Quadrupole pressure and shear forces inside baryons in the large $N_c$ limit

We derive number of relations between quadrupole energy, elastic pressure, and shear force distributions in baryons using the large $N_c$ picture of baryons as chiral solitons. The obtained large $N_c$ relations are independent of particular dynamics and should hold in any picture in which the baryon is the chiral soliton. One of remarkable qualitative predictions of the soliton picture is the nullification of the tangential forces acting on the radial area element for any tensor polarisation of the baryon. The derived relations provide a powerful tool to check the hypothesis that the baryons are chiral solitons, say using lattice QCD.


INTRODUCTION
The linear response of a hadron to a change of the external space-time metric is described by the gravitational form factors (GFFs). For the first time the GFFs for spin 0 and 1/2 were introduced and discussed in details in Refs [1,2], for spin-1 particles in Ref. [3] and for arbitrary spin hadrons in recent Ref. [4]. The GFFs contain rich information about the internal structure of hadrons, for a detailed review see Ref. [5]. Particular interest for us here are the energy distributions and mechanical properties -elastic pressure and shear force distributions inside the hadron. These fundamental distributions are encoded in the static energy momentum tensor (EMT) defined in the Breit frame as [6]: HereΘ µν QCD (0) is the QCD EMT operator which matrix element is computed between hadron states with spins projections σ, σ ′ and momenta p 0 = p 0′ = E = m 2 + ∆ 2 /4, and p i′ = −p i = ∆ i /2. The 00 component of the static EMT contains the information about the energy distribution inside the hadron, 0i components about the spin distribution, and ik components provide us the distribution of elastic pressure and shear forces inside the hadron [6].
Various components of the static EMT for arbitrary spin hadron can be decomposed in multipoles of the hadron's spin operator. The expansion to the quadrupole order has the following form [7-9] * : Also we introduce the irreducible (symmetric and traceless) tensor of n-th rank: Note that only monopole quantities ε 0 (r), p 0 (r), and s 0 (r) are left after the spin average. The functions ε 0 (r) and ε 2 (r) correspond to the spin averaged energy density and to the quadrupole deformation of the energy density in the hadron correspondingly. There is obvious relation d 3 r ε 0 (r) = m. Also it is obvious that ε 2 (r) = 0 for the hadrons of spin 0 and 1/2. That is why such hadrons can be called spherically symmetric. From the stability condition for the stress tensor ∂ i Θ ik (r) = 0 one can easily obtain the equations for the functions p n (r) and s n (r): d dr p n (r) + 2 3 s n (r) + 2 r s n (r) = 0, for n = 0, 2, 3.
These equations † have the form of the equilibrium relation between the elastic pressure distribution p n (r) and the shear force distribution s n (r), see e.g. Refs [5,6]. Therefore, we call the functions p 2 (r), p 3 (r) as the quadrupole elastic pressure distributions, and the functions s 2 (r), s 3 (r) as the quadrupole shear force distributions. The functions p 0 (r), s 0 (r) correspond to the spin averaged pressure and shear force distributions, they coincide with the distributions for spherically symmetric hadrons of the spin 0 and 1/2. The solution of the Eq. (7) can be written in terms of the 3D Fourier transform of (generalised) D-form factors: The form (8) of the quadrupole pressure and shear forces also ensures that all relations for the force distributions discussed in Sec. IX and App. of Ref. [5] are satisfied automatically. In particular, the (generalised) von Laue conditions are satisfied automatically: Note that the dimensionless constants (generalised D-terms): are characteristics of the elastic properties of the hadron which are as fundamental as other mechanical properties of the hadron such as the mass and the spin. In principle, they could be listed in PDG on equal footing with the mass and spin of particles. The first measurements of D 0 in hard QCD processes became available for the nucleon in Refs. [12,13] and in Ref. [14] for the pion. Profound studies of all subtleties in extraction of the D-term D 0 from hard exclusive processes can be found in Ref. [15]. The first studies of the quadrupole energy, elastic pressure, and shear force distributions were performed in Ref. [9] for the case of ρ-meson, where, the authors employed the light-cone constituent quark model. In the present paper we shall derive the relations between quadrupole energy, elastic pressure, and shear force distributions for the baryons in the large-N c limit. In the latter limit the baryons can be viewed as the chiral solitons. Our relations are independent of the dynamics (effective field theory) describing the chiral soliton and can be used as a strong criterion to check the hypothesis that the baryons are chiral solitons.

GRAVITATIONAL FORM FACTORS OF THE BARYON AS CHIRAL SOLITON
The most striking success of the old Skyrme idea [16] that baryons can be viewed as solitons of the pion (or chiral) field, is the classification of light baryons it suggests. This idea implies that various baryons are quantum excitations of the same classical object -the chiral soliton and, hence, the properties of baryons are interrelated. Quantum Chromodynamics has shed some light into why the chiral soliton picture is correct: we know now that † Such type of the equation can be called "hadron shape formation equation". Indeed, the non-trivial shape of the pressure distribution (hadron shape) appear due to non-trivial shear force distribution sn(r), the latter is also called pressure anisotropy [10]. Interestingly the pressure anisotropy (shear force distribution) plays an essential role in astrophysics [10], see the review [11] on the role of pressure anisotropy for self-gravitating systems in astrophysics and cosmology.
the spontaneous chiral symmetry breaking in QCD is, probably, the most important feature of strong interactions, determining to a great extent their dynamics, while the large N c (= numbers of colours) argumentation by Witten [17,18] explains why the pion field inside the nucleon can be considered as a classical one, i.e. as a "chiral soliton". Following Witten [18] we assume the self-consistent pseudoscalar field which binds up the N c quarks in the "classical" baryon (i.e. the soliton field) to be of the hedgehog form ‡ : U 0 (r) = exp (iτ a n a P (r)) , where the unit vector n a = r a /r, and the spherically-symmetric profile function P (r) is defined by dynamics. We shall not need the concrete form of this function in what follows-for us the particular form of the underlying effective field theory is not relevant. The only hypothesis we do here is that the baryon is the chiral soliton of the form (11). In order to provide the "classical" baryon with specific quantum numbers one has to consider an SU (2)-rotated pseudoscalar field: where R(t) is an unitary SU (2) matrix depending only on time and U 0 (r) is the static hedgehog field given by Eq. (11). Due to chiral symmetry the dynamics can depend only on the angular velocity of the rotation: Quantizing this rotation one gets the spectrum of baryons and relation of the angular velocity to the spin operator of the corresponding baryon Ω i =Ĵ i /I. Here I ∼ N c is the soliton moment of inertia, its particular value is not relevant for us here. We see that the expansion of any observable in the angular velocity corresponds to the 1/N c expansion. Looking on Eqs. (2,3) we come to our first conclusion that all quadrupole quantities appear in the second order of the angular velocity expansion and, hence, are 1/N 2 c suppressed relative to monopole one. Moreover, we make a key observation -the angular velocity dependence can enter any quantity only through the zero components of the left and right chiral currents: where the last proportionality follows from the hedgehog form of the chiral field of the soliton (11). It reflects the fact that for the hedgehog form of the chiral field the isospin rotations can be compensated by the rotation of the coordinate system. From this key observation we conclude that the static EMT in the soliton picture can depend on the baryon's spin operatorĴ i only through the vector product ˆ J × n i , independently of concrete dynamics.

Energy densities of rotating chiral soliton
Above we observed that the 1/N c corrections to the static EMT can be obtained as the expansion in the vector Ω × n i . Therefore we can write general form of the rotational corrections (up to ∼ Ω 2 order) to the static Θ 00 (r) as: Here the function of the radial coordinate F (r) depends on the concrete dynamics and again not relevant for derivations here. Quantizing the rotations (Ω i =Ĵ i /I) and comparing the obtained form to the general parametrisation (2) we obtain the first relation: ‡ We consider the case of two flavours and the small violation of the isospin symmetry is neglected. The generalisation to the three flavour case and inclusion of the flavour symmetry breaking terms are straightforward. ε 2 (r) = − 3 2 δ rot ε 0 (r). (17) Note that this relation (and the relations derived below) can have corrections of the order of ∼ 1/N 3 c . The rotational correction to the monopole energy density δ rot ε 0 (r) for the baryon excitation of the spin J has general form δ rot ε 0 (r) = J(J +1)/I 2 f (r). Therefore, we can relate the energy densities for ∆ baryon (J = 3/2) and the nucleon in the following way: This is the first example of relations between mechanical characteristics of different baryons which follows from the soliton nature of baryons in the large N c limit. To estimate the typical size of the quadrupole energy density ε ∆ 2 (r) we first use the obvious relation: Further, with help of Eq. (18), we obtain: This numerical value is about 10% of the integral d 3 r ε ∆ 0 (r) = m ∆ ≈ 1232 MeV, compatible with being the 1/N 2 c correction.
Quadrupole pressure and shear forces of rotating chiral soliton Again using the fact that the dependence of the static EMT of the chiral soliton can depend on the angular velocity only through the vector Ω × n i , we can write the general decomposition of the angular velocity correction to static stress tensor as: Here G 1,2,3 (r) are functions depending on the concrete dynamics. Quantizing the rotations (Ω i =Ĵ i /I) and comparing the obtained form to the general parametrisation (3) we obtain remarkable relation: The combination p 2 (r)+ 2 3 s 2 (r) enters the equilibrium equation (7) which under the condition (22) implies nullification of both p 2 (r) and s 2 (r). Taking into account this nullification we arrive eventually to the following non-trivial relations: The first of these relations is very interesting predictions of the large N c picture of baryons as the chiral solitons. Due to its simplicity it is the easiest to check, say on the lattice or in other QCD based models. These would be the nice check of the soliton nature of baryons.
We see that in contrast to the spherically symmetric hadron, the radial area element experiences not only normal forces, but the tangential one as well. The size of the tangential forces are governed by p 2 and s 2 as these forces are proportional to 2 3 s 2 (r) + p 2 (r). The absence of these tangential forces is the remarkable prediction of the soliton picture of baryons.
The rotational corrections to the monopole quantities p 0 (r) and s 0 (r) are proportional to ∼ J(J + 1)/I 2 , where J is the baryon spin. Therefore, we can relate the fundamental characteristics of the baryon elastic properties (10) (generalised D-terms) for different baryons: The same relations are valid for pressure and shear force distributions in the nucleon and in ∆ baryon. The rotational corrections to the monopole (spin averaged) pressure and shear force distributions were computed in the framework of the Skyrme model in Ref. [19] and studied in the same model in details in Ref. [20]. We refer the reader to these papers for the numerical values of the rotational corrections in the Skyrme model. Here we just give the value of the generalised D-terms (10) for ∆ baryon which we obtain using the Skyrme model results of Ref. [20]: The relative numerical values are compatible with the expectation from the 1/N c counting.

CONCLUSION AND OUTLOOK
We derive number of relations between quadrupole energy, elastic pressure, and shear force distributions in baryons using the large N c picture of baryons as chiral solitons (see Eqs. (17,18,23,26)). The obtained large N c relations are independent of particular dynamics and should hold in any picture in which the baryon is the chiral soliton. The relations provide a powerful tool to check the hypothesis that the baryons are chiral solitons, say using lattice QCD.
Probably the most remarkable (and the easiest to check) prediction of the soliton picture is the nullification of the generalised D-term D 2 . We think it should be rather easy to measure it on the lattice. Qualitatively this prediction of the soliton picture implies the nullification of the tangential forces acting on the radial area element for any tensor polarisation of the baryon, see Eqs. (25,35). It might be that this nullification is more general requirement, like the nullification of the graviomagnetic moment proven in [1]. At the moment we are not able to prove this conjecture beyond 1/N c expansion.
In the consideration here we restrict ourselves to two-flavour QCD and to the first rotational band of baryons like the nucleon and ∆ baryon. We think it is pretty straightforward to make the generalisation to the three-flavour case (see, e.g. Ref. [21]). Also the generalisation to the case of other rotational bands (excited resonance) can be performed with help of methods developed in Ref. [22].