Scattering amplitudes of regularized bosonic strings

TL;DR

This paper calculates scattering amplitudes of a regularized bosonic string using mean-field approximation, showing Regge behavior and stability of the effective action minimum in certain dimensions.

Contribution

It introduces a mean-field approach to compute string scattering amplitudes and demonstrates Regge behavior in the Lilliputian scaling limit.

Findings

Regge behavior of amplitudes recovered in the limit

Stability of the effective action minimum for dimensions less than 26

Use of mean-field approximation disregarding fluctuations

Abstract

We compute scattering amplitudes of the regularized bosonic Nambu-Goto string in the mean-field approximation, disregarding fluctuations of the Lagrange multiplier and an independent metric about their mean values. We use the previously introduced Lilliputian scaling limit to recover the Regge behavior of the amplitudes with the usual linear Regge trajectory in space-time dimensions d>2. We demonstrate a stability of this minimum of the effective action under fluctuations for d<26.

Equations

$$K_0\to K_{\ast }+\frac{K_R^2}{2{\Lambda }^2\sqrt{d^2-2d}},K_{\ast }=\left(d-1+\sqrt{d^2-2d}\right){\Lambda }^2,$$

$$L_R=\sqrt{\frac{d+\sqrt{d^2-2d}}{2K_R}}\Lambda L_0.$$

$$S_{\mathrm{NG}}$$

$$S_{\mathrm{eff}}=K_0\int {\mathrm{d}}^{2}z\rho$$

$$\mathrm{tr}\log \mathcal{O}=-{\int }_{a^2}^{\infty }\frac{\mathrm{d}\tau }{\tau }\int {\mathrm{d}}^2\omega ⟨\omega |\text{mbox}{e}^{-\tau \mathcal{O}}|\omega ⟩,$$

$$a^2=\frac{1}{4\pi {\Lambda }^2},$$

$$P(\tau )=\sum _k{p}_k\theta \left(\tau -{\tau }_k\right),\sum _k{p}_k=0$$

$$\mathcal{A}\left[P(\cdot )\right]=⟨\text{mbox}{e}^{\mathrm{i}\int P_{\mu }\mathrm{d}{x}^{\mu }}⟩=\int \mathcal{D}{X}^{\mu }\text{mbox}{e}^{-S_{\mathrm{NG}}+\mathrm{i}\int P_{\mu }\mathrm{d}{x}^{\mu }},$$

$$\int P_{\mu }\mathrm{d}{x}^{\mu }=-\sum _k{p}_k\cdot x_k,x_k=x({\tau }_k),$$

$$P(\tau )=\frac{1}{\pi }\sum _k{p}_k\arctan \left(\frac{\tau -{\tau }_k}{\sqrt{{\epsilon }_k}}\right),\sum _k{p}_k=0,$$

$${\epsilon }_k=\epsilon \frac{({\tau }_{k+1}-{\tau }_k)({\tau }_k-{\tau }_{k-1})}{({\tau }_{k+1}-{\tau }_{k-1})}$$

$$S_{\mathrm{eff}}$$

$$G_{\epsilon }(\sigma )=-\frac{1}{2\pi }\log \left({\sigma }^2+\epsilon (\sigma )\right),$$

$$A=\int {\mathrm{d}}^{2}z\bar{\rho }.$$

$$r=\frac{({\sigma }_4-{\sigma }_3)({\sigma }_2-{\sigma }_1)}{({\sigma }_4-{\sigma }_2)({\sigma }_3-{\sigma }_1)}.$$

$$\frac{{\omega }_L}{{\omega }_R}=\frac{K\left(\sqrt{r}\right)}{K\left(\sqrt{1-r}\right)},$$

$$\sum _{i,j=1}^4{p}_i{G}_{\epsilon }\left(\frac{{\sigma }_i-{\sigma }_j}{\sqrt{\bar{\lambda }}}\right)p_j$$

$$S_{\mathrm{eff}}$$

$$\frac{1}{2{\left(K\left(\sqrt{1-r}\right)\right)}^{1/2}}\eta \left(i\frac{K\left(\sqrt{1-r}\right)}{K\left(\sqrt{r}\right)}\right)$$

$$\prod _{m,n=1}^{\infty }\left(\frac{\pi m^2}{{\omega }_R^2}+\frac{\pi n^2}{{\omega }_L^2}\right)=\frac{1}{\sqrt{2{\omega }_L}}\eta \left(i\frac{{\omega }_R}{{\omega }_L}\right).$$

$$\bar{\lambda }=\frac{1}{2}+\frac{{\Lambda }^2}{2K_0}+\sqrt{\frac{1}{4}{\left(1+\frac{{\Lambda }^2}{K_0}\right)}^2-\frac{d{\Lambda }^2}{2K_0}}$$

$$A=\frac{1}{2\pi K_0^2}\frac{1}{(2\bar{\lambda }-1-{\Lambda }^2/K_0)\bar{\lambda }}\left[s\log r+t\log (1-r)\right].$$

$$S_{\mathrm{eff}}$$

$$\alpha (s)=\frac{1}{2\pi K_0\bar{\lambda }}s+\frac{d-2}{24},\alpha (t)=\frac{1}{2\pi K_0\bar{\lambda }}t+\frac{d-2}{24}$$

$$r_{\ast }=\frac{\alpha (s)}{\alpha (s)+\alpha (t)}$$

$$S_{\mathrm{eff}}=\alpha (s)\log \frac{\alpha (s)}{\alpha (s)+\alpha (t)}+\alpha (t)\log \frac{\alpha (t)}{\alpha (s)+\alpha (t)}.$$

$$\mathcal{A}\sim s^{\alpha (t)}$$

$$K_R=K_0\sqrt{{\left(1+\frac{{\Lambda }^2}{K_0}\right)}^2-\frac{2d{\Lambda }^2}{K_0}}=K_0\left(2\bar{\lambda }-1-\frac{{\Lambda }^2}{K_0}\right),$$

$$L_R=\sqrt{\frac{\bar{\lambda }}{2\bar{\lambda }-1-\frac{{\Lambda }^2}{K_0}}}{L}_0.$$

$$s=\frac{\bar{\lambda }}{2\bar{\lambda }-1-\frac{{\Lambda }^2}{K_0}}{s}_R,t=\frac{\bar{\lambda }}{2\bar{\lambda }-1-\frac{{\Lambda }^2}{K_0}}{t}_R$$

Full text

I Introduction

String theory emerged in very early 1970’s from dual resonance models which were introduced to explain linear Regge trajectories of hadrons. Canonical quantization of relativistic bosonic string is consistent only in $d=26$ dimensions and on mass shell. These restrictions can be potentially overcome by an alternative path-integral string quantization Pol81 , where the problem remains nonlinear in $d\neq 26$ and/or off mass shell even after a gauge fixing. More precisely, these nonlinearities are due to the fact that the ultraviolet cutoff depends on the metric on the string world sheet, as is prescribed by diffeomorphism invariance. They are natural when one uses the proper time regularization, but they are hardly seen for the zeta-function regularization which is intimately linked to the mode expansion used in canonical quantization.

In the recent papers AM15b ; AM17 we have analyzed the bosonic Nambu-Goto string in the mean field approximation, where the world sheet metric can be substituted by its mean value. This approximation becomes exact at large $d$ and is applicable for finite $d$. We considered the string with the Dirichlet boundary conditions and computed its ground state energy as a function of the string length. We showed that the results obtained by canonical quantization are reproduced if the bare string tension $K_{0}$ approaches its critical value $K_{*}$ from above:

[TABLE]

where $K_{R}$ stands for the renormalized string tension. Associated with this scaling limit was a renormalization of length scales AM15b ; AM17

[TABLE]

In the limit where the cutoff $\Lambda\to\infty$ and $K_{R}$ and $L_{R}$ stay finite, the ground state energy also stays finite and agrees with the one obtained using canonical quantization. Such a limit is possible because the bare metric on the worldsheet becomes singular. We called the scaling limit “Lilliputian” since the bare length scales as $L_{0}\sim 1/{\Lambda}$.

The goal of this Paper is to further understand the Lilliputian string world and, in particular, the meaning of the performed renormalization of the length scale by computing scattering amplitudes.

II Nambu-Goto string in the mean-field approximation

Our starting point is the Nambu-Goto string whose action is rewritten in the standard way by introducing a Lagrange multiplier $\lambda^{ab}$ and an independent metric $\rho_{ab}$ as

[TABLE]

The path integration goes independently over real values of $X^{\mu}$ and $\rho_{ab}$ and imaginary values of $\lambda^{ab}$. To obtain the effective action governing the fields $\lambda^{ab}$ and $\rho_{ab}$, we split $X^{\mu}=X^{\mu}_{\rm cl}+X^{\mu}_{\rm q}$ and perform the Gaussian path integral over $X^{\mu}_{\rm q}$. Fixing the conformal gauge $\rho_{ab}=\rho\delta_{ab}$, we find

[TABLE]

where the last term on the right-hand side comes from the ghost determinant.

For definiteness we can use the proper-time regularization of the traces in Eq. (LABEL:aux1)

[TABLE]

where the proper-time cutoff $a^{2}$ is related to the momentum cutoff $\Lambda^{2}$ by

[TABLE]

but the results will not depend on the regularization used.

In the mean-field approximation, which becomes exact at large $d$, we can disregard fluctuations of $\lambda^{ab}$ and $\rho$ about their saddle-point values, i.e. simply substitute them by the mean values. This is analogous to the study of the $N$-component sigma-model at large $N$, where we can disregard quantum fluctuations of the Lagrange multiplier.

III Mean field for scattering amplitude

To compute the scattering amplitude, we introduce a piecewise constant momentum loop

[TABLE]

and consider

[TABLE]

where $x^{\mu}(\tau)$ is the value of $X^{\mu}$ at the boundary modulo a reparametrization of the boundary. For the upper half-plane coordinates $z=\sigma+{\rm i}y$ the boundary is along the real axis, $y=0$, and we have explicitly $X^{\mu}(\sigma,0)=x^{\mu}(\tau(\sigma))$ with the nonnegative derivative ${\rm d}\tau/{\rm d}\sigma\geq 0$.

Integrating by parts in the exponent in Eq. (8) we obtain

[TABLE]

which reproduces the string vertex operators for scalars. The averaging in Eq. (8) can be also represented as the path integral over $\lambda^{ab}$ and $\rho$ with the effective action (LABEL:aux1). We shall return to this issue in Sect. V.

Let us regularize (7) as

[TABLE]

where

[TABLE]

to comply with diffeomorphism invariance. Then it becomes clear that (10) is a singular parametrization of a polygonal momentum loop. This is like the Wilson-loop/scattering-amplitude duality in the ${\cal N}=4$ supersymmetric Yang-Mills theory AM07 which was extended to QCD string in MO08 .

For the scattering amplitude it is convenient to use the upper half-plane coordinates, where the boundary of the string world sheet is parametrized by the real axis, resulting in the Koba-Nielsen variables. Repeating the technique of Sect. II for integrating over $X^{\mu}$, now with the Neumann boundary conditions, we obtain for constant $\bar{\lambda}$ (to be justified below) the effective action

[TABLE]

Here we have denoted $\sigma_{i}=\sigma(\tau_{i})$, $G_{\varepsilon}$ is a regularized Green function of the type

[TABLE]

and we have used the notation

[TABLE]

For the $2\to 2$ amplitude we have four $\sigma_{k}$’s ($k=1,\ldots,4$), but the amplitude depends only on the projective-invariant ratio

[TABLE]

From the conformal mapping of the upper half-plane onto a $\omega_{L}\times\omega_{R}$ rectangle we have

[TABLE]

where $K$ is the complete elliptic integral.

Noting that

[TABLE]

with $s=-(p_{1}+p_{2})^{2}$, $t=-(p_{1}+p_{4})^{2}$ being Mandelstam’s variables and dropping the last term on the right-hand side of Eq. (17) for lightlike momenta $p_{i}^{2}=0$, we find

[TABLE]

while the (ordered) integration over $\sigma_{i}$’s is inherited from reparametrizations of the boundary. It gives the volume of the projective group times the integration over $r$, which is equivalent to the integration over the ratio $\omega_{R}/\omega_{L}$. The boundary terms are negligible at least for lightlike momenta as we show below.

It is worth noting that our procedure of the smearing (10) is similar to the one introduced in DOP84 for computing the Lüscher term for a rectangular Wilson loop. Actually, the last term in Eq. (18), coming from the determinants, for $s\gg t$ makes sense of the momentum Lüscher term Jan01 ; Mak11a . However, it is exact for arbitrary $r$ in view of the identity AMS14

[TABLE]

for the Dedekind $\eta$-function:

[TABLE]

Minimizing (18) with respect to $A$, we find

[TABLE]

which is the same value as found in AM17 . Minimizing (18) with respect to $\bar{\lambda}$, we obtain

[TABLE]

For the saddle-point value of the effective action we thus have

[TABLE]

If we integrate $\,\mbox{e}^{-S_{\rm eff}}$ over $r$ as is prescribed by the path integral over reparametrizations, we get the Veneziano amplitude ${\cal B}\left(-\alpha(s),-\alpha(t)\right)$ with

[TABLE]

to all orders, not only semiclassically like in textbooks. This is a remarkable consequence of the identity (19).

Minimizing (23) with respect to $r$, we find

[TABLE]

and

[TABLE]

For $s\gg t$ this results in the Regge behavior

[TABLE]

with the linear Regge trajectory (24).

IV Scaling limit and renormalization

As mentioned in the introduction, the Lilliputian scaling regime was previously defined for a string with the Dirichlet boundary conditions by the following renormalization of string tension and the length scale AM15b ; AM17

[TABLE]

[TABLE]

Equations (28) and (29) are just slight rewritings of (1) and (2) and the scaling limit is $\Lambda\to\infty$ while $L_{R}$ and $K_{R}$ stay finite.

Motivated by the length-scale renormalization (29), we write

[TABLE]

in the scaling regime (1). Accounting for the renormalization of the string tension (28), we have

[TABLE]

resulting in the linear renormalized Regge trajectory

[TABLE]

with a finite slope $\alpha^{\prime}_{R}$.

V Metric and the boundary term

Equation (22) represents the mean area of fluctuating surfaces, while for the computation of the boundary term in the effective action we need the metric $\bar{\rho}(z)$ itself. It can be computed as an average of the induced metric

[TABLE]

To understand the structure of the boundary term, we shall compute (33) at the tree level, i.e. in the classical approximation.

The harmonic function in the upper half-plane with the boundary conditions (10) at the real axis is

[TABLE]

By T-duality the computation is the same as for for the world sheet

[TABLE]

Using (34), (35) we obtain for the classical induced metric

[TABLE]

As is shown in Mak11a , (36b) vanishes and (36a) coincides with (36c), provided $\sigma_{i}$’s satisfy

[TABLE]

for lightlike momenta. Then the metric becomes conformal (i.e. $\bar{\rho}_{ab}=\bar{\rho}\delta_{ab}$).

Equation (37) is the same condition as the recently advocated tree-level scattering equation SA . Because of the projective symmetry three equations in Eq. (37) are not independent, so for the case of four particles there is only one independent equation. This equation is nothing but the tree-level approximation of Eq. (25). Notice that Eq. (25) itself sums up all loops and guarantees that $\bar{\rho}_{ab}$ given by (33) is conformal in the mean-field approximation.

The classical metric at the boundary

[TABLE]

vanishes except near the points $x=\sigma_{k}$ associated with edges of the polygon. The integration along the boundary

[TABLE]

reproduces the length. Thus the boundary term in the effective action is proportional to (39). It vanishes for lightlike momenta $p_{k}^{2}=0$. Otherwise its contribution has to be analyzed and remains finite in the scaling limit (1).

We do not expect this boundary term (like the last term on the right-hand side of Eq. (17)) to have any effect on the Regge trajectory (24) because it depends on the masses $\sqrt{p_{k}^{2}}$ of colliding particles, while the Regge trajectory does not.

To avoid the problem with the boundary terms, it is tempting to consider the case of closed string scattering, when the boundary conditions are periodic along both axes (the torus topology). Then the boundary terms in the determinant are missing and what remains is four times larger than for the disk. This would change $(d-2)/24$ to $(d-2)/6$ in the above formulas as usual.

VI Stability of fluctuations

We can check if fluctuations about the mean values of $\lambda^{ab}$ and $\rho$ are stable in the quadratic approximation, following the considerations in AM17 . The only difference from AM17 is that $\bar{\rho}(z)$ is now coordinate-dependent, a dependence given by the right-hand side of Eq. (36a).

Let us first consider the divergent part of the effective action to quadratic order in fluctuations for the nondiagonal element of $\lambda^{ab}$. It is given by Eq. (25) of AM17 :

[TABLE]

Here $X_{\rm cl}$ in given by Eq. (35) for the present case.

Expanding to quadratic order

[TABLE]

we find from (40) for constant $\bar{\lambda}^{ab}=\bar{\lambda}\delta^{ab}$

[TABLE]

The first term on the right-hand side of Eq. (42) plays a very important role for dynamics of quadratic fluctuations. Because the path integral over $\lambda^{ab}$ goes parallel to imaginary axis, i.e. $\delta\lambda^{ab}$ is pure imaginary, the first term is always positive. Moreover, its exponential plays the role of a (functional) delta-function as $\Lambda\to\infty$, forcing $\delta\lambda^{ab}=\delta\lambda\,\delta^{ab}$.

For $s\gg t$ we keep only the bulk term to get the effective action to quadratic order in fluctuations

[TABLE]

for a certain $\epsilon\sim 1/\Lambda^{2}$. Notice the last term on the right-hand side is normal (and therefore regularization dependent) rather than anomalous as the third and fourth terms.

From Eq. (43) for the effective action to the second order in fluctuations we find the quadratic form

[TABLE]

with

[TABLE]

where

[TABLE]

is always positive.

Since $\delta\lambda(z)$ is pure imaginary, i.e. $\delta\lambda(-p)=-\delta\lambda^{*}(p)$, we find for the determinant associated with the matrix in Eq. (44)

[TABLE]

and the propagators corresponding to the action (VI) are given by

[TABLE]

The situation with stability of fluctuations is just the same as described in AM17 : they are unconditionally stable for $2<d<26$. For $d>26$ they are stable in the regularized case, where $\Lambda$ is large but finite, because the first term on the right-hand side of Eq. (46) then dominates. In the scaling regime (1), where $K_{R}$ is finite as $\Lambda\to\infty$, we have the situation where the first term on the right-hand side of Eq. (46) is finite after the renormalization of $\bar{\rho}$, so the second term dominates. The action (VI) thus becomes unstable for $d>26$ in the Lilliputian scaling limit.

VII Conclusion

We have computed scattering amplitudes of the regularized bosonic Nambu-Goto string in the mean-field approximation, disregarding fluctuations of the Lagrange multiplier and an independent metric about their mean values. We have recovered the Regge behavior of the amplitudes in the Regge regime with the usual linear Regge trajectory in space-time dimensions $d>2$. We have considered the Lilliputian scaling limit, previously introduced AM15b to recover the results of canonical quantization for the spectrum the Dirichlet string, and showed that it is the one where the slop of the Regge trajectory scales to a finite value. We have also demonstrated that the effective action indeed has a stable minimum given by the mean field and fluctuations about the mean values of $\lambda^{ab}$ and $\rho$ increase the effective action for $2<d<26$.

The fact that the mean-field approximation reproduces canonical quantization in $d=26$ is not surprising and is well understood. However, canonical quantization is not consistent in $2<d<26$, where the effective action depends nonlinearly on the world sheet metric $\rho$. An open interesting question is as to whether or not corrections to the mean field will change the above formulas in $2<d<26$.

Acknowledgments

The authors acknowledge support by the ERC Advanced grant 291092, “Exploring the Quantum Universe” (EQU). Y. M. thanks the Theoretical Particle Physics and Cosmology group at the Niels Bohr Institute for the hospitality.