Why Strings? Regge Behavior and Dual Resonance Models

String theory begins with an almost childlike modification of particle physics: instead of declaring the elementary object to be a point, declare it to be a one-dimensional object. That change sounds small. It is not.

A point particle sweeps out a worldline. A string sweeps out a two-dimensional worldsheet. At the free level there are two basic possibilities: an open string, with endpoints, and a closed string, a loop with periodic coordinate $\sigma$.

Point particles have worldlines. Strings have worldsheets. Interactions of strings are smooth joining and splitting processes rather than localized graph vertices.

The basic string scale is set by $\alpha'$:

$$\ell_s=\sqrt{\alpha'}, \qquad M_s\sim \frac{1}{\sqrt{\alpha'}}, \qquad T=\frac{1}{2\pi\alpha'}.$$

Here $T$ is the string tension. A large tension means a stiff, short string; a small tension means a floppy, extended string. At energies much lower than $M_s$, the extended object is hard to resolve, and its low-lying excitations look like ordinary particles. At energies comparable to $M_s$, the internal oscillator modes of the string become visible.

The central slogan is therefore

$$\text{one quantized string} \quad\Longrightarrow\quad \text{infinitely many particle states}.$$

This infinite tower is not a bug. Historically, it was one of the first clues.

1. Regge behavior: why strings know about spin

Long before QCD became the accepted theory of the strong interactions, hadronic resonances were observed to organize approximately along Regge trajectories:

$$J\simeq \alpha_0+\alpha_R' m^2,$$

where $J$ is angular momentum, $m$ is mass, $\alpha_0$ is an intercept, and $\alpha_R'$ is a slope. A relativistic string gives precisely this kind of relation for a simple reason: a long rotating string carries both energy and angular momentum, and both are distributed along its length.

For a classical rotating open string with massless endpoints, the result is

$$J=\alpha' E^2, \qquad \alpha'=\frac{1}{2\pi T}.$$

Quantum mechanically the exact trajectory is shifted by an intercept. For the open bosonic string, the leading trajectory will become

$$J=\alpha' M^2+1,$$

after quantization in the critical dimension. The important point here is not the intercept; it is the linearity. A one-dimensional relativistic object naturally produces states of increasing spin on approximately straight lines in the $J$—$M^2$ plane.

A rotating open string gives $J\propto E^2$. Quantization turns the classical line into an infinite tower of string states with a shifted intercept.

This is also why the same symbol $\alpha'$ appears in two guises. In old hadronic physics, it was the empirical Regge slope. In fundamental string theory, it is the inverse tension. The two meanings are historically linked, but conceptually one should distinguish a fundamental string from an effective QCD flux tube.

2. Gauge fields and gravity from one object

A second remarkable fact is that open and closed strings naturally produce the two kinds of massless particles needed for gauge theory and gravity.

Open strings have a massless spin-one excitation. In modern language this becomes a gauge boson. With Chan—Paton labels attached to the endpoints, open strings generate nonabelian gauge symmetry. We will derive this from the open-string spectrum.

Closed strings have a massless spin-two excitation. A consistent interacting massless spin-two particle couples universally to the stress tensor and is interpreted as the graviton. Thus closed strings do not merely coexist with gravity; perturbative closed-string theory contains gravity.

Schematically,

$$\text{open strings}\longrightarrow \text{gauge fields}, \qquad \text{closed strings}\longrightarrow \text{gravity}.$$

There is also a geometric reason string theory improves the ultraviolet behavior of perturbation theory. Point-particle Feynman diagrams contain local interaction vertices, where lines meet at a point. String interactions are described by smooth worldsheets. A splitting or joining event is not a new local vertex inserted by hand; it is part of the topology of a smooth surface.

This does not mean every ultraviolet question is automatically trivial. It means the fundamental perturbative objects are different: integrals over moduli spaces of Riemann surfaces replace sums of point-particle graphs with arbitrarily sharp local vertices.

3. The scale problem and compact dimensions

A useful reference scale is the Planck length,

$$\ell_P=\sqrt{G_N}\simeq 1.6\times 10^{-33}\,\mathrm{cm},$$

in units with $\hbar=c=1$. If $\ell_s$ is close to $\ell_P$, stringy excitations are far beyond currently accessible accelerator energies. But the relation between the string scale and the observed four-dimensional Planck scale depends on compact dimensions and on where gauge and matter fields live.

The simplest Kaluza—Klein idea is a field on a circle,

$$y\sim y+2\pi R.$$

A higher-dimensional scalar can be expanded as

$$\Phi(x,y)=\sum_{n\in \mathbb Z}\phi_n(x)e^{i n y/R}.$$

The momentum in the compact direction is quantized:

$$p_y=\frac{n}{R}.$$

A lower-dimensional observer therefore sees a tower of masses

$$m_n^2=m_0^2+\frac{n^2}{R^2}.$$

If $R$ is small, the nonzero modes are heavy. Strings add a second quantum number, winding, because closed strings can wrap compact directions. That fact will later lead to T-duality, one of the sharpest ways in which string geometry differs from point-particle geometry.

4. Dual resonance models

Historically, the next step was the dual-resonance description of hadron scattering. Consider four-particle scattering with all momenta incoming and mostly-plus metric. Define the Mandelstam invariants by

$$s=-(p_1+p_2)^2, \qquad t=-(p_1+p_3)^2, \qquad u=-(p_1+p_4)^2.$$

For external particles with $p_i^2=-m_i^2$, these obey

$$s+t+u=\sum_{i=1}^4 m_i^2.$$

In an ordinary tree-level field theory, different channels are represented by distinct diagrams. An $s$-channel exchange and a $t$-channel exchange are usually different terms in the amplitude.

The old Dolen—Horn—Schmid duality idea was different. It proposed that the sum over $s$-channel resonances and the sum over $t$-channel resonances should be two alternative expansions of the same analytic object, not two independent contributions to be added.

In a dual resonance model, the $s$-channel and $t$-channel resonance expansions are two limits of one amplitude. In modern open-string theory they come from different degeneration limits of the same disk worldsheet integral.

The Veneziano amplitude made this idea explicit. A useful prototype is

$$A(s,t)=g_o^2 B\bigl(-\alpha(s),-\alpha(t)\bigr) =g_o^2\frac{\Gamma[-\alpha(s)]\Gamma[-\alpha(t)]}{\Gamma[-\alpha(s)-\alpha(t)]},$$

where

$$\alpha(s)=\alpha_0+\alpha_R' s$$

is a linear Regge trajectory. Equivalently,

$$B\bigl(-\alpha(s),-\alpha(t)\bigr) =\int_0^1 dx\,x^{-\alpha(s)-1}(1-x)^{-\alpha(t)-1},$$

initially in the region where the integral converges, then by analytic continuation.

The variable $x$ is not yet a spacetime coordinate. In the later worldsheet interpretation, it becomes the remaining modulus of a four-punctured disk after using projective invariance to fix three insertion points. The two endpoint regions of the integral correspond to two different factorization channels:

$$x\to 0 \quad \Longleftrightarrow \quad s\text{-channel}, \qquad x\to 1 \quad \Longleftrightarrow \quad t\text{-channel}.$$

This is the analytic ancestor of a central string-theory fact: one worldsheet moduli space contains multiple particle-channel limits.

5. Poles and residues

The gamma function has poles at nonpositive integers. Therefore the Veneziano amplitude has poles when

$$\alpha(s)=0,1,2,\ldots,$$

and also when

$$\alpha(t)=0,1,2,\ldots.$$

Near an $s$-channel pole $\alpha(s)=n$, the amplitude behaves as

$$A(s,t)\sim \frac{g_o^2}{\alpha(s)-n}\,P_n\bigl(\alpha(t)\bigr),$$

where $P_n$ is a polynomial of degree $n$:

$$P_n\bigl(\alpha(t)\bigr)\propto \prod_{k=1}^{n}\bigl(\alpha(t)+k\bigr).$$

The beta function has an infinite tower of simple poles. The residue at level $n$ is a degree-$n$ polynomial in the crossed-channel variable, the analytic footprint of exchanged particles with spins up to $n$.

A polynomial of degree $n$ in the scattering angle is exactly what one expects from the exchange of particles with spins up to $n$. Thus the Veneziano amplitude packages infinitely many resonances on linear Regge trajectories into a single compact analytic expression.

From the modern point of view, the amplitude is the tree-level open-string four-tachyon amplitude. The tower of poles comes from the tower of string oscillator states. What began as a phenomenological ansatz for hadrons became the first exact formula of perturbative string theory.

Two lessons from the Veneziano amplitude

The Veneziano amplitude teaches two lessons that survive in modern string perturbation theory.

First, string amplitudes are analytic functions whose different particle-channel expansions are different limits of one worldsheet integral.

Second, the infinite tower of massive string states is not optional decoration. It is what makes duality and Regge behavior possible.

6. What to remember

This first lecture is motivational, but it already sets the agenda for the course.

• A string is an extended object with tension $T=1/(2\pi\alpha')$ and length scale $\ell_s=\sqrt{\alpha'}$.
• Quantizing a string gives an infinite tower of particle states.
• Rotating strings naturally produce approximately linear Regge trajectories.
• Open strings naturally contain gauge bosons; closed strings naturally contain gravitons.
• Compact dimensions are not an afterthought: they are required in consistent string theories and control the relation between higher-dimensional and lower-dimensional physics.
• The Veneziano amplitude is the prototype of stringy scattering: one analytic expression has infinitely many resonance poles and multiple channel expansions.

The next page turns this motivation into mechanics. We first study the relativistic point particle, because the string action is the two-dimensional generalization of the point-particle action.

Exercises

Exercise 1. Dimensions of $\alpha'$ and $T$

In units $\hbar=c=1$, show that $\alpha'$ has dimensions of length squared and that $T=1/(2\pi\alpha')$ has dimensions of energy per unit length.

Solution

In natural units, energy, mass, and inverse length have the same dimension:

$$[E]=[M]=[L]^{-1}.$$

Since $\ell_s=\sqrt{\alpha'}$ is a length,

$$[\alpha']=[L]^2=[M]^{-2}.$$

Tension is energy per unit length, so

$$[T]=\frac{[E]}{[L]}=[L]^{-2}=[M]^2.$$

Since

$$\left[\frac{1}{2\pi\alpha'}\right]=[L]^{-2},$$

the formula $T=1/(2\pi\alpha')$ has the correct dimensions.

Exercise 2. Kaluza—Klein masses on a circle

Let $y\sim y+2\pi R$ and consider a scalar satisfying

$$\left(-\partial_t^2+\nabla_x^2+\partial_y^2-m_0^2\right)\Phi=0.$$

Using

$$\Phi(x,y)=\sum_{n\in\mathbb Z}\phi_n(x)e^{iny/R},$$

derive the lower-dimensional mass formula.

Solution

For the $n$th Fourier mode,

$$\partial_y^2 e^{iny/R}=-\frac{n^2}{R^2}e^{iny/R}.$$

Substituting into the higher-dimensional equation gives

$$\left(-\partial_t^2+\nabla_x^2-m_0^2-\frac{n^2}{R^2}\right)\phi_n(x)=0.$$

This is the lower-dimensional Klein—Gordon equation with

$$m_n^2=m_0^2+\frac{n^2}{R^2}.$$

When $R$ is small, the nonzero Kaluza—Klein modes are heavy.

Exercise 3. Residues of the Veneziano amplitude

For

$$A(s,t)=g_o^2\frac{\Gamma[-\alpha(s)]\Gamma[-\alpha(t)]}{\Gamma[-\alpha(s)-\alpha(t)]},$$

show that near $\alpha(s)=n$ the residue is a polynomial in $\alpha(t)$ of degree $n$.

Solution

The gamma function has simple poles at nonpositive integers. Near $\alpha(s)=n$,

$$\Gamma[-\alpha(s)]\sim \frac{(-1)^{n+1}}{n!\,[\alpha(s)-n]}.$$

The remaining factor can be evaluated at $\alpha(s)=n$:

$$\frac{\Gamma[-\alpha(t)]}{\Gamma[-n-\alpha(t)]}.$$

Using

$$\frac{\Gamma(z)}{\Gamma(z-n)}=(z-1)(z-2)\cdots(z-n),$$

with $z=-\alpha(t)$, we find

$$\frac{\Gamma[-\alpha(t)]}{\Gamma[-n-\alpha(t)]} =(-1)^n\prod_{k=1}^{n}\bigl(\alpha(t)+k\bigr).$$

Therefore

$$A(s,t)\sim \frac{\text{constant}}{\alpha(s)-n} \prod_{k=1}^{n}\bigl(\alpha(t)+k\bigr),$$

so the residue is a degree-$n$ polynomial in $\alpha(t)$.

Exercise 4. From a rotating string to a Regge slope

Assume the classical rotating open string result

$$E=\frac{\pi T}{\omega}, \qquad J=\frac{\pi T}{2\omega^2}.$$

Eliminate $\omega$ and show that $J=\alpha' E^2$.

Solution

From the energy formula,

$$\omega=\frac{\pi T}{E}.$$

Substituting into the angular momentum gives

$$J=\frac{\pi T}{2}\frac{E^2}{\pi^2 T^2} =\frac{E^2}{2\pi T}.$$

Using

$$\alpha'=\frac{1}{2\pi T},$$

we obtain

$$J=\alpha' E^2.$$

The later quantum trajectory shifts this by an intercept, but the slope is already visible classically.

Exercise 5. Why duality is not ordinary diagram addition

Explain in your own words why the equality of $s$-channel and $t$-channel expansions in a dual resonance model is conceptually different from adding an $s$-channel Feynman diagram to a $t$-channel Feynman diagram.

Solution

In ordinary field theory, the amplitude is typically a sum of separate terms, for example

$$A=A_s+A_t+A_u+\cdots,$$

where each term comes from a different graph or class of graphs. The $s$-channel and $t$-channel exchanges are distinct contributions.

In a dual resonance model, the idea is that one analytic function has different expansions in different regions. An expansion near an $s$-channel pole displays a tower of $s$-channel resonances. An expansion in a different limit displays $t$-channel resonances. These are not two amplitudes to be added; they are two ways of expanding the same amplitude.

In modern string theory this happens because a single worldsheet moduli integral has different degeneration limits. Each degeneration limit looks like a different particle-exchange channel.