Figure 1

(a) The energy density ${\rho }_E(r)$ of the nucleon as function of $r$ for different pion masses. (b) The normalized energy density $4\pi r^2{\rho }_E(r)/M_N$ as function of $r$ for different pion masses. The curves are normalized such that one obtains unity upon integration over $r$.Reuse & Permissions

Figure 2

The pion mass dependence of (a) the energy density ${\rho }_E(0)$ in the center of the nucleon, and (b) the mean square radius of the energy density $\langle r_E^2\rangle$ defined in Eq. (23). The crosses indicate the physical point.Reuse & Permissions

Figure 3

(a) The angular momentum density ${\rho }_J(r)$ as function of $r$ for different pion masses $m_{\pi }$. (b) The mean square radius $\langle r_J^2\rangle$ of the angular momentum density as function of $m_{\pi }$.Reuse & Permissions

Figure 4

The distributions of (a) pressure $p(r)$ and (b) the shear forces $s(r)$ as functions of $r$ for different pion masses.Reuse & Permissions

Figure 5

The pion mass dependence of the pressure $p(0)$ in the center of the nucleon and of the position $r_0$ at which $p(r)$ vanishes.Reuse & Permissions

Figure 6

(a) $r^{2}p(r)$ as function of $r$ for different pion masses. The integrals over the regions where $r^{2}p(r)$ is, respectively, positive and negative cancel each other within numerical accuracy of better than $1\%$. This shows how the stability condition in Eq. (11) is satisfied; see text. (b) The pressure $p(0)$ as function of the energy density ${\rho }_E(0)$ in the center of the nucleon. Some pion masses to which the respective values refer are indicated. The physical point is marked by the cross.Reuse & Permissions

Figure 7

The form factors of the energy momentum tensor $M_2(t),J(t),$ and $d_1(t)$ as functions of $t$ for different pion masses. In the chiral limit $J(t)$ and $d_1(t)$ exhibit infinitely steep slopes at $t=0$, and $d_1(0)$ takes the value $-3.46,$ which does not fit on the scale in Fig. 7c.Reuse & Permissions

Figure 8

(a) The dipole masses as defined in Eq. (25) for the form factors $M_2(t),d_1(t),$ and $J(t)$ from the CQSM vs. $m_{\pi }$. Notice that $d_1(t)$ and $J(t)$ cannot be approximated by dipoles for $m_{\pi }=0$. (b) The constant $d_1$ as function of $m_{\pi }$ at low scale. (Solid line) The exact result from the CQSM. (Dotted line) The chiral expansion of $d_1(m_{\pi })$ to linear order in $m_{\pi }$ according to (26). The square marks the physical point.Reuse & Permissions

Figure 9

The mean square radius $\langle r_F^2\rangle$ of the trace of the total EMT operator vs. $m_{\pi }$.Reuse & Permissions

Figure 10

The form factors of the energy momentum tensor $A_{20}^{u+d}(t)\equiv A^Q(t)=M_2^Q(t)$ and $B_{20}^{u+d}(t)\equiv B^Q(t)=2J^Q(t)-M_2^Q(t)$ as functions of $t$ at the scale $\mu =2\mathrm{GeV}$ in a world with pion mass $m_{\pi }=900\mathrm{GeV}$. (See Appendix for the discussion of different notations.) The lattice data points are by the LHPC and SESAM Collaborations [26]. The solid curves are the results from the CQSM obtained here.Reuse & Permissions

Figure 11

The form factors of the energy momentum tensor $A_2^{u+d}(t)\equiv A^Q(t)=M_2^Q(t)$ and $B_2^{u+d}(t)\equiv B^Q(t)=2J^Q(t)-M_2^Q(t)$ and $C_2^{u+d}(t)\equiv C^Q(t)=\frac{1}{5}{d}_1^Q(t)$ as functions of $t$ at the scale $\mu =2\mathrm{GeV}$ in a world with pion mass $m_{\pi }=640\mathrm{GeV}$. (For notations see Appendix.) The lattice data points are from the QCDSF Collaboration [27]. The solid curves are the results from the CQSM obtained here.Reuse & Permissions

Figure 12

(a) The dipole mass $M_{\mathrm{dip}}$ of the form factor $d_1^Q(t)$ vs. $m_{\pi }$ as obtained from the CQSM. The linear $m_{\pi }$ dependence in the region $m_{\pi }\gtrsim 500\mathrm{MeV}$ is emphasized. (b) The pion mass dependence of the constant $d_1^Q$ as obtained from the CQSM at low scale. (Solid line) The exact result from the CQSM. (Dotted line) The chiral expansion of $d_1(m_{\pi })$ to linear order in $m_{\pi }$ according to Eq. (26). (Dashed line) The attempt of a “chiral extrapolation” from the region of large $m_{\pi }\gtrsim 400\mathrm{MeV}$. In both figures the cross marks the physical point.Reuse & Permissions