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The Width of the Roper Resonance in Baryon Chiral Perturbation Theory ∗ Jambul Gegelia A,B, Ulf-G

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The Width of the Roper Resonance in Baryon Chiral Perturbation Theory ∗ Jambul Gegelia A,B, Ulf-G Physics Letters B 760 (2016) 736–741 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb The width of the Roper resonance in baryon chiral perturbation theory ∗ Jambul Gegelia a,b, Ulf-G. Meißner a,c, De-Liang Yao a, a Institute for Advanced Simulation, Institut für Kernphysik and Jülich Center for Hadron Physics, Forschungszentrum Jülich, D-52425 Jülich, Germany b Tbilisi State University, 0186 Tbilisi, Georgia c Helmholtz Institut für Strahlen- und Kernphysik and Bethe Center for Theoretical Physics, Universität Bonn, D-53115 Bonn, Germany a r t i c l e i n f o a b s t r a c t Article history: We calculate the width of the Roper resonance at next-to-leading order in a systematic expansion of Received 17 June 2016 baryon chiral perturbation theory with pions, nucleons, and the delta and Roper resonances as dynamical Received in revised form 26 July 2016 degrees of freedom. Three unknown low-energy constants contribute up to the given order. One of them Accepted 26 July 2016 can be fixed by reproducing the empirical value for the width of the Roper decay into a pion and a Available online 29 July 2016 nucleon. Assuming that the remaining two couplings of the Roper interaction take values equal to those Editor: W. Haxton of the nucleon, the result for the width of the Roper decaying into a nucleon and two pions is consistent with the experimental value. © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license 3 (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . 1. Introduction Ref. [24]. The authors of Ref. [25] studied the impact of the ex- plicit inclusion of the Roper resonance in chiral EFT on the P 11 At low energies, chiral perturbation theory [1,2] provides a suc- pion–nucleon scattering phase shift, and Ref. [26] presented new cessful description of the Goldstone boson sector of QCD. It turns ideas on the extension of the range of applicability of chiral EFT out that a systematic expansion of loop diagrams in terms of small beyond the low-energy region. parameters in effective field theories (EFTs) with heavy degrees In this work we calculate the width of the Roper resonance in a of freedom is a rather complicated issue. The problem of power systematic expansion in the framework of baryon chiral perturba- counting in baryon chiral perturbation theory [3] may be solved tion theory with pions, nucleons, the delta and Roper resonances by using the heavy-baryon approach [4–6] or by choosing a suit- as explicit degrees of freedom. able renormalization scheme [7–10]. The resonance and (axial) The paper is organized as follows: in Section 2 we specify the vector mesons can also be included in EFT (see e.g. Refs. [11–20]). effective Lagrangian, in Section 3 the pole mass and the width On the other hand, the inclusion of heavier baryons such as the of the Roper resonance are defined and the perturbative calcula- Roper resonance is more complicated. tion of the width is outlined in Section 4. In Section 5 we discuss Despite the fact that the Roper resonance was found a long the renormalization and the power counting applied to the decay time ago in a partial wave analysis of pion–nucleon scattering data amplitude of the Roper resonance, while Section 6 contains the [21], a satisfactory theory of this state is still missing. The Roper numerical results. We briefly summarize in Section 7. is particularly interesting as it is the first nucleon resonance that 2. Effective Lagrangian exhibits a decay mode into a nucleon and two pions, besides the decay into a nucleon and a pion. Also, the Roper appears unexpect- We start by specifying the elements of the chiral effective La- edly low in the spectrum, below the first negative parity nucleon grangian which are relevant for the calculation of the width of the resonance, the S11(1535). It is therefore timely to address this Roper at next-to-leading order in the power counting specified be- state in a chiral EFT. First steps in this direction have been made low. We consider pions, nucleons, the delta and Roper resonances in Refs. [22–26]. In particular, the pion mass dependence of the as dynamical degrees of freedom. The corresponding most general pole mass and the width of the Roper resonance has been stud- effective Lagrangian can be written as ied in Refs. [22,23] and the magnetic moment was investigated in Leff = Lππ + Lπ N + Lπ + Lπ R + Lπ N + Lπ NR + LπR , (1) where the subscripts indicate the dynamical fields contributing to * Corresponding author. E-mail addresses: [email protected] (J. Gegelia), [email protected] a given term. From the purely mesonic sector we need the follow- (U.-G. Meißner), [email protected], [email protected] (D.-L. Yao). ing structures [2,27] http://dx.doi.org/10.1016/j.physletb.2016.07.068 0370-2693/© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. J. Gegelia et al. / Physics Letters B 760 (2016) 736–741 737 2 2 2 (2) F μ † F M † 3. The pole mass and the width of the Roper resonance Lππ = ∂μU∂ U + U + U, 4 4 1 1 The dressed propagator of the Roper resonance can be written L(4) = μ + + 2 ππ l4 u uμ χ+ (l3 l4) χ+ , (2) as 8 16 i where denotes the trace in flavor space, F is the pion decay = iSR (p) − − , (6) constant in the chiral limit and M is the leading term in the quark p/ mR0 R (p/) mass expansion of the pion mass [2]. The pion fields √are contained where −i R (/p) is the self-energy, i.e. the sum of all one-particle- × = in the unimodular unitary 2 2matrix U , with u U and irreducible diagrams contributing to the two-point function of the Roper resonance. The pole of the dressed propagator S R is ob- † † uμ = i u ∂μu − u∂μu , tained by solving the equation 2 −1 + M 0 S (z) ≡ z − mR0 − R (z) = 0 . (7) χ = u†χu† + uχ †u , χ = . (3) R 0 M2 We define the physical mass and the width of the Roper resonance by relating them to the real and imaginary parts of the pole The terms of the Lagrangian with pions and baryons contribut- ing to our calculation read: R z = mR − i . (8) 2 (1) ¯ 1 5 L = N iD/ − m + g u/γ N , The pertinent topologies of the one- and two-loop diagrams π N 2 contributing to the self-energy of the Roper resonance are shown (1) ¯ 1 5 in Fig. 1. We use BPHZ renormalization by subtracting real parts L = R iD/ − mR + gRu/γ R , π R 2 of loop diagrams in their chiral limit and replacing the parameters of the Lagrangian by their renormalized values. Imaginary parts L(2) = ¯ R + π R R c1 χ R , of loop diagrams remain untouched. All counter terms responsible for these subtractions are generated by the effective Lagrangian at (1) ¯ gπ NR μ L = R γ γ5uμ N + h.c., given order, however we do not show them explicitly. π NR 2 We solve Eq. (7) perturbatively order by order in the loop ex- 3 L(1) =−¯ i 2 jk − jk μν pansion. We parameterize the pole as π μξij iD/ mδ g z = m + h¯ z + h¯ 2 z + O h¯ 3 (9) μ ν, jk ν μ, jk μ jk ν jk μ ν 2 δ 1 δ 2 ( ), − i γ D + γ D + iγ D/ γ + mδ γ γ 0 R 2 0 where m2 = m + 4c M , with m the physical Roper mass in the g g R 1 R 1 jk μν 2 μ ν, jk ν, jk μ chiral limit, and substitute in Eq. (7) in which we write the self- + u/ γ5 g + (γ u + u γ )γ5 2 2 energy as an expansion in the number of loops 3 g3 + μ jk ν 2 l 2 3 γ u/ γ5γ ξ ν , = h¯ + h¯ + O h¯ (10) 2 kl R 1 2 ( ). 3 By expanding in powers of h¯ , we get L(1) = ¯ i 2 μα j + π N h μξij (z1) ωαN h.c., ¯ + ¯ 2 − ¯ − ¯ 2 3 hδz1 h δz2 h1(m2) h δz11(m2) L(1) = ¯ i 2 μα ˜ j + πR hR μξij (z) ωαR h.c., (4) 2 3 − h¯ 2(m2) + O(h¯ ) = 0 . (11) where N and R are isospin doublet fields with bare masses Solving Eq. (11) we obtain mN0 and mR0, corresponding to the nucleon and the Roper reso- nance, respectively. The vector–spinor isovector–isospinor Rarita– δz1 = 1(m2), Schwinger field ν represents the resonance [28] with bare 3 1 = + 2 i = i δz2 1(m2)1(m2) 2(m2). (12) mass m0, ξ is the isospin-3/2 projector, ωα 2 τ uα and μα = μα + μ α (z) g zγ γ , where z is a so-called off-shell param- Equations (8), (9) and (12) lead to the following expression for the eter. We fix the off-shell structure of the interactions involving the width delta by adopting g2 = g3 = 0 and z1 = z˜ = 0. Note that these off- shell parameters can be absorbed in LECs and are thus redundant = h¯ 2i Im [ (m )] R 1 2 [29–31].
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