3% ( E ≈ . .
3% ( E ≈ . ε ≈ . ε ≈ . ε ≈ . S - and P -wave contribu-tions), so the third- and fourth-order HS curves in Fig. 4are almost indistinguishable.The Bayesian uncertainty bands in Fig. 4 are similar tothose from the simple ε N +1 = ± max [ ε ν (cid:54) N ] analysis, seeFig. 2 of Ref. [27]. In both schemes, going to higher orders in the expansion reduces the width of the uncertaintybands for | δ | (cid:46)
1, and for | δ | (cid:46) . N = 4. This supports the conclusion that theexpansion is well-converged at fourth order for | δ | (cid:46) . | δ | (cid:38) Note that these results do notdepend on a s being of natural size; only k F a s has to besmall. B. Pad´e and Borel resummations
Resummation methods provide a means to extrapo-late a (truncated) series beyond the region where well-converged results are obtained, | δ | (cid:46) . δ ). We donot consider the HS case, as there higher-order EREparameters become relevant at stronger coupling. Re-garding Pad´e approximants, we restrict the discussionto those that give predictions for the Bertsch parame-ter ξ = E ( −∞ ). Regarding the more extensive Borelmethods, we focus on the region of weak-to-intermediatecoupling since only there (i.e., for | δ | (cid:46)
1) the extrapola-tions are well-converged.
1. Pad´e approximants
For a given formal power series E ( δ ) = 1 + ∞ (cid:88) ν =1 ε ν δ ν , (119)the Pad´e[ n, m ] approximant is the rational functionPad´e[ n, m ]( δ ) = 1 + (cid:80) nk =1 a k δ k (cid:80) ml =1 b l δ l , (120)whose Maclaurin expansion matches the series up to or-der N = n + m . Only “diagonal” Pad´es with n = m havea nontrivial unitary limit, i.e., Pad´e[ n, n ] −→ a n /b n for δ → −∞ . To have meaningful results in the strong-coupling regime thus mandates the restriction to even N = 2 n , i.e., ( N, n ) = (2 ,
1) and (
N, n ) = (4 , More precisely, the Fermi-momentum expansion is an asymptoticseries that diverges for N → ∞ for all | δ | >
0, see Sec. II C.“Convergence” is therefore meant to indicate that the result atoptimal truncation is adequately close to the exact result. Pad´e predictions for the Bertsch parameter were previously stud-ied by Baker [51, 52], see also Ref. [27]. For investigations ofextrapolations of the Fermi-momentum expansion that incorpo-rate strong-coupling constraints, see Refs. [71, 72]. Note thatin this work, and in Fig. 5, we use the same labels (CB andPCB) for (unconstrained) conformal Borel approximants as usedin Ref. [72] for the corresponding strong-coupling constrainedBorel approximants. k F a s ) E / E g = 2, a p = r s = 0 fourth orderthird ordersecond orderPadé [1/1]Padé [2/2]QMC k F a s E / E g = 2, a s = r s = a p Figure 4. Convergence behavior of the Fermi-momentum expansion for the ground-state energy
E/E of a dilute Fermi gasof spin one-half fermions with a p = r s = 0 (LO, left panel) and a s = 3 r s / a p (HS, right panel) at negative and positive k F a s , respectively. Note that the x -axes in the two panels are different. The respective uncertainty bands correspond the the68% credibility intervals from our Bayesian estimation of the next-higher coefficient in the k F a s expansion. In the LO case wealso show results obtained from two different Pad´e approximants, Pad´e[1 ,
1] (gray line) and Pad´e[2 ,
2] (black line). Finally, thethick red dots in each panel correspond to results from nonperturbative QMC calculations [7, 63, 64]. See the text for moredetails.
The results obtained from the Pad´e[1 ,
1] and [2 ,
2] ap-proximants (which were already studied in Ref. [27]) areshown in the left panel of Fig. 4. One sees that thePad´e[2 ,
2] approximant is very close to the QMC resultsfor δ (cid:46) − .
2, while Pad´e[1 ,
1] is in better agreement closeto the unitary limit δ → −∞ . Note however that pair-ing correlations become relevant for larger values of − δ ,and it is questionable that Pad´es can capture pairing ef-fects (which are expected to be encoded in the high-orderbehavior of the k F a s expansion [37]) at low truncationorders. The range for the Bertsch parameter obtainedfrom Pad´e[1 ,
1] and [2 , ξ Pad´e ∈ [0 . , . ξ ≈ .
376 extracted from experimentswith cold atomic gases, and also with the extrapolatedvalue for the normal (i.e., non-superfluid) Bertsch pa-rameter ξ n ≈ .
45 [1]. Altogether, these results seem toindicate that Pad´e approximants converge in a larger re-gion, compared to the Fermi-momentum expansion.
2. Borel resummation
Borel methods are based on the Borel(-Leroy) trans-form of Eq. (119), i.e., B ( t ) = 1 + ∞ (cid:88) k =1 c k Γ( k + 1 + β ) t k , (121)where the standard Borel transform corresponds to β = 0. In contrast to the perturbation series, the Borel For a more extensive study of the Pad´e[1 ,
1] approximant, seeRef. [73]. transformed series has a finite convergence radius. Thatis, the large-order behavior c k k →∞ ∼ a k Γ( k + 1 + β ) , (122)together with the choice of β , determines the nature ofthe leading singularity of B ( t ) at t = 1 /a , see Refs. [68,74, 75].There are several methods that allow one to constructapproximants B N ( t ) from the truncated (at order N ) ver-sion of the Borel series [Eq. (121)]. The most straightfor-ward is the Pad´e-Borel method, which constructs B N ( t )in terms of Pad´e approximants matched to the Borel se-ries. From B N ( t ), approximants B N ( δ ) for E ( δ ) are thenobtained in terms of the inverse Borel transform B N ( x ) = (cid:90) ∞ dt e − t t β B N ( tx ) . (123)Note that the inverse Borel transform of the truncatedBorel series just gives the truncated original series, i.e.,an essential part of Borel resummation is the analyticcontinuation (via Pad´e or conformal methods, see below)of the Borel series beyond its radius of convergence.A more sophisticated approach is to re-expand theBorel series in terms of a conformal mapping w ( t ), lead-ing to B N ( t ) = N (cid:88) k =0 s k [ w ( t )] k . (124)The principal analytic information that determines thechoice of the mapping w ( t ) is the coefficient a in6 PCB[2]CB[2]Padé-Borel[2]PCB[3]CB[3]Padé-Borel[3]PCB[4]Padé-Borel[4]CB[4] E / E k F a s −2−1.5−1−0.50 Figure 5. Results for the ground-state energy
E/E of adilute Fermi gas of spin one-half fermions with a p = r s = 0(LO case) obtained from various Borel resummations, see thetext for details. The numbers in squared brackets denotethe underlying truncation order N . For comparison, we alsoshow the perturbative results at second (dotted blue line) andfourth order (dashed red line), see also Fig. 4. The PCB[2] andCB[2] as well as the PCB[4] and Pad´e-Borel[4] approximantsoverlap at the present resolution scale. Eq. (122). That is, given knowledge of a one choosesthe mapping w ( t ) = √ − at − √ − at + 1 (125)that projects the cut Borel t plane to the interior ofthe unit disc [67, 75]. The purpose of the conformalre-expansion is twofold. First, given its larger con-vergence radius, the re-expanded series [Eq. (124)] isexpected to converge faster, compared to the originalBorel series [Eq. (121)]. Second, Eq. (124) by itselfprovides an analytic continuation of the Borel series;we denote the corresponding B N ( t ) as CB (“conformal-Borel”) approximants. Pad´e approximants can also bematched to Eq. (124), corresponding to PCB (“Pad´e-conformal-Borel”). Finally, regarding the choice of β inEq. (121): the CB and PCB approximants have a square-root branch point at t = 1 /a , which motivates the choice β = β + 3 /
2, since then the exact Borel transform hasthe same feature [68].The results from several Borel approximants forthe weak-to-intermediate coupling regime are shownin Fig. 5. Specifically, we show the Pad´e-Borel, CBand PCB approximants obtained for truncation orders N = 2 , ,
4. In the Pad´e-Borel and PCB case we use thestandard choices for Pad´e approximants, n = m for even N and n = m − N . Further, for Pad´e-Borel we use the standard β = 0, and for CB and PCB we em-ploy the conjectured large-order behavior a = − /π and β = 0 (see Ref. [37] and Sec. II C) and set β = 3 /
2. Onesees that, similar to the Pad´es, at fourth order the vari-ous Borel approximants are well-converged and close tothe QMC data for | δ | (cid:46) | δ | (cid:46) C. Beyond fourth order
The first complication regarding the calculation of co-efficients beyond fourth order is the rapid increase ofthe number of Hugenholtz diagrams with N . Graphtheory methods allow one to automatically generate di-agrams [76–78], from which one finds that the num-ber of diagrams without single-vertex loops increasesas (1 , , , , , , . . . ) for N = (1 , , , , , , . . . ),where the number of the relevant normal diagrams in-creases as (1 , , , , , , . . . ).For a given set of higher-order diagrams, the evaluationof those without UV divergences and those that have onlysimple ladder-type divergences (which are renormalizedby two-body counterterms) would be relatively straight-forward. That is, for a given diagram the only complica-tion compared to a fourth-order diagram of similar typewould be additional three-momentum integrals.The main complication concerning higher-order calcu-lations lies (as in the fourth-order case) with UV diver-gences that are not renormalized by two-body countert-erms. For instance, at fifth order one encounters sev-eral three-body scattering diagrams of the form of di-agrams of Fig. 2 but with two additional intermediatestates. These diagrams have logarithmic subdivergencesthat cancel if the diagrams are summed. The remain-ing linear UV divergence cancels for g = 2 and is oth-erwise renormalized by a momentum-independent three-body counterterm. For the next diagonal Pad´e approx-imant (Pad´e[3 , VII. SUMMARY
In this paper we have discussed high-order perturba-tive EFT calculations for fermions at very low energy7scales. In particular the issue of renormalization hasbeen investigated in detail. We have then elaborated andexpanded on our recent calculation [27] of the fourth-order term in the Fermi-momentum or k F a s expansionfor the ground-state energy of the general dilute Fermigas. The result for the complete (i.e., including both an-alytic and logarithmic terms) fourth-order coefficient hasbeen given for two different regularization and renormal-ization schemes: cutoff regularization (with divergencesubtraction) and dimensional regularization (with mini-mal subtraction).The central results for the Fermi-momentum expan-sion are summarized in Sec. V, where in Table I thevarious contributions to the regular (i.e., nonlogarith-mic) ( k F a s ) part of the fourth-order term are listed.In Sec. VI we have then investigated the convergencebehavior of the expansion for the case of spin one-half fermions. Using Bayesian methods and compar-ing against results from nonperturbative QMC compu-tations, we found that the expansion is well-converged atfourth order for | k F a s | (cid:46) .
5, and exhibits divergent be-havior for | k F a s | (cid:38)
1, see Fig. 4. (To be precise, the k F a s expansion is a divergent asymptotic series; by “divergentbehavior” we mean that the accuracy of the result atoptimal truncation is deficient.)Further, we have shown that Pad´e-Borel resummations(of the a s -only part of the expansion) improve the con-vergence and give well-converged results at fourth orderin the region | k F a s | (cid:46)
1, see Fig. 5. Accurate resultsthroughout the entire BCS regime with negative k F a s (and into the BEC region) can however be obtained viaresummations that incorporate explicit strong-couplingconstraints [71, 72]. Given the technical challenges that arise beyond fourth order, it is unlikely that the k F a s ex-pansion will be evaluated to even higher precision in thenear future.Our results for the Fermi-momentum expansion atfourth order provide important constraints for ultracoldatoms and dilute neutron matter. Specifically, our resultsserve as useful benchmarks for future QMC simulationsof dilute Fermi systems and may be used to construct im-proved models of neutron star crusts. Future work maybe targeted at high-order calculations of the dilute Fermigas expansion at finite temperature. ACKNOWLEDGMENTS
We thank T. Duguet, H.-W. Hammer, J.A. Melendez,and S. Wesolowski for useful discussions, and P. Arthuisfor sending us his list of fifth- and sixth-order diagrams.We are also grateful to S. Gandolfi and S. Pilati forsending us their QMC results. This work is supportedin part by the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation) – Project-ID 279384907 –SFB 1245, the US Department of Energy, the Office ofScience, the Office of Nuclear Physics, and SciDAC un-der awards DE-SC00046548 and DE-AC02-05CH11231.C.D. acknowledges support by the Alexander von Hum-boldt Foundation through a Feodor-Lynen Fellowship.This material is based upon work supported by the U.S.Department of Energy, Office of Science, Office of Nu-clear Physics, under the FRIB Theory Alliance awardDE-SC0013617. Computational resources have been pro-vided by the Lichtenberg high performance computer ofthe TU Darmstadt. [1] M. J. H. Ku, A. T. Sommer, L. W. Cheuk, and M. 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