We present the solution to the recently derived Wilsonian renormalization
group (RG) equation for nuclear current operators. In order to eliminate the
present ambiguity in the RG equation itself, we introduce a new condition
specifying the cutoff independence of the five point Green function
corresponding to the two-body propagator with current operator insertion. The
resulting effective current operator is then shown to obey a modified
Ward-Takahashi identity which differs from the usual one, but that nevertheless
leads to current conservation.Comment: 12 page

Blankleider, B.

Kvinikhidze, A. N.

English

arXiv

Kvinikhidze, Alexander N

Blankleider, Boris

Flinders Academic Commons

Renormalization Group Equation For Nuclear Current
            Operators

Archived at the Flinders Academic Commons  http://dspace.flinders.edu.au/dspace/  This is the publisher’s copyrighted version of this article.  Publisher’s site: http://prc.aps.org/ PHYSICAL REVIEW C 76, 064003 (2007)Wilsonian renormalization group equation for nuclear current operatorsA. N. Kvinikhidze*A. Razmadze Mathematical Institute, Georgian Academy of Sciences, Aleksidze Str.1, Tbilisi 0193, Georgia andDepartment of Physics, Flinders University, Bedford Park, South Australia 5042, AustraliaB. Blankleider†Department of Physics, Flinders University, Bedford Park, South Australia 5042, Australia(Received 3 July 2007; revised manuscript received 7 November 2007; published 28 December 2007)We present the solution to the recently derived Wilsonian renormalization group (RG) equation for nuclearcurrent operators. To eliminate the present ambiguity in the RG equation itself, we introduce a new conditionspecifying the cutoff independence of the five-point Green function corresponding to the two-body propagatorwith current operator insertion. The resulting effective current operator is then shown to obey a modifiedWard-Takahashi identity that differs from the usual one, but that nevertheless leads to current conservation.DOI: 10.1103/PhysRevC.76.064003 PACS number(s): 25.10.+s, 05.10.Cc, 21.30.Fe, 21.60.−nI. INTRODUCTIONThe use of the Wilsonian renormalization group (RG)method [1–3] to impose a cutoff  on the momenta of virtualstates is an important tool for studying various aspects ofnuclear effective field theory (EFT) [4–9]. In this contextit has mostly been used to study the strong interactions ofnonrelativistic two-nucleon systems where the central startingpoint is the RG equation for the two-body effective potentialV [4]. Recently, however, Nakamura and Ando (NA) [10]have extended the scope of such studies by deriving theRG equation for the two-body effective current operatorOµ. The main purpose of the present article is to presentthe unambiguous solution to this equation. As our solutiondiffers from the one given by NA, we have endeavored togive a detailed account of both the solution and the RGequation itself. In particular, we present an off-shell cutoff-independence condition that leads necessarily to NA’s RGequation. By contrast, NA derived their equation as onlya sufficient condition for an on-shell cutoff-independencecondition. In this way we eliminate the consequent ambiguityof NA’s RG equation. Last, we examine the question of currentconservation for the derived effective current operator Oµ. Wefind that even in the best case where Oµ is obtained (via theRG equation) from the full field-theoretic current operator Oµ∞that satisfies the usual Ward-Takahashi (WT) identity [11], Oµwill not obey this identity (it will instead satisfy a modifiedWT identity). Nevertheless, the same operator Oµ is shown toconserve current in matrix elements.II. SOLUTION TOTHE CURRENT OPERATOR RG EQUATIONWe consider a nonrelativistic two-body system for whichthe RG method is used to introduce a momentum cutoff of. For this purpose it is convenient to use the projection*sasha kvinikhidze@hotmail.com†boris.blankleider@flinders.edu.auoperators [10]η =∫d3k(2π )3 |k〉〈k| θ ( − k), (1)λ =∫d3k(2π )3 |k〉〈k| θ (¯ − k)θ (k − ), (2)where ¯ > . The RG equation for the reduced space effectivepotential V [4] can then be written as∂V∂= VG0 ∂λ∂V, (3)where G0 = (E − H0)−1 is the two-body free propagator and∂λ∂= −∫d3k(2π )3 |k〉〈k| δ(k − ). (4)Equation (3) can be derived from the reduced space Lippmann-Schwinger equationT = V + VηG0T (5)by using the fact that the off-shell scattering amplitude, T ,does not depend on .Although Eq. (3) has been used to study nuclear EFT [4–9],such investigations have been limited to the purely hadronicsector. However, in a recent work NA have extended the scopeof such studies by deriving the corresponding RG equation forthe reduced space effective current operator Oµ [10]. Writingthis current operator asOµ = ηµη, (6)the RG equation derived by NA can be expressed as∂µ∂= VG0 ∂λ∂µ + µ∂λ∂G0V. (7)Here we provide the solution to Eq. (7), noting that thesolution given in Refs. [10,12,13] differs from ours. We find,unambiguously, thatOµ = η(1 − V ¯G0λ)−1Oµ¯(1 − λG0V ¯)−1η= η[1 + V¯λG0(1 − V ¯G0λ)−1]Oµ¯0556-2813/2007/76(6)/064003(5) 064003-1 ©2007 The American Physical SocietyA. N. KVINIKHIDZE AND B. BLANKLEIDER PHYSICAL REVIEW C 76, 064003 (2007)× [1 + (1 − λG0V ¯)−1G0λV ¯]η= η[1 + V¯λ(E′ − H0 − V ¯λ)−1]Oµ¯× [1 + (E − H0 − λV ¯)−1λV ¯]η, (8)where Oµ is given at  = ¯ by its starting value Oµ¯ and V ¯is the two-body interaction defined in the model space with thecutoff ¯.1 The last line of Eq. (8) has been written in a formthat is most easily compared with Refs. [10,12,13].To prove Eq. (8), we show explicitly that the correspondingµ,µ = (1 − V ¯G0λ)−1Oµ¯(1 − λG0V ¯)−1, (9)satisfies Eq. (7).2We first use Eqs. (A2) to write Eq. (9) asµ = (1 + VG0λ)Oµ¯(1 + λG0V), (10)and therefore∂µ∂= ∂VG0λ∂Oµ¯(1 + λG0V) + (1 + VG0λ)×Oµ¯∂λG0V∂. (11)The use of the RG equation for V, Eq. (3), further gives∂VG0λ∂= ∂V∂G0λ + VG0 ∂λ∂= VG0 ∂λ∂VG0λ + VG0 ∂λ∂= VG0 ∂λ∂(VG0λ + 1). (12)Using this in Eq. (11) then gives the RG equation for thecurrent operator, Eq. (7).Our solution, Eq. (8), should be compared with the solutionfirst given by NA in Ref. [10] and then used for RG analysesin Refs. [10,12,13]:Oµ = η[1 + V ¯λG0(1 − V ¯G0)−1λ]Oµ¯× [1 + λ(1 − G0V ¯)−1G0λV ¯]η= η[1 + V¯λ(E′ − H0 − V ¯)−1λ]Oµ¯× [1 + λ(E − H0 − V ¯)−1λV ¯]η. (13)It is seen that our solution differs substantially from the oneof Ref. [10]; in particular, the interaction operator V¯ inthe denominators (E′ − H0 − V ¯λ)−1 and (E − H0 − λV ¯)−1of Eq. (8) is projected by λ, so that each intermediatestate in the perturbation series for (E′ − H0 − V ¯λ)−1 and(E − H0 − λV ¯)−1 involves relative momenta restricted tothe interval  < k < ¯. By contrast, no such restriction onmomenta appears in the corresponding intermediate states ofEq. (13).1It is worth noting that Eq. (8) is just the expression for aneffective operator in the Bloch-Horowitz approach [14] or Feshbach’sprojection formalism [15] (with P = η and Q = λ).2Equation (8) can also be derived directly from our definition ofµ, Eq. (22), by making use of Eqs. (A7) and Eqs. (A8).III. UNAMBIGUOUS DERIVATION OFTHE CURRENT OPERATOR RG EQUATIONThe RG equation for µ, Eq. (7), was derived in Ref. [10] asonly a sufficient condition for the  invariance of the physicalmatrix element of Oµ:∂〈Oµ〉∂= ∂∂¯ψβηµ(Eβ,Eα)ηψα = 0. (14)That is, the equation used to define µ was chosen to be3〈Oµ〉 ≡ ¯ψβηµ(Eβ,Eα)ηψα = ¯ψβµ(Eβ,Eα)ψα, (15)whereµ(Eβ,Eα) ≡ Oµ¯(Eβ,Eα) = ηµ(Eβ,Eα)η∣∣= ¯ (16)can be identified with the current vertex function of the fullspace [16] in the limit ¯ → ∞. The sandwiching two-bodywave functions ¯ψβ and ψα include bound states and scatteringstates whose relative momenta, p′ and p, respectively, aresmaller than the cutoff parameter: p′, p < . Although notemphasized in Ref. [10], this restriction is essential for thederivation of Eq. (7) in the case of scattering states, and it alsoprevents Eq. (15) from a possible mathematical inconsistencyof having more equations than the number of unknownvariables. This point is clarified under Eq. (22).We stress that Eq. (7) is only a sufficient condition forthe  independence of ¯ψηµηψ , as expressed by Eq. (14),even though this equation involves matrix elements betweenall states ψ (bound and scattering). The exact nature ofthis ambiguity can be illustrated by substituting either thescattering state equation,ψp(k) = (2π )3δ3(p − k) + G0(Ep, k)×∫V(Ep;k,k′)θ ( − k′)ψp(k′) d3k′(2π )3 , (17)or the corresponding bound state equation, into Eq. (14) (hereEp′ and Ep are the on-shell energies), thereby resulting in¯ψp′η∂µ(Ep′ , Ep)∂ηψp= ¯ψp′ηV(Ep′ )G0(Ep′ ) ∂λ∂µ(Ep′ , Ep)ηψp+ ¯ψp′ηµ(Ep′ , Ep)∂λ∂G0(Ep)V(Ep)ηψp. (18)Clearly Eq. (7) provides only a sufficient condition to guaran-tee Eq. (18).Here we eliminate the ambiguity in the validity ofEq. (7) by showing that this RG equation is a sufficient andnecessary condition for  independence of the five-pointfunction ηGηµηGη. In other words, rather than basingthe RG approach to the current operator on the condition[Eq. (15)]¯ψηµηψ = ¯ψµψ (19)3To save on notation we suppress total momentum variables fromthe argument of µ.064003-2WILSONIAN RENORMALIZATION GROUP EQUATION FOR . . . PHYSICAL REVIEW C 76, 064003 (2007)for all  < ¯, we suggest that it be based on the conditionηGηµηGη = ηGµη (20)for all  < ¯, where Gµ is the five-point function defined asGµ = GηµηG∣∣¯= = GµG. (21)We note that Gµ corresponds to the two-body Green functionG with all possible insertions of a current [16]. In the five-point function ηGµη, neither the incoming nor the outgoingtwo-body states are on the energy shell; by contrast, boththese states are on the energy shell in ¯ψµψ . At the sametime, such five-point Green functions are necessary ingredientsfor three-body currents where two-body subsystems are off-shell. In this sense the use of ηGµη for the RG approach tothe current operator is naturally related to the RG approachto the two-body interaction, where the cutoff independenceof the fully off-shell two-body scattering amplitude is used[4]. Equation (20) thus defines the effective current vertexµ so that the five-point Green function GηµηG coincideswith the five-point Green function Gµ = GµG if the relativemomenta of incoming and outgoing nucleons are below .Showing the two-body energy arguments, Eq. (20) isηG(E′)η µ(E′, E)ηG(E)η = ηGµ(E′, E)η= ηG(E′)µ(E′, E)G(E)η, (22)where the external η’s ensure the above-mentioned restrictionon the relative momenta of incoming and outgoing particles.Without this restriction, one would have the self-consistencyconstraint µ = ηµη, which would simply mean that themodel is cutoff by  from the very beginning, leaving us withnothing further to be done.The  independence specified by Eq. (22) leads to the RGequationηG(E′)∂ηµ(E′, E)η∂G(E)η = 0. (23)The cutoff-independence condition of Eq. (15) differsfrom the one of Eq. (22). It is also a weaker condition asEq. (15) follows from Eq. (22); i.e., Eq. (15) is a necessarybut not a sufficient condition for Eq. (22) to be satisfied.The essential difference between these two conditions is thatEq. (22) involves off-shell scattering amplitudes throughηG(E)η, whereas Eq. (15) involves half-on-shell amplitudesthrough the scattering statesψp(k) = (2π )3δ(p − k) + G0(Ep, k)T (Ep;k,p). (24)In this sense Eq. (15) looks more like an extension of theRG approach discussed in Ref. [17] (which is based on theindependence of the half-on-shell scattering amplitude) tothe case of current operators. Moreover, Eq. (22) definesµ(E′, E) unambiguously whereas there is an ambiguity inits definition by Eq. (15) (this ambiguity has already beenpointed out in Ref. [10]).To see the difference between these two conditions yet moreprecisely, let’s write Eq. (23) in expanded form for the case ofscattering states:∂∂∫ [(2π )3δ(p′ − k′) + T (E′;p′,k′)G0(E′, k′)]× d3k′(2π )3 θ ( − k′)µ(E′, E,k′,k)θ ( − k)d3k(2π )3× [(2π )3δ(k − p) + G0(E, k)T (E;k,p)] = 0, (25)where no restriction is put on E′ and E. Writing Eq. (14) ina similar way, it becomes clear that only that part of Eq. (25)corresponding to E = Ep, and E′ = Ep′ , reproduces Eq. (14).With no restriction being put on the external relative momentap and p′ of Eq. (25) (apart from p′, p < ), one can invert theexternal Green functions to obtain the RG equation for µ.That is why the RG equation for µ is not only a sufficientbut also a necessary condition for Eq. (23), whereas it is onlya sufficient condition for Eq. (14).To show explicitly how one obtains the RG equationunambiguously, we use Eqs. (A6) and the shorthand notationδ ≡ ∂η/∂ = −∂λ/∂ in the following:0 = ηG∂ηµη∂Gη = ηG(η∂µ∂η + δµη + ηµδ)Gη= ηGη∂µ∂ηGη + ηGδµηGη + ηGηµδGη= ηGη∂µ∂ηGη + (ηG0 + ηGηVG0)δµηGη+ ηGηµδ(ηG0 + G0VηGη)= ηGη∂µ∂ηGη + ηGηVG0δµηGη+ ηGηµδG0VηGη + ηG0δµηGη + ηGηµδηG0= ηGη(∂µ∂+ VG0δµ + µδG0V)ηGη+ ηG0δµηGη + ηGηµδηG0. (26)Furthermore, as we are interested in external relative momentastrictly below , the last two terms of Eq. (26) are zero sinceδη = 0. One can then invert ηGη in the reduced subspace byacting on Eq. (26) with 1 − VG0η from the right side andwith 1 − ηG0V from the left:0 = (1 − ηG0V)ηGη(∂µ∂+ VG0δµ + µδG0V)× ηGη(1 − VG0η)= G0η(∂µ∂+ VG0δµ + µδG0V)ηG0, (27)where Eqs. (A7) have been used. In this way we derive the RGequation for the current operator, Eq. (7), unambiguously.IV. CURRENT CONSERVATIONThe question of how to properly implement current conser-vation in EFT with a cutoff, so that gauge invariance is ensuredin practical calculations, is a subtle one [18]. Here we showthat the problem of current conservation in the RG approach is064003-3A. N. KVINIKHIDZE AND B. BLANKLEIDER PHYSICAL REVIEW C 76, 064003 (2007)likewise not so simple (it is certainly not as simple as presentedin Ref. [10]).To avoid the well-known problems of current conservationin theories with a finite cutoff, we consider the simple casewhere the starting cutoff is taken to infinity, ¯ = ∞. Then inthe best case we will have the usual two-body Ward-Takahashi(WT) identities [11]qµGµ(E′, E) = 00G(E) − G(E′)00, (28a)qµµ(E′, E) = G−1(E′)00 − 00G−1(E), (28b)where 00 is the zero’th component of the current operatorµ0 of two noninteracting particles and is specified for initial(final) total four-momentum P = p1 + p2 (P ′ = p′1 + p′2)and relative momentm p (p′) as〈p′|00(P ′, P )|p〉= i(2π )3[e1δ(p′2 − p2) + e2δ(p′1 − p1)]= i(2π )3[e1δ(p′ − p − q/2) + e2δ(p′ − p + q/2)]. (29)It is important to realize that the WT identities of Eqs. (28) aredamaged after the introduction of a finite momentum cutoff. In particular, introducing the cutoff into Eq. (28a) givesqµηGµ(E′, E)η = η00G(E)η − ηG(E′)00η. (30)Because η00G(E)η = η00ηG(E)η (the cutoff η does notcommute with 00), it is evident that Eq. (30) is not of the sameform as Eq. (28a), so it is not a usual WT identity. Similarly,to see how the WT identity for current operator Oµ = ηµη(which does depend on ) differs from the usual one, we useEq. (8) and Eq. (28b) to writeqµηµ(E′, E)η= η[1 − V¯G0(E′)λ]−1[G−1(E′)00 − 00G−1(E)]× [1 − λG0(E)V ¯]−1η= η[1 + VG0(E′)λ][G−1(E′)00 − 00G−1(E)]× [1 + λG0(E)V]η. (31)This expression can be simplified usingG−1(1 + λG0V) = G−10 − ηV (32a)(1 + VG0λ)G−1 = G−10 − Vη, (32b)which follow from the equations for G, Eqs. (A5a) and (A5b).Thus,qµηµ(E′, E)η= η[G−10 (E′) − V(E′)]η00[1 + λG0(E)V(E)]η− η[1 + V(E′)G0(E′)λ]00η[G−10 (E) − V(E)]η. (33)Although Eq. (33) is not a usual WT identity, it still leads toa conserved current due to the operators in the curly brackets,[G−10 − V]:qµ ¯ψp′ηµ(E′, E)ηψp = 0. (34)Having derived the modified WT identity, Eq. (33),it is easy to realize that there was no obligation of pushing thestarting cutoff to infinity. We could have started with a finitecutoff ¯; however, our starting WT identities would then needto be Eqs. (30) and (33) (with  replaced by ¯), instead of theusual ones, Eqs. (28). In this way we would come to the sameresult [Eqs. (30) and (33) for any  < ¯].It is important to note that the modified WT identity,Eq. (33), relates the reduced space effective current vertexµ, only to the corresponding effective potential V (V ¯is not involved), and that it enters the WT identity onlywith relative momenta below  for all physically interestinglow energy transitions. These properties are indispensable forconstructing a self-contained EFT in the reduced momentumspace [18].ACKNOWLEDGMENTSThe research described in this publication was madepossible in part by GNSF Grant GNSF/ST06/4-050.APPENDIX: USEFUL EQUATIONSHere we gather together some standard equations of the RGapproach that are made use of in the main text. First, we notethat the solution of the RG equation for the effective potential,Eq. (3), can be formally written in terms of the initial potentialV¯ asV = (1 − V ¯G0λ)−1V ¯, (A1a)= V¯(1 − λG0V ¯)−1. (A1b)These equations then give the useful relations(1 − V¯G0λ)−1 = 1 + VG0λ, (A2a)(1 − λG0V ¯)−1 = 1 + λG0V. (A2b)Second, we note that in this article we assume that a finitevalue of ¯ defines the full model space. That is, all relativemomenta are assumed to lie within a sphere of radius ¯ sothatη¯ ≡ η|= ¯ = 1, ¯λ ≡ λ|= ¯ = 0, (A3)andλ = 1 − η. (A4)We also note that Eq. (5), and its reverse form T =V + T G0ηV, implies that the two-body Green functionG ≡ G0 + G0T G0 satisfies the equationsG = G0 + (λG0 + Gη)VG0, (A5a)G = G0 + G0V(λG0 + ηG), (A5b)G = G0 + GV ¯G0, (A5c)G = G0 + G0V ¯G. (A5d)These then imply the following equations (together with theirreversed forms):ηG = ηG0 + ηGηVG0, (A6a)ηG = ηG0 + ηGV ¯G0, (A6b)ηGη = ηG0 + ηGV ¯G0η. (A6c)064003-4WILSONIAN RENORMALIZATION GROUP EQUATION FOR . . . PHYSICAL REVIEW C 76, 064003 (2007)The first of these equations, and its reversed form, thengivesηGη(1 − VG0η) = ηG0, (A7a)(1 − ηG0V)ηGη = G0η, (A7b)while the last two equations of Eqs. (A6), and their reversedforms, giveηG = ηGη (1 − V¯G0λ)−1 , (A8a)Gη = (1 − λG0V ¯)−1 ηGη. (A8b)[1] K. G. Wilson and J. B. Kogut, Phys. Rep. 12, 75 (1974).[2] J. Polchinski, Nucl. Phys. B231, 269 (1984).[3] T. R. Morris, Prog. Theor. Phys. Suppl. 131, 395 (1998).[4] M. C. Birse, J. A. McGovern, and K. G. Richardson, Phys. Lett.B464, 169 (1999).[5] M. C. Birse, Phys. Rev. C 74, 014003 (2006).[6] K. Harada, K. Inoue, and H. Kubo, Phys. Lett.B636, 305 (2006).[7] K. Harada and H. Kubo, Nucl. Phys. B758, 304 (2006).[8] K. Harada, H. Kubo, and A. Ninomiya (2007), arXiv:nucl-th/0702074.[9] S. X. Nakamura, Prog. Theor. Phys. 114, 77 (2005).[10] S. X. Nakamura and S.-I. Ando, Phys. Rev. C 74, 034004(2006).[11] W. Bentz, Nucl. Phys. A446, 678 (1985).[12] S. X. Nakamura and S.-i. Ando, Talk given at the 5thInternational Workshop on Chiral Dynamics, Theory andExperiment, 18–22 September 2006, Durham/Chapel Hill, NC,USA, arXiv:nucl-th/0611052.[13] S. X. Nakamura and A. Ga˚rdestig (2007), arXiv:0704.3757[nucl-th].[14] C. Bloch and J. Horowitz, Nucl. Phys. 8, 91 (1958).[15] H. Feshbach, Ann. Phys. 5, 357 (1958); 19, 287(1962).[16] A. N. Kvinikhidze and B. Blankleider, Phys. Rev. C 60, 044003(1999).[17] S. K. Bogner, T. T. S. Kuo, A. Schwenk, D. R. Entem, andR. Machleidt, Phys. Lett. B576, 265 (2003).[18] A. N. Kvnikhidze et al. (in preparation).064003-5

Physical Review C

Wilsonian renormalization group equation for
            nuclear current operators

On the Wilsonian renormalization group equation for nuclear current operators

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