Are occupation numbers observable?

Physics Letters B 531 (2002) 203–208

www.elsevier.com/locate/npe

Are occupation numbers observable?

R.J. Furnstahl, H.-W. Hammer

Department of Physics, The Ohio State University, Columbus, OH 43210, USA

Received 30 August 2001; accepted 21 December 2001

Editor: W. Haxton

Abstract

The question of whether occupation numbers and momentum distributions of nucleons in nuclei are observables is consideredfrom an effective field theory perspective. Field redefinitions lead to variations that imply the answer is negative, as illustratedin the interacting Fermi gas at low density. Implications for the interpretation of(e, e′p) experiments with nuclei are discussed. 2002 Elsevier Science B.V. All rights reserved.

PACS:05.30.Fk; 21.10.Pc; 21.65.+f; 25.30.Fj

Keywords:Effective field theory; Field redefinition; Occupation number; Momentum distribution;(e, e′p) experiments

The advent of new continuous beam electron ac-celerators (CEBAF, Mainz, MIT-Bates, NIKHEF) per-mits experiments that probe hadronic matter in bothinclusive and exclusive reactions with unprecedentedprecision. These experiments are expected to deepenour understanding of nuclear structure and reactionmechanisms by measuring the response of nuclei toelectroweak probes over a wide range of energy andmomentum transfers [1,2]. Exclusive electron scatter-ing experiments at large momentum transfer, such asthe knockout of a proton in(e, e′p) on a nucleus, areparticularly important.

It is often claimed that occupation numbers and/ormomentum distributions of nucleons in nuclei canbe extracted from these experiments. Their extraction

E-mail addresses:[email protected] (R.J. Furnstahl),[email protected] (H.-W. Hammer).

from the data is built on the extreme impulse approx-imation but is obscured by the need to consider fi-nal state interactions and meson exchange currents.The interpretation and comparison with theory is madebased on a given nuclear Hamiltonian, but from theperspective of the underlying theory of the strong in-teraction, QCD, there is no unique or preferred Hamil-tonian. Rather, there are infinitely many such low-energy effective Hamiltonians that are related by fieldredefinitions that leave observables unchanged (e.g.,see Ref. [3]). What happens to occupation numbersunder such transformations?

A powerful framework to study low-energy phe-nomena in a model-independent way is given by ef-fective field theory (EFT) [3–5]. The underlying ideais to exploit a separation of scales in the system. Forexample, if the typical momentak are small comparedto the inverse range of the interaction 1/R, low-energyobservables can be described by a controlled expan-

0370-2693/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0370-2693(01)01504-0

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204 R.J. Furnstahl, H.-W. Hammer / Physics Letters B 531 (2002) 203–208

sion in kR. All short-distance effects are systemati-cally absorbed into low-energy constants using renor-malization. The EFT approach allows for accurate cal-culations of low-energy processes and properties withwell-defined error estimates.

In this Letter, we explore from an EFT perspectivethe question of whether occupation numbers andmomentum distributions of nucleons in nuclei areobservable. In an EFT, observables are characterizedby invariance under local field redefinitions. If aquantity depends on the particular representation ofthe LagrangianL (beyond the level of truncationerrors), it is not an observable. Off-shell Green’sfunctions for scattering processes in the vacuum, forexample, can be changed by field redefinitions. On-shell Green’s functions, which correspond to S-matrixelements, however, are unchanged [6,7].

To address the issue most cleanly, we focus on thequestion of whether occupation numbers in a homoge-nous medium at finite density are observables in theframework of EFT. In a general finite system, occu-pation numbers and momentum distributions are verydifferent quantities. In a homogeneous system, how-ever, they are equivalent. Since there are no asymptoticstates in an infinite medium, however, the usual analy-sis for field redefinitions does not directly carry overto in-medium observables. While the analysis can beextended to thermodynamic observables like the en-ergy density or particle number [8], this extension isnot obvious for other quantities, such as the momen-tum distribution or occupation numbers.

We use the interacting Fermi gas at low density asa laboratory to illustrate a fundamental and genericproblem with the definition of momentum occupationnumbers. For a given representation of the Hamil-tonian, one can define an operatora

†kak that counts

particles/holes with momentumk and gives the ex-pected result in the noninteracting limit. As discussedbelow, this operator is not derived from a global sym-metry, in contrast to the total number operator. In theEFT framework this is a problem, as there is no pre-ferred form of the effective Lagrangian.

To highlight this problem, we consider both the par-ticle numberN and the momentum distributionn(p).We describe the system using a local Lagrangian for anonrelativistic fermion field with spin independent in-teractions that is invariant under Galilean, parity, and

time-reversal transformations:

L=ψ†[i∂t +

−→∇2

2M

]ψ − C0

2

(ψ†ψ

)2

(1)+ C2

16

[(ψψ)†

(ψ←→∇ 2ψ

)+ H.c.

]+ · · · ,

where←→∇ =←−∇ −−→∇ is the Galilean invariant derivative

and H.c. denotes the Hermitian conjugate. A conve-nient and transparent regularization scheme is dimen-sional regularization with minimal subtraction; see,e.g., Ref. [9] for details. In this scheme the coefficientsare simplyC0 = 4πa/M andC2 = C0are/2, wherea is thes-wave scattering length andre the effectiverange.

We generate an infinite, equivalent class of La-grangiansLα by performing the field redefinition

ψ −→ ψ + 4πα

Λ3

(ψ†ψ

)ψ,

(2)ψ†−→ψ†+ 4πα

Λ3ψ†(ψ†ψ

),

in L. The factor 1/Λ3 is introduced to keep the arbi-trary parameterα dimensionless.Λ is the breakdownscale of the EFT and the additional factor of 4π is in-troduced for convenience [8]. We obtain forLα :

Lα =L− 4πα

Λ3 2C0(ψ†ψ

)3

+ 4πα

Λ3

{(ψ†ψ

)ψ†(i∂tψ)

− 1

2M

[ψ†(−→∇ψ) ·ψ†(−→∇ψ)+ 2

(−→∇ψ†)ψ ·ψ†(−→∇ψ)]+ H.c.

}(3)+ · · · +O

(α2),

where higher-order two- and three-body terms, allfour- and higher-body terms, and terms ofO(α2) havebeen omitted.

For α = 0 we recover the originalL. The La-grangianLα contains additional vertices, including anoff-shell vertex, but gives exactly the same energydensity and particle number as the LagrangianL. InRef. [8] it was illustrated how the necessary cancella-tions occur in general and for this particular example.Furthermore, it was shown how the number operatormust be constructed as the conserved charge of the

R.J. Furnstahl, H.-W. Hammer / Physics Letters B 531 (2002) 203–208 205

Noether current associated with theU(1) phase sym-metry:

(4)ψ(x)→ e−iφψ(x) and ψ†(x)→ eiφψ†(x),

under whichLα is invariant. One can convenientlyidentify the Noether current by promotingφ to a func-tion of x and considering infinitesimal transformationswith

(5)Lα→ Lα

[ψ,ψ†;φ(x)].

Then the number density operator is given by [8]:

Nα ≡ δ

δ(∂tφ)Lα

[ψ,ψ†;φ(x)]

(6)=ψ†ψ + 4πα

Λ3 2(ψ†ψ

)2.

The particle number itself is given by the spatialintegral of Nα . In a uniform system, however, thedifference is simply a factor of the volume and it isconvenient to refer toNα as the number operator.Note that Eq. (6) differs from the naive expectationψ†ψ for α �= 0. In the Appendix of Ref. [8], itwas demonstrated how the contributions from theadditional vertices inLα and the additional term inNα cancel order-by-order inα. Consequently, the totalparticle numberN is unchanged by field redefinitions,as expected for an observable.

Matters become more complicated for the momen-tum distributionn(k) since the corresponding opera-tor is not simply related to the conserved charge of aNoether current. In the literature, the operator

(7)nk = a†kak,

where ak destroys a fermion of momentumk, isuniversally adopted [1,2,10,11]. This definition givesthe correct result in the noninteracting limit,n(k) =gθ(kF − k), whereg is the spin degeneracy factor. Ifthere were a preferred form of the Lagrangian/Hamil-tonian, as is usually assumed, Eq. (7) would uniquelydetermine the momentum distribution. In terms of thefield operators

ψ(x)=∫

d3k

(2π)3eik·xak,

(8)ψ†(x)=∫

d3k

(2π)3e−ik·xa†

k,

we can writea†kak =

∫d3x nk(x) with

(9)nk(x)=∫

d3y eik·yψ†(x+ y)ψ(x),

which is nonlocal in coordinate space. Using thedefinition of the one-particle Green’s function [12],

(10)

iG(x, t;x′, t ′)= 〈"0|T [ψH (x, t)ψ†H (x′, t ′)]|"0〉

〈"0|"0〉 ,

with |"0〉 the exact ground state of the system and

(11)ψH (x, t)= eiH t ψ(x)e−iH t ,

a Heisenberg operator, the expectation value ofnk(x)can be written as [11]

(12)

n(k)= ⟨nk(x)

⟩= limη→0+

(−i)g∫

dω

2πeiωηG(ω,k).

The momentum distributionn(k) for the hard sphereFermi gas to second order inkFa has been calculatedusing this definition in Refs. [13,14].

With the definition in Eq. (12), the explicit expres-sions forn(k) in Refs. [13,14] are reproduced in theEFT by settingα = 0 and calculating the one-particleGreen’s function toO(C2

0). The one-particle Green’sfunctionG(ω,k) is related to the proper self energyΣ∗(ω,k) via [12]

(13)G(ω,k)= 1

ω− k2/(2m)−Σ∗(ω,k),

where the spin indices have been suppressed. ToO(C2

0) there are only two diagrams for the properself energy, which are shown in Fig. 1(a) and (b). Thefilled circle represents theC0 interaction from Eq. (1).The Feynman rules for evaluating these diagrams canbe found in Refs. [8,9]. Fig. 2 shows schematicallyhow theO(C2

0) contribution modifies the distribution.Since particles can be kicked out of the Fermi seaby the interaction, some occupation probability ismoved to states above the Fermi surface. (Note that thesecond-order diagram shown is a Feynman diagramand so includes modifications of both particle and holestates.)

If we use the equivalent LagrangianLα with α �=0, n(k) differs already atO(α). The additional self-energy diagrams up toO(αC0) are shown in Fig. 1(c)–(h). Here the empty circle represents the induced three-body vertex∝ αC0 and the empty triangle the induced


Fig. 1. Diagrams contributing toΣ∗(ω,k) to O(C20) [(a) and (b)]

and atO(αC0) [(c) to (h)].

Fig. 2. Schematic picture of the occupation number as a function ofmomentum in a uniform Fermi system with no interactions (dashedline) and including leading correction from interactions (solid line)(cf. Ref. [10]). The square with a cross denotes an insertion ofEq. (14).

off-shell vertex proportional toα in Eq. (3). (TheFeynman rules for these vertices can again be found inRefs. [8,9].) Equivalently, the occupation numbers canbe calculated from energy diagrams with an operatorinsertion corresponding to Eq. (7). The appropriateinsertion fora†

kak on a fermion line with energyω andmomentump is

(14)(−i)(2π)3 limη→0+

eiωηδ3(p− k)δαβ ,

whereα andβ are spin indices and the factor exp(iωη)

ensures the correct ordering of operators. The cor-

Fig. 3. Feynman diagrams for the occupation numbern(k). (a) gives

the contribution froma†kak while (b) shows contributions from the

additional term forα �= 0. Note that diagram (a)(iii) represents only

one of three possible insertions ofa†kak . The square with a cross

and the square with a circled cross denote insertions of Eqs. (14)and (16), respectively.

responding diagrams up toO(αC0) are shown inFig. 3(a), where we have indicated the insertion ofEq. (14) by the square with the cross. (Note that thediagram in Fig. 3(a)(iii) represents only one of threepossible insertions of the operatora

†kak.) Only the first

two diagrams in Fig. 3(a) are nonvanishing. We find

*n(k)(i) =−2(g− 1)ρ4πα

Λ3θ(kF− k),

(15)

*n(k)(ii) = 2ig(g − 1)4πα

Λ3

C0

(2π)9

× limη→0+

∫d4p

∫d4l

∫d4q eiωη

× δ3(p− k)G0(p)G0(l)

×G0(p− q)G0(l + q).

This discrepancy is not surprising, since the definitionin Eq. (12) corresponds to the operator given inEq. (7). Even the operator for the total particle numbercorresponding to the LagrangianLα is different fromthe naive expectation. For the occupation number,however, there is no symmetry that can be used toconstruct the operatornαk and so an ambiguity occurs.This simple exercise can be continued to higher order,with increasingly sophisticated diagrams. The samepattern recurs, with additional diagrams depending onα induced at each order, generatingα dependence inn(k). This implies thatn(k) is not an observable.

We might take Eq. (7) together withL as adefinitionand transform the operatornk at the sametime asL. For finite α this implies that one has to

R.J. Furnstahl, H.-W. Hammer / Physics Letters B 531 (2002) 203–208 207

calculate the additional diagrams shown in Fig. 3(b)where the square with the circled cross denotes aninsertion of

(−i)(2π)34πα

Λ3 (δα1α3δα2α4 + δα1α4δα2α3)

(16)× limη→0+

4∑j=1

eiωj ηδ3(pj − k),

whereωj , pj , andαj label the energy, momentum,and spin of the linej , respectively. The occupationnumbers defined this way are independent ofα byconstruction. The first two diagrams in Fig. 3(b)exactly cancel the contributions from the first twodiagrams in Fig. 3(a) (cf. Eq. (15)), while the thirddiagram vanishes.However, we had no basis for theoriginal definition, since EFT is model-independentand makes no a priori assumptions on the dynamics.Thus we conclude that occupation numbers (or evenmomentum distributions) cannot be uniquely definedin general.

How does this conclusion fit in with the standardanalysis of(e, e′p) experiments, where the cross sec-tions measured are, by definition, observables? Thereare further complications in a finite system, wherethe momentum distribution differs from the occupa-tion numbers, but we can illustrate the analogous sit-uation in our model problem by introducing an exter-nal sourceJ (x) coupled to the fermion number. [ThusJ (x) plays the role of the Coulomb field of the virtualphoton in(e, e′p).] If we were constructing an EFTof an underlying theory (such as QCD), we would ex-pect to need the most general coupling consistent withthe symmetries, but for simplicity we will assume thatJ (x) has nonzero coupling only toψ†ψ in the originalrepresentationL.

The noninteracting cross section forα = 0 corre-sponds to the diagram in Fig. 4(a). The same crosssection is obtained forα �= 0, but only if we includethe contributions from the induced vertex to the fi-nal state interaction in Fig. 4(b) [and to the initialstate interaction, which is not shown]and the vertexcontribution from the modified operator in Fig. 4(c).In general, there arealwayscontributions of all threetypes, dressed with additional interactions order-by-order, which are mixed up under field redefinitions.Isolating Fig. 4(a) is a model-dependent proceduresince it depends onα.

Fig. 4. Schematic diagrams for the interaction of an external sourceJ(x) [wavy line] with the fermion system, “knocking out” a particle.

Note that the ambiguities have a natural size, as dis-cussed for an analogous shifting of contributions be-tween two-body off-shell and three-body vertices inRef. [8]. An interesting question is whether the starkdifference in occupation numbers between nonrela-tivistic and relativistic Brueckner calculations [15] canbe explained by the ambiguity. Similar ambiguities oc-cur in other areas of physics as well. In deep inelasticscattering, the physical cross section can be written asa convolution of quark and gluon distributions with co-efficient functions determined by perturbative QCD.It is well known, however, that the distributions andthe coefficient functions are individually scheme andscale dependent. The scheme and scale-dependence ofauxiliary quantities such as the pion distribution in anucleon was recently clarified [16]. In Ref. [17], itwas demonstrated that the nucleon occupation num-ber cannot be extracted from mesonic0 decays inhypernuclei if the calculation is properly done usingspectral functions. Another question of current interestis whether the condensate fraction in a Bose–Einsteincondensate, which is essentially the occupation num-ber of the condensate, can be measured [18].

The conclusion that the extraction of a momentumdistribution from (e, e′p) cross sections is ambigu-ous because of final state interactions and vertex cor-rections (e.g., meson exchange currents) is not a sur-prise. While it is well known that such ambiguities arepresent, the usual assumption is that there is a “cor-rect” answer that can be extracted from experiment. Ina similar spirit, different ways of implementing the im-pulse approximation for the response of many-fermionsystems have been analyzed in Ref. [19]. In contrast,the EFT perspective clearly implies that ambiguities inthe extraction of momentum distributions in(e, e′p)cannot be resolved by experiment. It is not only thatthe momentum distribution is difficult to extract but


that it cannot be isolated in principle within a calcula-tional framework based on low-energy degrees of free-dom. Rather, such auxiliary quantities can only be de-fined in a specific convention, like a particular form ofthe Hamiltonian, regularization scheme and so on. Itcan still be useful to discuss such quantities within agiven convention. All true observables, however, canjust as well be described in a different framework thatadheres to different conventions.

Acknowledgements

We thank E. Braaten, H. Grießhammer, S. Jeschon-nek, X. Ji, R. Perry, S. Puglia, and B. Serot for usefulcomments. This work was supported in part by the USNational Science Foundation under Grant Nos. PHY-9800964 and PHY-0098645.

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Are occupation numbers observable?