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Baryon self energies in the chiral loop expansion A. Semkea and M.F.M. Lutza aGesellschaft fu¨r Schwerionenforschung (GSI) 6 Planck Str. 1, 64291 Darmstadt, Germany 0 0 2 Abstract n a J Wecomputetheselfenergiesofthebaryonoctetanddecupletstates attheone-loop 7 level applying the manifestly covariant chiral Lagrangian. It is demonstrated that 1 expressions consistent with the expectation of power counting rules arise if the self 2 energies are decomposed according to the Passarino-Veltman scheme supplemented v by a minimal subtraction. This defines a partial summation of the chiral expansion. 1 6 A finite renormalization required to install chiral power counting rules leads to 0 the presence of an infrared renormalization scale. Good convergence properties for 1 1 the chiral loop expansion of the baryon octet and decuplet masses are obtained 5 for natural values of the infrared scale. A prediction for the strange-quark matrix 0 element of the nucleon is made. / h t - l c u 1 Introduction n : v i X TheapplicationofchiralperturbationtheorytotheSU(3)flavorsectorofQCD r is hampered by poor convergence properties for processes involving baryons a [1,2,3,4,5]. The original computation of the nucleon self energy by Gasser, Sainio and Svarc [6] was performed as an application of the manifestly covariant chiral Lagrangian. It was observed that the MS scheme [7] leads to results that contradict the power counting rules. Subsequently the heavy-baryon formulation of the chiral Lagrangian was suggested by Jenkins and Manohar [8]. Whereas the chiral power counting rules are realized transparently, manifest Lorentz invariance is given up in that scheme. Computations for the baryon octetmasses[2,3]donotappeartobeconvergent intheheavy-baryonformula- tion once the strange quark sector is included. The convergence was improved by introducing a finite cutoff into the loop functions [9,10,11]. Clearly, alter- native schemes are desirable. This work aims at introducing a partial summation scheme, the construction of which is guided by covariance and analyticity. Various manifestly Lorentz Preprint submitted to Elsevier Science 9 February 2008 invariant formulations of chiral perturbation theory were suggested that re- cover the power counting rules [12,13,14,15,16,17,18,19]. All such schemes are bound to reproduce computations performed within the heavy-baryon formal- ism.Themotivationforthesearchforalternativesstemsinpartfromthequest of summation schemes that enjoy improved convergence properties. Some of the proposed schemes have been applied to the evaluation of the baryon octet masses at the one-loop order. The infrared scheme (IR) introduced by Becher andLeutwyler [13]wasusedbyEllisandTorikoshi[4],however, findingnocon- vincing convergence properties. Similarly the extended on-mass shell scheme (EOMS) introduced by Gegelia and Japaridze [15] suffers from unacceptably large subleading order terms [5]. It is the purpose of the present work to perform computations based on the scheme proposed in [14,17]. We will evaluate the baryon octet and decuplet self energies at the one-loop level and study the convergence properties of the minimal chiral subtraction scheme (χ-MS) [14,17]. The latter is based on the Passarino-Veltman reduction [20] supplemented by a minimal subtraction scheme. It suggests a natural partial summation of the chiral expansion. It is proven in the Appendix of the present work that given any one-loop integral that arises when computing one-baryon processes it is sufficient to renormalize the scalar master-loop functions of the Passarino-Veltman reduction in a manner that the latter are compatible with the expectation of chiral counting rules. Within the χ-MS scheme the empirical octet and decuplet masses can be reproduced accurately with a small residual dependence on an infrared renormalization scale only. Goodconvergence properties are found for natural values of the infrared scale. A prediction for the strange-quark matrix element of the nucleon is made. 2 Relevant chiral interaction terms WecollectthetermsofthechiralLagrangianthatdeterminetheleadingorders of baryon octet and decuplet self energies [21,22]. Up to chiral order Q2 the baryon propagators follow from =tr B¯[i∂/ M◦ ]B [8] L − (cid:16) (cid:17) tr ∆¯ [i∂/ M◦ ]gµν i(γµ∂ν +γν∂µ)+γµ[i∂/+ M◦ ]γν ∆ µ [10] [10] ν − · − − 2d(cid:16) tr ∆¯(cid:16) ∆µ tr χ 2d tr (∆¯ ∆µ)χ (cid:17) (cid:17) 0 µ 0 D µ 0 − · − · +2b tr(cid:16)B¯B tr(cid:17)χ (cid:16)+(cid:17)2b tr B¯[χ(cid:16) ,B] +2b (cid:17)tr B¯ χ ,B , 0 0 F 0 D 0 { } (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) 2 m2 0 0 π   χ = 0 m2 0 . (1) 0 π      0 0 2m2 m2   K − π   We assume perfect isospin symmetry through out this work. The fields are decomposed into isospin multiplets Φ=τ π +α K +K α+ηλ , † † 8 · · · √2B=α N +λ Λ+τ Σ+Ξtiσ α, (2) † 8 2 · · · withtheGell-Mannmatrices,λ ,andtheisospindoublet fieldsK = (K+,K0)t i and Ξ = (Ξ0,Ξ )t. The isospin Pauli matrices σ = (σ ,σ ,σ ) act exclusively − 1 2 3 in the space of isospin doublet fields (K,N,Ξ) and the matrix valued isospin doublet α, α = 1 (λ +iλ ,λ +iλ ) , τ = (λ ,λ ,λ ) . (3) † √2 4 5 6 7 1 2 3 The tree-level expression for the baryon mass shifts are recalled. For the octet and decuplet states (1) implies ∆M(2) = 2m2 (b +2b ) 4m2 (b +b b ), N − π 0 F − K 0 D − F ∆M(2) ∆M(2) = +16 b (m2 m2), Σ − Λ 3 D K − π ∆M(2) ∆M(2) = 8b (m2 m2), Ξ − N − F K − π ∆M(2) ∆M(2) = 4(b +b )(m2 m2), (4) Ξ − Σ − D F K − π and ∆M(2) = 2(d +d )m2 4d m2 , ∆ − 0 D π − 0 K ∆M(2) ∆M(2) = 4 d (m2 m2), Σ − ∆ −3 D K − π ∆M(2) ∆M(2) = 4 d (m2 m2), Ξ − Σ −3 D K − π ∆M(2) ∆M(2) = 4 d (m2 m2). (5) Ω − Ξ −3 D K − π At tree level the parameters b ,b and d can be determined by the mass D F D differences of the baryon states: b +0.06GeV 1, b 0.21GeV 1, d 0.48GeV 1. (6) D − F − D − ≃ ≃ − ≃ − It is anamazing result of the tree-level chiral analysis that it yields parameters (6) that are quite consistent with expectations from the large-N operator c 3 analysis [23,24]. At leading order there are three independent parameters, b ,b and b . It holds 0 D F b +b = 1 d , d = b . (7) D F 3 D 0 0 TheevaluationofthebaryonselfenergiestoorderQ3 probesthemeson-baryon vertices F D = tr B¯γ γµ[∂ Φ, B] + Tr B¯γ γµ ∂ Φ,B 5 µ 5 µ L 2f 2f { } (cid:16) (cid:17) (cid:16) (cid:17) C 1 tr ∆¯ (∂ Φ)[gµν Zγµγν]B +h.c. µ ν − 2f · − 2 (cid:16) (cid:17) H tr [∆¯µ γ γ ∆ ](∂νΦ) , (8) 5 ν µ − 2f · (cid:16) (cid:17) where we apply the notations of [17]. We use f = 92.4 MeV in this work. The values of the coupling constants F,D,C and H may be correlated by a large-N operator analysis [23,25]. At leading order the coupling constants can c be expressed in terms of F and D only. We employ the values for F and D as suggested in [26,17]. All together we use F = 0.45, D = 0.80, H = 9F 3D, C = 2D, (9) − in this work. We take the parameter Z = 0.72 from a detailed coupled-channel studyofmeson-baryonscatteringthatwasbasedonthechiralLagrangian[17]. 3 Baryon octet self energies It is straightforward to evaluate the one-loop fluctuation of the baryon octet states as implied by the interaction vertices specified in (8). There are two types of contributions that are characterized by intermediate states involving the octet [8] or decuplet [10] baryons. For an arbitrary dimension d we write: ddk iµ4 d G(B) 2 Σloop (p) = U−V QR γ k/S (p k)γ k/ B∈[8] QX∈[8]Z (2π)d k2 −m2Q +iǫ "RX∈[8] 2f  5 R − 5 (B) 2   G + QR γ Γ (k)γ Sµν(p k)Γν(k) ,  2f  0 †µ 0 R − # RX∈[10]   4 G(N)=√3(D+F) G(Λ)=2D G(Σ)= 2D G(Ξ)= √3(D F) πN πΣ πΛ √3 πΞ − − G(ηNN)=−D−√33F G(K¯ΛN) =− 23(D+3F) G(πΣΣ)=−√8F G(K¯ΞΛ) =−D−√33F G(KNΛ)=−D+√33F G(ηΛΛ)=−2p√D3 G(K¯ΣN) =√2(D−F) G(K¯ΞΣ) =−√3(D+F) G(N)=√3(D F) G(Λ) = 2(D 3F) G(Σ)= 2D G(Ξ)= D+3F KΣ − KΞ 3 − ηΣ √3 ηΞ − √3 p G(Σ)=√2(D+F) KΞ G(N)=2C G(Λ)= √3C G(Σ)= 2C G(Ξ)= C π∆ πΣ − πΣ − 3 πΞ − G(KNΣ)=C G(KΛΞ) =−√2C G(K¯Σ∆) =−p83C G(ηΞΞ)=−C G(ηΣΣ)=Cp G(K¯ΞΣ) =C G(Σ)= 2C G(Ξ) = √2C KΞ − 3 KΩ − Table 1 p (B) Meson-baryon coupling constants G with B [8] defined with respect to isospin QR ∈ states [17]. The upper blocks specify the coupling constants for R [8], the lower ∈ blocks the ones for R [10]. ∈ 1 γµγν (d 2)pµpν pµγν pνγµ Sµν(p) = − gµν − + − , R /p−MR +iǫ − d−1 − (d−1)MR2 (d−1)MR ! 1 Z S (p) = , Γ (k) = k γ k/, (10) R µ µ µ /p M +iǫ − 2 R − (B) with the notation for the meson-baryon coupling constants G suggested in QR [17]. We assume perfect isospin symmetry in this work. All coupling constants required in (10) are recalled in Tab. 1 where we apply the phase convention for the isospin states given in [17,27]. The meson and baryon masses m and Q M in the propagators are assumed to be physical, i.e. a partial summation R is assumed for the propagators. The parameter µ is the ultraviolet scale of UV dimensional regularization. It is long known that the expression (10) as it stands is at odds with chiral power counting rules [6]. Close to the baryon mass the one-loop expression should carry minimal chiral order Q3. The application of dimensional regularization in combination with the MS renormalization scheme leads to contributions of order Q0 and Q2. Any manifest Lorentz invariant formulation of chiral perturbation theory takes (10) as the starting point of the renormalization program [13,14,15]. Therefore it us useful to simplify first the expression (10). Applying the Passarino-Veltman reduction [20] we obtain the following form (B) 2 G Σloop (p) = QR a[8] (p)I +b[8] (p)I +c[8] (p)I (p2) B [8]  2f  QR R QR Q QR QR ∈ Q∈[8X],R∈[8] h i (B) 2  G + QR a[10](p)I +b[10](p)I +c[10](p)I (p2) ,(11)  2f  QR R QR Q QR QR Q∈[8X],R∈[10] h i   in terms of the invariant master loop functions 5 ddk iµ4 d ddk iµ4 d I = U−V , I = U−V , Q (2π)d k2 m2 +iǫ R (2π)d k2 M2 +iǫ Z − Q Z − R ddk iµ4 d 1 I (p2) = − U−V . (12) QR (2π)d k2 m2 +iǫ(p k)2 M2 +iǫ Z − Q − − R The coefficient functions are readily derived. The baryon-octet intermediate states define M2 +p2 M2 p2 a[8] (p) = M R /p, b[8] (p) = R − /p, QR − R − 2p2 QR 2p2 (M2 p2)2 m2 (M2 +p2) c[8] (p) = m2 M R − − Q R /p. (13) QR Q R − 2p2 The baryon-decuplet intermediate states lead to d 2 a[10](p) = − 2M p2 m2 M2 p2 QR 8(d 1)M2 p2 R Q − R − − R h (cid:16) (cid:17) 4 + m4 +2(M2 +p2)m2 (M2 p2)2 M2 p2 /p − Q R Q − R − − d R n o i d 2 4Z d(Z2 3) 6 b[10](p) = − 2M p2 − − − m2 +M2 p2 QR 8(d 1)M2 p2 R d 2 Q R − − R h (cid:16) − (cid:17) 4(Z 1)2 + m4 + − 2(Z2 2) p2 2M2 m2 +M4 (p2)2 /p Q d − − − R Q R − n (cid:16)(cid:16) (cid:17) (cid:17) o i d 2 c[10](p) = − 2M p2 m4 +2(M2 +p2)m2 QR 8(d 1)M2 p2 R − Q R Q − R h (cid:16) (M2 p2)2 + m6 3(M2 +p2)m4 − R − Q − R Q (cid:17) n + 3M4 +2p2M2 +3(p2)2 m2 (M2 p2)2(M2 +p2) /p . (14) R R Q − R − R (cid:16) (cid:17) o i 4 Baryon decuplet self energies We turn to the one-loop fluctuations of the baryon decuplet states. There are two terms induced by intermediate baryon octet and decuplet states. We write ddk iµ4 d G(B) 2 Σµν,loop(p) = − U−V QR γ k/Sµν(p k)γ k/ B∈[10] QX∈[8]Z (2π)d k2 −m2Q +iǫ "RX∈[10] 2f  5 R − 5 (B) 2   G + QR Γµ(k)S (p k)γ Γν, (k)γ , (15) R 0 † 0  2f  − # RX∈[8]   6 G(∆)=√2C G(Σ)= C G(Ξ)= C G(Ω) = 2C πN πΛ − πΞ − K¯Ξ − G(∆)= √2C G(Σ)= 2C G(Ξ) =C KΣ − πΣ − 3 K¯Λ G(K¯ΣN) = p23C G(K¯ΞΣ) =C G(Σ)=Cp G(Ξ)= C ηΣ ηΞ − G(Σ)= 2C KΞ − 3 G(∆)= 5H G(Σ)= √p8H G(Ξ)= 1H G(Ω) = 2 H π∆ − 3 πΣ 3 πΞ − 3 K¯Ξ −√3 G(η∆∆)=−p13H G(K¯Σ∆) =−√38H G(K¯ΞΣ) =−p√23H G(ηΩΩ)= √23H G(∆)= p2H G(Σ)=0 G(Ξ) = 2H KΣ − 3 ηΣ KΩ − 3 p G(Σ)= √8H G(Ξ)= 1pH KΞ 3 ηΞ √3 Table 2 (B) Meson-baryon couplingconstantsG withB [10] definedwithrespecttoisospin QR ∈ states [17,27]. Theupperblocks specify thecouplingconstants forR [8], thelower ∈ blocks the ones for R [10]. ∈ wherethebuilding blocks of(15)arespecified in(10).Alistwithcoupling con- (B) stants, G , required in (15) is provided in Tab. 2. Perfect isospin symmetry QR is assumed. The self energy tensor, Σ (p), determines the dressed propagator, S (p), by µν µν means of the Dyson equation S (p) = S(0)(p)+S(0)(p)Σαβ(p)S (p), (16) µν µν µα βν where the bare propagator, S(0)(p), follows from the expression in (10) upon µν using the bare decuplet mass. The Dirac-Lorentz structure of a spin three-half particle causes a little complication. It is convenient to decompose the self energy into a complete set of tensors [28] defined for arbitrary dimension d: pµpν Pµν(p) = gµν P (p) Vµ(p)P (p)Vν(p), 23± − p2 ± − ∓ (cid:16) (cid:17) pν Pµν = Vµ(p)P (p)Vν(p), Pµν = Vµ(p)P (p) , 12±,11 ∓ 21±,12 ∓ √p2 pµ pµ pν Pµν = P (p)Vν(p), Pµν = P (p) , 12±,21 √p2 ∓ 12±,22 √p2 ∓ √p2 1 /p 1 p/pµ P (p) = 1 , Vµ(p) = γµ . (17) ± 2 ± √p2 √d 1 − p2 (cid:16) (cid:17) − (cid:16) (cid:17) Any Dirac-Lorentz tensor, A (p), that depends on a single 4-momentum only µν can be represented as follows Aµν(p) = A23±(p2)P3µν(p)+ Ai12j±(p2)P1µν,ij(p), (18) 2± ij, 2± X± X± 7 A32±(p2) = d(d2 2) trP3µν(p)Aνµ(p), Ai21j±(p2) = d2 trP1µν,ji(p)Aνµ(p). 2± 2± − The information on the decuplet masses is encoded in the spin three-half components of the self energy. Owing to the projector properties of the tensors introduced in (17) those components are determined by the corresponding components of the self energy tensor. It holds 3 3 1 S2±(p2) = p2 M◦ Σ2±(p2) − , B − ∓ [10] − B M =M◦ +hqReΣ23+(M2), i (19) B [10] B B where we apply the quasi-particle definition of the decuplet masses. Like for the baryon-octet self energies it is useful to derive simplified and explicit representations of the spin three-half components of the decuplet self energies. Applying the Passarino-Veltman reduction we seek a representation for the one-loop contribution to the decuplet self energy of the form (B) 2 G Σloop (p) = QR a[8] (p)I +b[8] (p)I +c[8] (p)I (p2) B [10]  2f  QR R QR Q QR QR ∈ Q∈[8X],R∈[8] h i (B) 2  G + QR a[10](p)I +b[10](p)I +c[10](p)I (p2) ,(20)  2f  QR R QR Q QR QR Q∈[8X],R∈[10] h i   where the master loop functions were already introduced in (12). For nota- tional convenience we suppress the index 3+ in the self energies. It is straight- 2 forward to derive the various components M a[8] (p) = R 2p2 m2 M2 p2 QR 8(d 1)p2p2 Q − R − − h (cid:16) (cid:17) 4M2 p2 √p2 + m4 +2(M2 +p2)m2 (M2 p2)2 R , − Q R Q − R − − d M R n o i M b[8] (p) = R 2p2 m2 +M2 p2 QR 8(d 1)p2p2 − Q R − − h (cid:16) (cid:17) 2p2 √p2 + m4 2 M2 +2p2 m2 +M4 p2p2 , Q − R − d Q R − M R n (cid:16) (cid:17) o i M c[8] (p) = R 2p2 m4 +2(M2 +p2)m2 (M2 p2)2 QR 8(d 1)p2p2 − Q R Q − R − − h (cid:16) (cid:17) + m6 3(M2 +p2)m4 +(3M4 +2M2 p2 +3p2p2)m2 Q − R Q R R Q n √p2 (M2 p2)2(M2 +p2) , (21) − R − R M R o i 8 and 1 a[10](p) = 2p2 (d 4)(M2 +p2)m2 (d 2)m4 QR 8(d 1)2M p2p2 − R Q − − Q R − h (cid:16) 8M2 p2 +4d(3 2d)M2 p2 +2(M4 6p2M2 +p2p2)+ R − R R − R d (cid:17) + 2(d 2)(M2 +p2)m4 +2 (d 2)(M4 +p2p2) 4p2M2 m2 − − R Q − R − R Q n (cid:16) (cid:17) (M2 +p2) 4d2M2 p2 +d(M4 14p2M2 +p2p2) − R R R − R (cid:16) 8p2M2 √p2 2(M4 6p2M2 +p2p2) R , − R − − d M R (cid:17)o i 1 b[10](p) = 2p2 (d 2)m4 + (4 d)M2 QR 8(d 1)2M p2p2 − Q − R R − h (cid:16) (cid:16) 8 +(d 8+ )p2 m2 2(M4 p2p2) + (d 2)(M2 +p2)m4 − d Q − R − − R Q (cid:17) (cid:17) n 4p2M2 2 2 (d 2)M4 +(d 8)p2M2 + R 2(d 3+ )p2p2 m2 − − R − R d − − d Q (cid:16) (cid:17) √p2 +(M2 p2) (d 2)M4 +2d(2d 5)p2M2 +(d 2)p2p2 , R − − R − R − M R (cid:16) (cid:17)o i 1 c[10](p) = 2p2 (d 2)m6 2(d 3)(M2 +p2)m4 QR 8(d 1)2M p2p2 − Q − − R Q R − h (cid:16) + (d 6)(M4 +p2p2)+2d(2d 5)p2M2 m2 − R − R Q (cid:16) (cid:17) +2(M2 p2)2(M2 +p2) + (d 2)(M2 +p2)m6 R − R − R Q (cid:17) n 3(d 2)(M4 +p2p2)+2(d 6)p2M2 m4 − − R − R Q +((cid:16)M2 +p2) 3(d 2)(M4 +p2p2)+2d(2(cid:17)d 7)p2M2 m2 R − R − R Q (cid:16) (cid:17) √p2 (M2 p2)2 (d 2)(M4 +p2p2)+2d(2d 5)p2M2 . (22) − R − − R − R M R (cid:16) (cid:17)o i 5 Renormalization and power counting It is important to discriminate carefully two different issues. First, are the chi- ralWard identities satisfied and second are the power counting rules manifest? We point out that the one-loop expressions for the self energies (11, 20) are consistent with all chiral Ward identities simply because whatever symmetry the Lagrangian enjoys dimensional regularization preserves those at the level of the Green functions. The loop expansion does not cause a violation of the Ward identities. The task is to devise a renormalization scheme that preserves the Ward identities but leads at the same time to manifest power counting for the renormalized loop functions. 9 We focus on the renormalization for the one-loop expressions I , I and Q N I (p2) introduced in (12). It is emphasized that those master loops are the QR only ultraviolet divergent objects that arise in the computation of any one- loop diagramif the Passarino-Veltman reduction is applied. Any scalar master loop function that arises in the Passarino-Veltman reduction that is finite and non-trivial in the chiral domain behaves as dictated by power counting rules. The latter statement is almost trivial since for finite integrals dimensional counting is justified as long as performing the loop integrations commutes with taking the limit of large baryon masses. Scalar integrals that are trivial in the chiral domain, i.e. those ones that can be Taylor expanded in the soft momenta may violate the counting rules. This expectation was confirmed by explicit computations [14]. It is proven in the Appendix of the present work that given any one-loop integral that arises when computing one-baryon processes it is sufficient to renormalize the scalar master-loop functions of the Passarino-Veltman reduction in a manner that the latter are compatible with the expectation of chiral counting rules. Thus it is of central importance to consider the ultraviolet divergent master loop function. We first recall their well-known properties for arbitrary space- time dimension d. The tadpole loop has the form Γ(1 d/2) M2 (d−4)/2 I = M2 − R R R (4π)2 4πµ2! M2 2 M2 = R +γ 1 ln(4π)+ln R + (4 d) , (23) (4π)2 −4 d − − µ2 ! O − ! − UV where γ is the Euler constant. The expression for the mesonic tadpole, I , fol- Q lows by replacing the mass M in (23) by the meson mass m . In dimensional R Q regularization the divergent part of the master loop I (p2) is determined QR unambiguously by the tadpole specified in (23). The algebraic identity I I R Q I (0) = − , (24) QR m2 M2 Q − R holds for arbitrary values of d. If we slightly rewrite the Passarino-Veltman representation, using the subtracted master loop ∆I (p2) = I (p2) I (0), (25) QR QR QR − rather than the original loop function I (p2), the renormalization of the QR ultraviolet divergencies is reduced to the consideration of the tadpole terms only. We write 10