Lagrangian alternative to QCD string

Abstract

The spectrum of radially excited hadrons provides much information about the confinement forces in QCD. The confinement is realized most naturally in terms of the QCD string whose quantization gives rise to the radially excited modes. We propose an alternative framework for the description of the excited spectrum. Namely we put forward some effective field models in which the hadrons acquire masses due to interaction with a scalar field modeling the non-perturbative gluon vacuum. The effective potential for this field is periodic with an infinite number of non-equivalent vacua. The radially excited hadrons emerge as elementary excitations over different vacua. We construct explicit examples for such effective theories in the meson sector. An interesting byproduct of the considered models is the existence of a classical field configuration in each vacuum. Depending on the model, it can represent a domain wall, Nielsen–Olesen vortex or ’t Hooft–Polyakov monopole. In the pure scalar sector, it is shown that the first quantum correction leads to splitting of the radially excited modes into two nearly degenerate states. Such a phenomenon has some phenomenological support. The presented approach may also be viewed as an alternative framework to the bottom-up AdS/QCD models where the radial excitations appear on the classical level as well.

Appendices

Appendix A: Meson clusters

In this appendix, we reproduce the figure from review [18] showing the known spectrum of the light non-strange mesons [28]. The experimental uncertainties are indicated and purely established states are not shaded. The positions of meson clusters (first described in Ref. [17]) are marked by the vertical dashed lines.

Appendix B: Non-Abelian extensions

The construction of the non-Abelian extensions for the model of Sect. 3 is straightforward. We will consider the case of the gauge group SU(2). Let the scalar field φ transform according to the real vector representation. The extension of the Lagrangian (11) takes then the form

$$\begin{aligned} \mathcal{L}&=-\frac{1}{4}F_{\mu\nu}^a \bigl(F^{\mu\nu}\bigr)^a+\frac{1}{2}\bigl(D^\mu \varphi^a\bigr) \bigl(D_\mu \varphi^a\bigr) \\ &\quad{} +\frac{\mu^4}{2\lambda} \biggl[\cos \biggl(\frac{\lambda}{\mu^2}\varphi ^a \varphi^a-2\pi b \biggr)-\cos(2\pi b) \biggr], \end{aligned}$$

(29)

where a=1,2,3 and the field strength and the covariant derivative are defined by

$$\begin{aligned} &F_{\mu\nu}^a=\partial_\mu A_\nu^a- \partial_\nu A_\mu^a+g\varepsilon ^{abc} A_\mu^b A_\nu^c, \\ &D_\mu\varphi^a=\partial_\mu \varphi^a+g\varepsilon^{abc}A_\mu^b \varphi^c. \end{aligned}$$

The scalar field acquires the non-zero v.e.v.

$$\varphi^a\varphi^a=\langle\varphi\rangle_{k}^2. $$

Consider the small fluctuation of the field φ _a near its v.e.v.

$$\varphi_a=(\varphi_0)_a+ \sigma_a. $$

The gauge freedom allows to point the vector φ ₀ along the third axis,

$$ (\varphi_0)_a=\langle\varphi\rangle_{k} \delta_{a3}. $$

(30)

Substituting (30) into the Lagrangian (29) and keeping only the terms quadratic in fields, we arrive at

$$\begin{aligned} \mathcal{L}_{\sigma,A}^{(2)}&=\frac{1}{2}\partial_\mu \sigma \partial^\mu\sigma-\frac{1}{2} M^2_\sigma \sigma^2-\frac{1}{4}{\mathcal{F}_{\mu\nu}^a} \bigl({\mathcal{F}^{\mu\nu}}\bigr)^a \\ &\quad{} +\frac{1}{2}M^2_{A} \bigl(\bigl(A_\mu^1 \bigr)^2+\bigl(A_\mu^2\bigr)^2 \bigr)+\text{const}, \end{aligned}$$

(31)

where

$$\begin{aligned} &\mathcal{F}_{\mu\nu}^a=\partial_\mu A_\nu^a-\partial_\nu A_\mu^a, \\ &M^2_{A}=M^2_{A^1}=M^2_{A^2}=g^2 \langle\varphi\rangle_{k}^2, \end{aligned}$$

and the gauge boson $A_{\mu}^{3}$ (corresponding to the unbroken symmetry with respect to rotation around the third axis) remains massless.

Another realization of the Higgs model with the SU(2) gauge group is given by the complex spinor representation of the scalar field,

$$\varphi=\left ( \begin{array}{c} \frac{\varphi_1+i\varphi_2}{\sqrt{2}} \\ \frac{\varphi_3+i\varphi_4}{\sqrt{2}} \\ \end{array} \right ). $$

The corresponding Lagrangian is

$$\begin{aligned} \mathcal{L}&=-\frac{1}{4}F_{\mu\nu}^a \bigl(F^{\mu\nu}\bigr)^a+\bigl(D^\mu \varphi \bigr)^\dagger(D_\mu \varphi) \\ &\quad{} +\frac{\mu^4}{2\lambda} \biggl[\cos \biggl(\frac{2\lambda}{\mu^2}\varphi \varphi^\dagger -2\pi b \biggr)-\cos(2\pi b) \biggr], \end{aligned}$$

(32)

where

$$\begin{aligned} &F_{\mu\nu}^a=\partial_\mu A_\nu^a- \partial_\nu A_\mu^a+g\varepsilon ^{abc} A_\mu^b A_\nu^c, \\ &D_\mu\varphi=\partial_\mu\varphi-ig\frac{\tau^a}{2}A_\mu^a \varphi. \end{aligned}$$

The classical configurations delivering minimum to the effective potential satisfy the condition

$$\varphi\varphi^\dagger=\frac{\langle\varphi\rangle_{k}^2}{2}. $$

The v.e.v.’s of the fields are $A_{\mu}^{a}=0$ and

$$\varphi=\left ( \begin{array}{c} 0 \\ \frac{\langle\varphi\rangle_{k}}{\sqrt{2}} \\ \end{array} \right ). $$

Consider the small scalar excitations over the vacua (the unitary gauge is chosen),

$$\varphi=\left ( \begin{array}{c} 0 \\ \frac{\langle\varphi\rangle_{k}+\eta}{\sqrt{2}} \\ \end{array} \right ). $$

The quadratic in fields part of the Lagrangian takes the form

$$\begin{aligned} \mathcal{L}_{\sigma,A}^{(2)}&=\frac{1}{2}\partial_\mu \sigma \partial^\mu\sigma-\frac{1}{2} M^2_\sigma \sigma^2-\frac{1}{4}{\mathcal{F}_{\mu\nu}^a} \bigl({\mathcal{F}^{\mu\nu}}\bigr)^a \\ &\quad{} +\frac{1}{2}M^2_{A^a}A_\mu^a \bigl(A^\mu\bigr)^a+\text{const}, \end{aligned}$$

(33)

where the spectrum for $M^{2}_{\sigma}$ coincides with (7) and the vector spectrum is

$$ M^2_{A^a}=\frac{g^2 \langle\varphi\rangle_{k}^2}{4}, \quad k=0,1,2,\ldots,\ a=1,2,3. $$

(34)

The analysis of classical solution in the isovector case results in the ’t Hooft–Polyakov monopoles. We will reproduce briefly the standard derivation of those solutions.

Consider the static case and set $A_{0}^{a}(\vec{x})=0$ for any $\vec{x}$ and a. The equations of motion following from the Lagrangian (29) are

$$\begin{aligned} &D^i D_i\varphi^a= \mu^2\varphi^a \sin \biggl(\frac{\lambda}{\mu^2} \varphi^b\varphi^b-2\pi b \biggr), \end{aligned}$$

(35)

$$\begin{aligned} &D_i\bigl(F^{i j}\bigr)^a= -g \varepsilon^{abc}\bigl(D^j\varphi^b\bigr) \varphi^c. \end{aligned}$$

(36)

Let us look for the solutions in the form of the hedgehog ansatz [35],

$$ \varphi_a=x_a \frac{u(r)}{r}, \qquad A_i^a=a(r)\varepsilon_{iab}x_b, $$

(37)

which after the substitution to Eqs. (35) and (36) leads to the equations

$$\begin{aligned} &a''+4r^{-1}a'-3ga^2-g^2r^2a^3-g^2u^2a=gu^2r^{-2}, \end{aligned}$$

(38)

$$\begin{aligned} &u''+2r^{-1}\bigl(u'- r^{-1}u\bigr)-2gau-2g^2a^2r^2u \\ &\quad{} -\mu^2u\sin \biggl(\frac{\lambda}{\mu^2}u^2-2\pi b \biggr)=0. \end{aligned}$$

(39)

These equations cannot be solved analytically but it is easy to find the asymptotics of the solutions at large distances,

$$ r\rightarrow\infty{:}\quad u(r)\propto\langle\varphi\rangle_{k}, \qquad a(r)\propto-\frac{1}{gr^2}. $$

(40)

In order to extract the physical fields ’t Hooft introduced the following gauge-invariant definition for the field strength tensor [36]:

$$ \mathcal{F_{\mu\nu}}=\frac{1}{|\varphi|} \varphi_a \bigl(F^{\mu\nu}\bigr)^a -\frac{1}{g|\varphi|^3} \varepsilon^{abc}\varphi_a(D_\mu \varphi_b) (D_\nu \varphi_c). $$

(41)

If one has φ _a =|φ|(0,0,1) in some domain for any gauge transformation then $\mathcal{F_{\mu\nu}}=\partial_{\mu}A_{\nu}^{3}-\partial_{\nu}A_{\mu}^{3}$. The asymptotic of $\mathcal{F_{\mu\nu}}$ at spatial infinity is

$$\mathcal{F_{\mu\nu}}\rightarrow- \frac{1}{g r^3}\varepsilon_{\mu\nu a}x_a. $$

Here the symbol ε is zero if any of four-dimensional indices takes the value 4. We thus obtain the expression for the magnetic field corresponding to the magnetic charge g _m =1/g,

$$ B_a(\vec{x})=\frac{x_a}{g r^3}, \quad r\rightarrow\infty. $$

(42)

It is seen that the vector field extends to spatial infinity. This is related with the existence of the massless mode (see (31)) which can penetrate via the Bose-condensate of the scalar field. The given observation allows to prove (see, e.g., Ref. [35]) that the hedgehog solution represents the Dirac monopole, i.e. the vector-potential has the form

$$A_r=A_\theta=0, \qquad A_\varphi=\frac{g_m}{r} \frac{1-\cos\theta}{\sin \theta}. $$

As in Sect. 3, we thus conclude that for each vacuum there exists an associated non-trivial classical field configuration—the ’t Hooft–Polyakov monopole in the case under consideration. The magnetic field created by these monopoles at spatial infinity is, however, equal for any vacuum.

In the case of the isospinor Higgs field—the model (32)—all gauge fields are massive and located in a domain of the size approximately 1/M _A . The “magnetic” field at spatial infinity does not appear in this model and the isospinor hedgehog is not a magnetic monopole. In addition, the isospinor hedgehog is unstable [35].