Towards baryon-baryon scattering in manifestly Lorentz-invariant formulation of SU(3) baryon chiral perturbation theory

TL;DR

This paper develops a Lorentz-invariant framework for baryon-baryon scattering in SU(3) chiral perturbation theory, deriving integral equations and demonstrating perturbative renormalizability at leading order.

Contribution

It introduces a coupled-channel integral equation approach with renormalizable potentials in a Lorentz-invariant formulation of baryon chiral perturbation theory.

Findings

Leading-order potentials are perturbatively renormalizable.

Integral equations have unique solutions in all partial waves.

Finite subtractions improve ultraviolet convergence.

Abstract

We study baryon-baryon scattering by applying time-ordered perturbation theory to the manifestly Lorentz-invariant formulation of SU(3) baryon chiral perturbation theory. We derive the corresponding diagrammatic rules paying special attention to complications caused by momentum-dependent interactions and propagators of particles with non-zero spin. We define the effective potential as a sum of two-baryon irreducible contributions of time-ordered diagrams and derive a system of integral equations for the scattering amplitude, which provides a coupled-channel generalization of the Kadyshevsky equation. The obtained leading-order baryon-baryon potentials are perturbatively renormalizable, and the corresponding integral equations have unique solutions in all partial waves. We discuss the issue of additional finite subtractions required to improve the ultraviolet convergence of (finite) loop integrals on the example of nucleon-nucleon scattering in the $^3P_0$ partial wave. Assuming that corrections beyond leading order can be treated perturbatively, we obtain a fully renormalizable formalism which can be employed to study baryon-baryon scattering.

Equations

$${\mathcal{L}}_{\mathrm{eff}}={\mathcal{L}}_{\varphi }+{\mathcal{L}}_{\varphi \mathrm{B}}+{\mathcal{L}}_{\mathrm{BB}}+\cdots .$$

$${\mathcal{L}}_{\varphi \mathrm{B}}^{(1)}=\text{mbox}Tr\left\{\bar{\mathrm{B}}\left(i{\gamma }_{\mu }{D}^{\mu }-m\right)\mathrm{B}\right\}+\frac{D/F}{2}\text{mbox}Tr\left\{\bar{\mathrm{B}}{\gamma }_{\mu }{\gamma }_5[u^{\mu },\mathrm{B}{]}_{\pm }\right\},$$

$${\mathcal{L}}_{\mathrm{BB}}^{(0)}=C_i^1\text{mbox}Tr\left\{{\bar{\mathrm{B}}}_{\alpha }{\bar{\mathrm{B}}}_{\beta }{\left({\Gamma }_i\mathrm{B}\right)}_{\beta }{\left({\Gamma }_i\mathrm{B}\right)}_{\alpha }\right\}+C_i^2\text{mbox}Tr\left\{{\bar{\mathrm{B}}}_{\alpha }{\left({\Gamma }_i\mathrm{B}\right)}_{\alpha }{\bar{\mathrm{B}}}_{\beta }{\left({\Gamma }_i\mathrm{B}\right)}_{\beta }\right\}+C_i^3\text{mbox}Tr\left\{{\bar{\mathrm{B}}}_{\alpha }{\left({\Gamma }_i\mathrm{B}\right)}_{\alpha }\right\}\text{mbox}Tr\left\{{\bar{\mathrm{B}}}_{\beta }{\left({\Gamma }_i\mathrm{B}\right)}_{\beta }\right\},$$

$$\frac{g_A^2}{4F_0^2}\int \frac{d^{4}k}{(2\pi {)}^4}\frac{{\bar{u}}_{\alpha }(p_3){\left[{\gamma }_{5}k/\right]}_{\alpha \lambda }(p_3/+k/+m_N{)}_{\lambda \beta }{C}_{\beta \delta ,\mu \nu }{\bar{u}}_{\gamma }(p_4){\left[{\gamma }_{5}k/\right]}_{\gamma \sigma }(p_4/-k/+m_N{)}_{\sigma \delta }{u}_{\mu }(p_1)u_{\nu }(p_2)}{\left(k^2-M_{\pi }^2+iϵ\right)\left((p_3+k{)}^2-m_N^2+iϵ\right)\left((p_4-k{)}^2-m_N^2+iϵ\right)},$$

$$u(q)=\left(\frac{\omega(q,m_{N})+m_{N}}{2m_{N}}\right)^{1/2}\left(\begin{array}[]{c}\chi\\ \frac{\vec{\sigma}\cdot\vec{q}\ \chi}{\omega(q,m_{N})+m_{N}}\end{array}\right),$$

$$(p/+m_N{)}_{\mu \nu }\equiv 2m_N\sum u_{\mu }(p){\bar{u}}_{\nu }(p)+\left[p_0-\omega (p,m_N)\right][{\gamma }_0{]}_{\mu \nu },$$

$$\frac{1}{q^2-m^2+iϵ}$$

$$\frac{q^0}{q^2-m^2+iϵ}$$

$$\frac{{(q^0)}^2}{q^2-m^2+iϵ}$$

$$S_{\beta \alpha }$$

$$(2\omega (p_i,M_i){)}^{-1}$$

$$\frac{m_j}{\omega (p_j,m_j)}\sum u(p_j)\bar{u}(p_j),$$

$$\frac{m_i}{\omega (p_i,m_i)}\sum u(p_i)\bar{u}(p_i)-{\gamma }_0,$$

$${\left[E-\sum _{\gamma }\omega (p_{\gamma },m_{\gamma })+iϵ\right]}^{-1},$$

$$M_{BB}=Z_B^2\bar{u}(p_3)\bar{u}(p_4){\tilde{\Gamma }}_{4B}u(p_1)u(p_2)\equiv Z_B^2\tilde{\mathcal{M}},$$

$$\tilde{\mathcal{M}}=\tilde{V}+\bar{V}G\bar{V}+\bar{V}GVG\bar{V}+\bar{V}GVGVG\bar{V}+\cdots =\tilde{V}+\bar{V}G\bar{V}+\bar{V}G\mathcal{M}G\bar{V},$$

$$\mathcal{M}=V+VG\mathcal{M}.$$

$$\mathcal{M}$$

$$V$$

$$G$$

$$G^{IJ}(E)=\frac{1}{{\omega }_I{\omega }_J}\frac{m_I{m}_J}{E-{\omega }_I-{\omega }_J+iϵ},$$

$$T_{l^{\prime }l,s^{\prime }s,j}^{IJ,KL}\left(E,p^{\prime },p\right)$$

$$V_{0,M_P}^{IJ,KL}$$

$$\sqrt{\omega \left(p,m\right)+m}=\sqrt{2m}+\left[\sqrt{\omega \left(p,m\right)+m}-2m\right]=\sqrt{2m}+\mathcal{O}(p^2/m),$$

$$V_{\mathrm{LO},M_P}^{IJ,KL}$$

$$\sum _{Q,R}\int \frac{d^3\vec{k}}{(2\pi {)}^3}{X}_{VGV}\equiv \sum _{Q,R}\int \frac{d^3\vec{k}}{(2\pi {)}^3}{V}_{\mathrm{LO},M_P}^{IJ,QR}{G}^{QR}(E)V_{\mathrm{LO},M_P}^{QR,KL},$$

$$X_{VGV}$$

$$-\frac{g_A^2}{4F_0^2}\frac{{\vec{\tau }}_1\cdot {\vec{\tau }}_2}{\omega (p-p{}^{\prime },M_{\pi })}\frac{4m_N^2}{(m_N+\omega (p,m_N))(m_N+\omega (p^{\prime },m_N))}\frac{\left[{\vec{\sigma }}_1\cdot \left(\vec{p}-\vec{p}{}^{\prime }\right)\right]\left[{\vec{\sigma }}_2\cdot \left(\vec{p}-\vec{p}{}^{\prime }\right)\right]}{\omega (p-p{}^{\prime },M_{\pi })+\omega (p,m_N)+\omega (p^{\prime },m_N)-E-iϵ},$$

$$V_{\mathrm{LO}}^{{}^3{P}_0}\left(p{}^{\prime },p\right)=Cp{}^{\prime }p+V_{\pi }\equiv V_C+V_{\pi },$$

$$T=V+VGT,$$

$$V_C(p^{\prime },p)=\bar{\xi }(p^{\prime })\mathcal{C}\xi (p),$$

$$T=T_{\pi }+(1+T_{\pi }G)\bar{\xi }\mathcal{X}\xi (1+G{T}_{\pi }),$$

Full text

I Introduction

Nuclear systems with non-vanishing strangeness provide an important connection between nuclear, particle and astrophysics. Hypernuclei emerging from replacing one or several nucleons in a nucleus by hyperons serve as a testing ground for effects of strange quarks in nuclear matter. In addition to the nuclear forces, hyperon-nucleon (YN) interactions play a crucial role for understanding hypernuclear binding. Experiments with hypernuclei aiming at the determination of the YN, hyperon-hyperon (YY) and cascade-nucleon ($\Xi$N) interactions are carried out at various laboratories world-wide such as CERN, DA$\Phi$NE, GSI, JLab, J-PARC, KEK, MAMI, RHIC and will also be performed at the future FAIR facility, see Refs. Pochodzalla:2011rz ; Esser:2013aya ; Feliciello:2015dua ; Gal:2016boi for review articles of hypernuclear physics.

Given that experiments involving hypernuclei are rather challenging, a particularly valuable source of information on YN and YY interactions is provided by lattice QCD. Several lattice-QCD groups including HAL QCD Doi:2017zov ; Nemura:2017vjc ; Sasaki:2018mzh and NPLQCD Beane:2013br ; Beane:2012ey ; Beane:2012vq ; Beane:2010em ; Beane:2008dv Collaborations, Yamazaki et al. Yamazaki:2012hi ; Yamazaki:2015asa and the Mainz group Hanlon:2018yfv have already produced interesting results on baryon-baryon (BB) systems, and more results will become available in the near future. Although some lattice simulations are already approaching the physical values of the light quarks, most of the available lattice QCD calculations still correspond to unphysically large values of quark masses and require a reliable theoretical framework to perform extrapolations to their physical values.

Chiral effective field theory (ChEFT) for few-baryon systems offers a natural approach to analyze low-energy properties of (hyper)nuclei and to perform chiral extrapolations. It goes back to the seminal papers by Weinberg, who proposed a way of extending chiral perturbation theory to systems involving two and more nucleons Weinberg:rz ; Weinberg:um . In the resulting ChEFT approach the power counting rules are applied to the effective BB potential defined as a sum of contributions of two-baryon-irreducible diagrams. The scattering amplitude is then obtained by solving the Lippmann-Schwinger (LS) or Schrödinger equations. For reviews of ChEFT in the few-body sector see Refs. Bedaque:2002mn ; Epelbaum:2005pn ; Epelbaum:2008ga ; Machleidt:2011zz ; Epelbaum:2012vx .

The Bonn-Jülich and later Munich groups have pioneered chiral EFT for BB interactions in the strange sector using the non-relativistic formulation Meissner:2016ood ; Haidenbauer:2015zqb ; Petschauer:2015nea ; Haidenbauer:2014rna ; Haidenbauer:2013oca ; Haidenbauer:2011za ; Haidenbauer:2011ah ; Haidenbauer:2009qn ; Polinder:2007mp ; Haidenbauer:2007ra ; Polinder:2006zh ; Haidenbauer:2018gvg ; Haidenbauer:2014uua ; Haidenbauer:2017sws . In these studies the standard Weinberg power counting for nucleon-nucleon (NN) interactions extended to the strangeness-$S=-1$ (i.e. $\Lambda$N, $\Sigma$N) and strangeness-$S=-2$ (i.e. $\Lambda\Lambda$, $\Sigma\Sigma$ and $\Xi$N) systems is utilized. At leading order (LO) in the Weinberg power counting, the two-baryon potentials consist of four-baryon contact interactions without derivatives and the one-pseudoscalar-meson exchange. At next-to-leading order (NLO) one has to take into account the contributions from two-pseudoscalar-meson exchange diagrams and from four-baryon contact interactions with two derivatives or a single insertion of the quark masses. To regularize the ultraviolet divergences appearing from iterations of the LS equation already at leading order, a finite-cutoff regularitazion has been employed using exponential cutoffs in the range of $500\ldots 700$ MeV.

A modified approach to NN scattering has been proposed in Ref. Epelbaum:2012ua . This novel framework employs time-ordered perturbation theory (TOPT) and relies on the manifestly Lorentz-invariant effective Lagrangian. It offers a perturbatively renormalizable modification of Weinberg’s approach and has already been explored in the non-strange sector Epelbaum:2013ij ; Epelbaum:2013naa ; Epelbaum:2015sha . Dirac spinors have been kept in their full form and an alternative power counting has been suggested in Ref. Ren:2016jna . First applications of the formalism based on the Lorentz-invariant Lagrangian to BB systems with non-zero strangeness can be found in Refs. Li:2016mln ; Ren:2018xxd ; Li:2016paq ; Li:2018tbt ; Song:2018qqm .111The first application of the modified Weinberg approach to hadronic molecules can be found in Ref. Baru:2015tfa . In these exploratory studies, the BB scattering amplitudes were obtained by solving the (generalized) Kadyshevsky equation kadyshevsky , however, a systematic approach to the SU(3) sector using TOPT, which would allow for a straightforward generalization beyond the leading order, has not been formulated yet. In this paper we fill this gap by generalizing the modified Weinberg approach of Ref. Epelbaum:2012ua to the SU(3) sector and work out in detail the TOPT diagrammatic rules for particles with non-zero spin and interactions involving time derivatives. To achieve this goal, we start with the Lorentz-invariant effective Lagrangian and derive the diagrammatic rules of TOPT by integrating over zeroth components of loop momenta in Feynman diagrams for BB scattering. Special care is taken to deal with complications caused by momentum-dependent interactions and propagators of particles with non-zero spin. We provide details which are not given in Ref. Epelbaum:2012ua and following papers. The obtained rules of TOPT can be applied systematically to all orders in the loop expansion. Using the standard Weinberg power counting for diagrams contributing to BB scattering, one has to take into account an infinite number of graphs already at LO. We define the effective potential as a sum of all possible two-baryon-irreducible TOPT diagrams and obtain the scattering amplitudes by solving the corresponding integral equations. The resulting formulation permits a systematic investigation of few-baryon systems using both a renormalizable approach, which relies on a perturbative treatment of corrections beyond LO and allows one to completely eliminate the ultraviolet cutoff, and a conventional scheme based on iterating a truncated potential to all orders, which requires the ultraviolet cutoff to be chosen of the order of the hard scale of the problem Lepage:1997cs ; Gegelia:1998iu ; Park:1998cu ; Lepage:1999kt ; Epelbaum:2004fk ; Gegelia:2004pz ; Epelbaum:2006pt ; Epelbaum:2018zli . In the latter case, the manifestly Lorentz-invariant formulation of chiral EFT is expected to permit a larger cutoff variation as compared to the conventional non-relativistic framework, which is especially important for the SU(3) sector.

Our paper is organized as follows: in section II we work out the rules of TOPT for a system of baryons interacting with pseudoscalar mesons including momentum-dependent vertices. A system of integral equations for BB scattering is derived in section III. Next, in section IV we discuss the LO BB potential and the renormalization. The results of our work are summarized in section V.

II Diagrammatic rules in time-ordered perturbation theory

To formulate the theoretical framework describing BB scattering in SU(3) baryon chiral perturbation theory (BChPT) by applying the rules of TOPT we start with the manifestly Lorentz-invariant effective Lagrangian. It consists of the purely mesonic, single-baryon, two-baryon, $\ldots$ parts,

[TABLE]

The effective Lagrangian is organised as an expansion in powers of the quark masses and derivatives. The lowest-order mesonic Lagrangian can be found in Ref. Gasser:1984yg . The lowest-order Lagrangian in the single-baryon sector is given by

[TABLE]

where $D$ and $F$ are coupling constants, $D_{\mu}{\rm B}=\partial_{\mu}{\rm B}+[[u^{\dagger},\partial_{\mu}u],{\rm B}]$ denotes the covariant derivative with $u_{\mu}=iu^{\dagger}\partial_{\mu}Uu^{\dagger},u^{2}=\exp\left(\sqrt{2}\,iP/F_{0}\right).$ Next, $P$ and $B$ are the irreducible octet representations of ${\rm SU}(3)_{f}$ for the Goldstone bosons and baryons, respectively (see, e.g. Ref. Petschauer:2015nea ), and $F_{0}$ is the meson decay constant in the chiral limit and we consider the isospin-symmetric case.

The effective BB Lagrangian contains terms with an increasing number of derivatives acting on the baryon field. Using field redefinitions and re-organizing certain terms one can achieve that the four-baryon effective Lagrangian contributing to the LO BB potential involves only the following terms without derivatives Polinder:2007mp :

[TABLE]

where $C_{i}^{1}$, $C_{i}^{2}$ and $C_{i}^{3}$ are coupling constants, $\alpha$ and $\beta$ are the Dirac spinor indices and $\Gamma_{1}=1,\ \ \ \Gamma_{2}=\gamma^{\mu},\ \ \ \Gamma_{3}=\sigma^{\mu\nu},\ \ \ \Gamma_{4}=\gamma^{\mu}\gamma_{5},\ \ \ \Gamma_{5}=\gamma_{5}.$ Notice that the $\Gamma_{5}$-term yields a vanishing contribution at LO, i.e. it starts contributing at NLO.

Our aim is to obtain diagrammatic rules of TOPT. Notice that the systematic technique of Ref. Sterman:1994ce leads to the TOPT rules, which are not directly applicable to vertices involving time components of momenta. Additional complications emerge from treating particles with non-zero spin. In particular, zeroth components of momenta in the numerators of propagators of particles with non-zero spin cannot be substituted by their on-shell values. To our surprise, we were unable to find in the literature a more complete treatment of TOPT applicable to the case at hand.

To obtain the rules of TOPT we follow the usual procedure by first drawing all possible Feynman diagrams (in principle, an infinite number of them) relevant for the process of interest, assigning the momenta to propagators associated with internal lines and performing the trivial momentum integrations using the delta functions appearing at the vertices. The remaining overall delta function ensures momentum conservation for the external legs of a diagram. Next, we perform integrations over the zeroth components of the loop momenta. This leads to a decomposition of each Feynman diagram into a sum of time-ordered diagrams.

To demonstrate the above procedure we consider a Feynman diagram contributing to the NN scattering amplitude and depicted in the third line of Fig 1. Omitting SU(3) coefficients, the corresponding expression reads

[TABLE]

where $g_{A}=D+F$ is the axial coupling, $p_{1}$ and $p_{2}$ ($p_{3}$ and $p_{4}$) are the four-momenta of incoming (outgoing) nucleons with $p_{i}^{2}=m_{N}^{2}$ and $C_{\beta\delta,\mu\nu}$ stands for the four-nucleon interaction with the Dirac-spinor indices $\beta\delta,\mu\nu$.

In our calculations we use the Dirac spinors with the four-momentum $q$

[TABLE]

where $\omega(q,M):=\left(\vec{q}\,{}^{2}+M^{2}\right)^{1/2}$, $\chi$ is the two-component spinor, and for the nucleon propagator we apply

[TABLE]

where the summation is done over polarizations. Next, we make use of the following identities for the propagator with the momentum $q$ and the mass $m$

[TABLE]

These relations turn out to be particularly useful for deriving the diagrammatic rules of TOPT. Then, we apply the relations given by Eqs. (II) to the pion propagator and simplify the expression for the amplitude by collecting terms with equal denominators. Those terms which have all poles on the same side of the real axis in the complex $k_{0}$-plane vanish after the integration over $k_{0}$ as a consequence of the Cauchy theorem. For the remaining terms, closing the contour of integration on the sides with single poles and picking up the corresponding residues we obtain eight contributions which can be represented as time-ordered diagrams shown in first two lines in Fig 1. We stress that due to the cancellation of the $k_{0}$-dependent vertices with the pion propagator, see the first term (i.e. the unity) on the right-hand side of the last line of Eq. (II), the Feynman diagram shown in Fig. 1 contains also purely short-range TOPT contributions. It is however important to understand that a set of TOPT contributions, as shown by the diagrams in Fig. 1, is not unique and can be identically regrouped in such a way that the individual contributions are different but their sum is the same. Indeed, instead of applying Eqs. (II) to the pion propagator we could first use the identity $k^{0}=(p_{3}^{0}+k^{0})-p_{3}^{0}$ and then use Eqs. (II) for the nucleon propagators, which would result in different topologies. In particular, no purely short-range TOPT contributions would emerge in this case. While in general the choice of TOPT topologies is just a matter of convenience, in some cases it is beneficial to employ Eqs. (II) to the nucleon propagator to separate the irreducible contribution (a contribution which does not possess the NN cut) from the Feynman diagram, see e.g. Ref. Lensky:2005jc where this trick was applied to identify the irreducible one-loop diagrams contributing to the reaction $NN\to NN\pi$.

For the sake of compactness, we do not provide the explicit TOPT expressions corresponding to the amplitude given by Eq. (4) but rather formulate the general rules of TOPT for processes with baryons and pseudoscalar mesons in the initial and final states. 222Modifications are required for processes involving anti-baryons in initial and final states. In the absence of fermion fields and interaction terms with time derivatives, the rules given below reduce to the ones of Ref. Weinberg1 .

The $S$ matrix for a process $\alpha\to\beta$ can be written as

[TABLE]

where $P^{\mu}$ is the total four-momentum and $\Pi_{B/F}^{\alpha,\beta}$ denotes a product of the expressions following these symbols over all bosons (labelled by B)/fermions (labelled by F) in the initial and final states. The invariant amplitude $M_{\beta\alpha}$ is obtained by using the following diagrammatic rules:

• •

Draw all possible time-ordered diagrams for the process $\alpha\to\beta$.333Do not include diagrams with closed baryon lines - for low-energy processes their contributions are taken into account by redefining coupling constants of the effective Lagrangian. That is, draw each Feynman diagram with $N$ vertices $N!$ times while ordering the vertices in every possible way in a sequence running from right to left. Label each line with a four-momentum $p=(p_{0},\vec{p})$ as prescribed by the corresponding Feynman diagram.

• •

Include a factor

[TABLE]

for every internal line corresponding to a pseudoscalar meson with the mass $M_{i}$ and four-momentum $p_{i}$.

• •

For every internal baryon line with the momentum $p_{j}$ and mass $m_{j}$ include a factor

[TABLE]

where summation is done over polarizations.

• •

For every internal anti-baryon line (i.e. the line with baryon number flowing opposite to the time direction) with the momentum $p_{i}$ and mass $m_{i}$ include a factor

[TABLE]

where summation is done over polarizations.

• •

For every incoming (outgoing) external fermion with momentum $p$ ($p^{\prime}$) include $u(p)$ ($\bar{u}(p^{\prime})$).

• •

For interaction vertices, use the ones of the standard Feynman rules. Care has to be taken of zeroth components of momenta appearing in the vertices. Indeed, as follows from Eq. (II), for interactions containing one power of the zeroth component of a momentum, this zeroth component has to be replaced by the energy of the particle carrying this momentum if it corresponds to a particle and by $-1$ times the energy if the line corresponds to an antiparticle. If a Feynman diagram involves two vertices containing zeroth component of a momentum corresponding to the same line, the rule of the previous sentence applies to each of the vertices. However, in addition, one has to include also the TOPT diagrams where the two vertices and the connecting propagator cancel each other thus resulting in an effective contact contribution – see the unity in the very last line of Eq.(II). We also note that interaction terms containing two or more time derivatives acting on the same field are always eliminated from the effective Lagrangian by using suitable field redefinitions.

• •

For every intermediate state, i.e. a set of lines between any two vertices corresponding to particles enumerated with $\gamma$, include an energy denominator

[TABLE]

where $E$ is the total energy of the system, i.e. the sum of the energies of all particles in initial/final state.

• •

Integrate over all internal momenta ($\int d^{3}\vec{k}_{i}/(2\pi)^{3}$ for each $k_{i}$), and add together the contributions from all time-ordered diagrams.

III Integral equations for baryon-baryon scattering

The BB scattering amplitude $M_{BB}$ is obtained from the four-point vertex function $\tilde{\Gamma}_{4B}$ by applying the standard LSZ formula

[TABLE]

where $Z_{B}$ is the residue of the dressed baryon propagator and $u$, $\bar{u}$ are Dirac spinors corresponding to the incoming and outgoing baryons. The on-shell amplitude $\tilde{\cal M}$ is given as a sum of an infinite number of TOPT diagrams. Notice that it does not include diagrams with corrections on external legs.

For BB scattering, the purely two-baryon intermediate states are enhanced Weinberg:rz . Therefore, it is convenient to define the effective potential as a sum of all possible two-particle irreducible TOPT diagrams. The amplitude $\tilde{\cal M}$ is then given by an infinite series

[TABLE]

where $G$ is the two-baryon Green’s function and $\tilde{\cal M}$, ${\cal M}$, $\tilde{V}$, $\bar{V}$ and $V$ are the on-shell amplitude, the off-shell amplitude, the on-shell potential, the half-off-shell potential and the off-shell potential, respectively. The on-shell potential $\tilde{V}$ does not include diagrams with corrections on external legs. The half-off-shell potential $\bar{V}$ does not include diagrams with corrections on external legs with on-shell momenta while the off-shell potential $V$ also includes diagrams with corrections on external legs.

The off-shell amplitude ${\cal M}$ can be obtained by solving the following equation:

[TABLE]

To cover all different processes with strangeness $S=0,-1,-2$ in the isospin limit, Eq. (15) has to be understood as a $7\times 7$ matrix equation, (a generalization of the Kadyshevsky equation kadyshevsky ) where

[TABLE]

and the two-body Green functions read

[TABLE]

where $m_{I}$ and $\omega_{I}$ are the mass and energy of the $I$-th baryon.

We calculate the BB scattering amplitude in the center-of-mass system (CMS) and denote the three-momenta of the incoming and outgoing baryons by $\vec{p}$ and $\vec{p}\,^{\prime}$, respectively. In the partial wave basis, Eq. (15) leads to the following coupled-channel equations with the partial wave projected potential $V^{IJ,KL}_{l^{\prime}l,s^{\prime}s,j}\left(p^{\prime},p\right)$,

[TABLE]

where $IJ,KL$ and $PQ$ denote initial, final and intermediate particle channels, $l$, $l^{\prime\prime}$, $l^{\prime}$ and $s$, $s^{\prime\prime}$, $s^{\prime}$ correspond to their orbital angular momentum and their spin, respectively, while $j$ refers to the total angular momentum of BB states. A standard UV counting shows that in the limit of large integration momenta, considered for the same potential, Eq. (32) with the Green functions of Eq. (31) has a milder UV behaviour than the corresponding Lippmann-Schwinger equation. Therefore, its solutions are expected to show less sensitivity to the variation of the cutoff parameter.

IV LO Baryon-baryon potential and renormalization

We apply the standard Weinberg power counting to the BB interaction potential, thus representing it as an expansion with a finite number of TOPT diagrams at any given order. Diagrams contributing to the LO effective potential $V_{\rm LO}$ for BB scattering are shown in Fig. 2. The potential consists of the short-range contact interaction part $V_{0,C}^{IJ,KL}$, the long-range part generated by two one-pseudoscalar-meson-exchange time-ordered diagrams and the additional short-range contact term due to the momentum-dependent interactions as explained in the diagrammatic rules. Adding together the last three contributions (i.e. all but $V_{0,C}^{IJ,KL}$) and performing simplifications, up to higher-order effects, we obtain the following expression for the one-meson exchange contribution

[TABLE]

where $q=p_{I}-p_{K}=p_{L}-p_{J}$. The isospin factors ${\cal I}_{IJ,KL}$ and the values of $f_{IKP}$ can be found in Refs. Haidenbauer:2011ah ; Haidenbauer:2007ra .

The expressions for the LO contact interactions relevant for our TOPT approach ($V_{0,C}^{IJ,KL}$) are derived straightforwardly from the Lagrangian of Eq.(3) and can be found in Refs. Ren:2016jna ; Li:2016mln ; Li:2018tbt . It is important to emphasize that both the one-meson exchange of Eq.(33) and the contact interaction potentials contain also higher-order contributions according to Weinberg’s power counting. For example, for the contact interaction those emerge from the relativistic energy-dependent normalization factors of the nucleon spinors, as given by Eq. (5). To single out LO contact interactions, we rewrite the relativistic expressions with baryon energies as

[TABLE]

and shift the term in the square brackets to the higher-order BB potential. By doing so we obtain contact interactions which are identical to those of the non-relativistic approach, see Refs. Polinder:2006zh ; Polinder:2007mp . We also perform such splitting for the terms in the numerator of the expression in Eq. (33), that is we replace the particle energies by their masses up to higher-order effects. However, in full analogy to the treatment of the Green function (see Eq. (31)), we keep the full energy expressions in the denominator in Eq. (33), because performing their non-relativistic expansion is not commutative with loop integration Epelbaum:2012ua . While after renormalization relativistic effects as such are not expected to play a significant role at low energies, keeping the full unexpanded expressions in the denominators allows one to carry out renormalization of the LO BB amplitude by obtaining cutoff independent results – see further discussion below. Thus, for the one-meson-exchange contribution to the LO potential we finally obtain

[TABLE]

The difference between $V_{0,M_{P}}^{IJ,KL}$ and $V_{{\rm LO},M_{P}}^{IJ,KL}$ (cf. Eqs. (33) and (IV)) is included in the higher-order corrections. The resulting LO long-range potential has a milder ultraviolet behaviour than its analog obtained by using the LO approximation for Dirac spinors as done in Ref. Epelbaum:2012ua . A distinctive feature of the potential of Eq. (IV) as compared to its non-relativistic analog is that its iterations within the integral equations (32) lead to ultraviolet finite diagrams. To demonstrate this feature, consider e.g. the one-loop integral

[TABLE]

where $\vec{p}_{Q}=\vec{p}_{I}-\vec{k}$ and $\vec{p}_{R}=\vec{p}_{J}+\vec{k}$. For $k\equiv|\vec{k}|\to\infty$, we obtain for the integrand

[TABLE]

where two factors of $1/k^{3}$ stem from the denominators of the one-meson exchange potentials of Eq. (IV) and an additional factor of $1/k^{3}$ represents the UV behavior of the Green function of Eq.(31). The integrand $X_{VGV}$ behaves as $\sim 1/k^{5}$ and thus leads to an UV convergent one-loop integral. Analogously, it can be easily shown that all iterations of the one pseudoscalar meson-exchange potential lead to UV finite diagrams. Then, the full LO potential, which also contains the contact interactions, is perturbatively renormalizable since all divergences appearing from its iterations can be absorbed in the coupling constant of the contact interaction. As a consequence, the ultraviolet cutoff can be safely removed (set to infinity) at LO which allows one to avoid finite-cutoff artefacts inherent to the conventional non-relativistic framework. Also, in this approach, one does not face a well-known issue of the integral equation having non-unique solution for singular attractive potentials (this is e.g. the case for the sufficiently strong $1/r^{2}$ attractive potential substituted in the Lippmann-Schwinger equation) – see, e.g., Ref. Epelbaum:2012ua for more details. For our LO potential the integral equations (32) for the BB scattering amplitudes have unique solutions for all partial waves.

Although the LO potential can be properly renormalized as discussed above, in certain channels of BB scattering one may still run into the situation that corrections beyond LO are large enough to require their nonperturbative treatment. Below, we address this issue in detail on the example of NN scattering.

The LO NN potential consists of two momentum-independent contact interactions contributing to S-waves and the one-pion exchange potential (OPEP) corresponding to Eq. (IV), which has the form

[TABLE]

where $E=2\sqrt{m_{N}^{2}+q_{\rm on}^{2}}$ with $q_{\rm on}$ being the absolute value of the three-momentum of nucleons in the center-of-mass frame. The LO potential, substituted in the equations of (32), is perturbatively renormalizable and the integral equations have unique solutions. As a general trend, the calculations done at LO provide a reasonable description of the empirical phase shifts as shown in Fig. 3.

However, one recognizes large deviations from the Nijmegen partial wave analysis (PWA) for the cases of the 1S0 and 3P0 partial waves. In the 1S0 channel, the observed discrepancy can be traced back to the large (as compared to the inverse range of the OPEP) experimental value of the effective range. In the 3P0 channel, the OPEP is known to become non-perturbative at rather low momenta Birse:2005um . Given that in this channel the loop integrals in our Lorentz-invariant formulation are effectively cut off at momenta of the order of the nucleon mass and there is no contact interaction at LO, the observed discrepancy does not come as a surprise. For both the 1S0 and 3P0 partial waves, the large differences between the LO results and the empirical phase shifts suggest that at least a part of the subleading corrections must be treated non-perturbatively. While the non-perturbative inclusion of pion-exchange potentials beyond LO would generally destroy the explicit renormalizability feature of our approach and thus prevent one from eliminating the cutoff, it is still possible to treat the sub-leading contact interactions non-perturbatively within a cutoff-independent approach in the way consistent with the principles of EFT. For the 1S0 channel, it was already demonstrated in Ref. Epelbaum:2015sha that the proper inclusion of the NLO contact interactions in the considered non-perturbative approach results in the significant improvement for the phase shift, see also Ref. Baru:2015ira for related discussion in the context of low-energy theorems. The use of the modified OPE potential from Eq. (38) in the ${}^{1}S_{0}$ case is expected to produce results similar to those of Ref. Epelbaum:2015sha .

In the following, we consider in detail the 3P0 partial wave. To improve the description of the 3P0 phase shift, we add the lowest-order contact interaction term to the potential which is treated non-perturbatively thus obtaining

[TABLE]

where $V_{\pi}$ stands for the OPEP of Eq. (38) projected onto the ${}^{3}P_{0}$ partial wave. For the above potential it is possible to write the solution to the integral equation in such a form (analogously to Ref. Kaplan:1996xu ), which allows one to carry out the subtractive renormalization. For this purpose, we write the integral equations (32) symbolically as

[TABLE]

and, analogously to Ref. Epelbaum:2015sha , present their solution, for a separable contact interaction potential

[TABLE]

as

[TABLE]

where

[TABLE]

and the amplitude $T_{\pi}$ satisfies the equation

[TABLE]

In a close analogy to Ref. Epelbaum:2015sha , we apply the subtractive (BPHZ-type) renormalization, i.e. we subtract all divergences in loop diagrams and replace the coupling constants by their renormalized, finite values (see, e.g., Ref. Collins:1984xc for further details of BPHZ renormalization). Subtractive renormalization of the considered problem corresponds to the inclusion of contributions of an infinite number of counter terms generated by bare parameters of the effective Lagrangian Epelbaum:2018zli .

We need to apply subtractive renormalization to the expression of Eq. (42), where for $V_{C}$ from Eq. (39) we have

[TABLE]

Furthermore, by analysing the asymptotic behavior of the OPEP of Eq.(38) in the 3P0 channel,

[TABLE]

one is led to conclude that the amplitude $T_{\pi}$ is finite, and so are $\bar{\Xi}(p^{\prime})=(1+T_{\pi}\,G)\,\bar{\xi}$ and $\Xi(p)=\xi(1+G\,T_{\pi})$. All (multi-loop) sub-diagrams contained in $\xi\,G\,T_{\pi}G\,\bar{\xi}$ are also finite, so that this quantity contains only the overall logarithmic divergence which stems from the regime $p\to\infty,\,p^{\prime}\to\infty$ in Eq. (46). On the other hand, the term $\xi\,G\,\bar{\xi}$ is quadratically divergent and therefore requires additional BPHZ subtractions. Indeed, in the limit when the cutoff $\Lambda$ is much larger than the nucleon mass $\xi\,G\,\bar{\xi}$ can be written as

[TABLE]

where $f(E)$ is a finite function. Applying the subtractive renormalization, the final renormalized expression reads

[TABLE]

where the $\alpha$ subtracts the overall divergence of $\xi\,G\,T_{\pi}G\,\bar{\xi}$. The subtracted expression of $(\xi\,G\,\bar{\xi})^{R}$ is given in Ref. Epelbaum:2015sha

[TABLE]

where $\mu$ is the renormalization scale. In practice, we fix the bare constant $1/C=1/C_{R}+\alpha$ as a function of the cutoff numerically in such a way that it cancels the divergent part of $\xi\,G\,T_{\pi}G\,\bar{\xi}$ and the resulting cutoff-independent scattering amplitude describes the phase shift for a fixed value of the energy chosen to be 20 MeV. The resulting NN phase shift in the ${}^{3}P_{0}$ channel is plotted in Fig. 4 (right panel) and shows a very good description of the data. The renormalized amplitude shows weak dependence on the renormalization scale $\mu$ - the variation of $\mu$ from the pion mass to the nucleon mass results in an effect of $\sim 0.6$ degrees in the phase shift for $E_{\rm lab}\sim$ 100 MeV.

It is instructive to compare the above subtractive renormalization with the so-called “non-perturbative renormalization”, where the contact interaction is tuned to reproduce the empirical value of the scattering amplitude at a given energy without subtracting all ultraviolet divergences as advocated e.g. in Refs. Nogga:2005hy ; PavonValderrama:2005uj ; Long:2011qx within the nonrelativistic framework. Such “non-perturbative renormalization” does remove the quadratic divergence in $\xi\,G\,\bar{\xi}$ and the logarithmic divergence in $\xi\,G\,T_{\pi}G\,\bar{\xi}$ with the energy-independent prefactors, while the linear and logarithmic divergences with energy-dependent coefficients still survive in $\xi\,G\,\bar{\xi}$ (see Eq. (47)). In Fig. 4 we confront the cutoff dependence of the 3P0 phase shift for such a “non-perturbatively renormalized” scattering amplitude (left panel) with the residual cutoff dependence in the properly renormalized approach as discussed above (right panel). In line with the reasoning discussed above, one may conclude that the approach without explicit subtractions of divergences does not lead to a properly renormalized result for the scattering amplitude. This conclusion is in full agreement with Refs. Epelbaum:2009sd ; Epelbaum:2018zli .

V Summary

In this paper we considered the baryon-baryon (BB) scattering problem in the framework of manifestly Lorentz-invariant formulation of SU(3) BChPT by applying time-ordered perturbation theory. By integrating over zeroth components of loop momenta in Feynman diagrams we formulated the diagrammatic rules of time-ordered perturbation theory, which can be applied to momentum-dependent interactions and particles with non-zero spin. For the case of BB scattering, the importance of time-ordered diagrams can be determined using the Weinberg’s power counting rules Weinberg:rz ; Weinberg:um . An infinite number of diagrams contributes to the BB scattering amplitude at any finite order. To sum up the relevant contributions it is convenient to define the effective potential as a sum of all two-baryon irreducible contributions to the scattering amplitude within TOPT. In a full analogy with the conventional nonrelativistic framework, the scattering amplitudes are obtained as solutions to a system of coupled-channel integral equations with the potentials at the corresponding order. These equations represent a coupled-channel generalization of the Kadyshevsky equation kadyshevsky and feature a milder ultraviolet behaviour as compared to their non-relativistic analogs. We obtained new perturbatively renormalizable LO BB potential which leads to unique solutions of the integral equations for scattering amplitudes in all partial waves. On the example of NN scattering we addressed the issue of the non-perturbative inclusion of the leading short-range interaction in the ${}^{3}P_{0}$ partial wave. For this purpose, we carried out subtractive renormalization in a way consistent with EFT. We also considered the “non-perturbative renormalization” approach for the problem at hand as advocated, e.g., in Refs. Nogga:2005hy ; PavonValderrama:2005uj ; Long:2011qx to determine the value of the contact interaction. The resulting cutoff dependence of the amplitude supports the conclusions of Refs. Epelbaum:2009sd ; Epelbaum:2018zli about the incompatibility of such “non-perturbative renormalization” with the principles of EFT.

The established formalism can be used to study BB scattering in the SU(3) sector based on a renormalizable formulation with the corrections beyond LO treated perturbatively. Such a framework permits a complete removal of the ultraviolet cutoff $\Lambda$ by taking the limit $\Lambda\to\infty$, see Refs. Epelbaum:2012ua ; Epelbaum:2013ij ; Epelbaum:2013naa for applications in the non-strange sector. Alternatively, one may follow a more traditional approach by solving the integral equations for a truncated potential without relying on a perturbative treatment of higher-order contributions as it is usually done for NN scattering, see e.g. Epelbaum:2014efa ; Epelbaum:2014sza ; Reinert:2017usi ; Entem:2017gor . In that case the $\Lambda\to\infty$ limit is not legitimate anymore, but one may still expect to benefit from the milder ultraviolet behavior of the integral equations in the Lorentz-invariant formulation, which should provide more flexibility in the choice of the ultraviolet cutoff. Work along these lines is in progress.

Acknowledgments

We are grateful to Ulf-G. Meißner for useful comments on the manuscript. This work was supported in part by the Georgian Shota Rustaveli National Science Foundation (Grant No. FR17-354), by DFG and NSFC through funds provided to the Sino-German CRC 110 “Symmetries and the Emergence of Structure in QCD” (NSFC Grant No. 11621131001, DFG Grant No. TRR110), the BMBF (Grant No. 05P18PCFP1) and the Russian Science Foundation (Grant No. 18-12-00226).