On fine structure of strings: the universal correction to the Veneziano amplitude
HJE
On fine structure of strings: the universal correction
Amit Sever 0 1 2 3 5
Alexander Zhiboedov 0 1 2 4
0 17 Oxford Street, Cambridge, MA 20138 , U.S.A
1 Ramat Aviv 69978 , Israel
2 1211 Geneva 23 , Switzerland
3 School of Physics and Astronomy, Tel Aviv University
4 Department of Physics, Harvard University
5 CERN, Theoretical Physics Department
We consider theories of weakly interacting higher spin particles in flat spacetime. We focus on the four-point scattering amplitude at high energies and imaginary scattering angles. The leading asymptotic of the amplitude in this regime is universal and equal to the corresponding limit of the Veneziano amplitude. In this paper, we find that the first sub-leading correction to this asymptotic is universal as well. We compute the correction using a model of relativistic strings with massive endpoints. We argue that it is unique using holography, effective theory of long strings and bootstrap techniques.
1/N Expansion; Confinement
-
Meson spectrum
Meson scattering
Polchinski-Strassler mechanism
Massive ends approximation
Massive ends in the conformal gauge
Effective theory of long strings
Solving for the boundary metric Correction to the amplitude
Classical scattering of strings with massive ends
Discontinuity, Lorentzian segments and emergence of the s − u crossing 15
4.1
Derivation
Bootstrap
1 Introduction 2
Holographic setup 3 4 5
6
A
2.1
2.2
2.3
2.4
2.5
2.6
3.1
3.2
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
Discontinuities of the massive endpoints correction
Bootstrapping the correction
– 1 –
5.10 Fixing the solution
5.11 The role of crossing
Conclusions
A.1 Order m0
A.2 Order m1/2
A.3 Order m1
A.4 Order m3/2
A.5 Evaluation of A4
Mass perturbation for the four-point amplitude
B Bootstrap in Mellin space 40
Introduction
Weakly interacting higher spin particles (WIHS) appear in very different physical
contexts [1]. One is the Yang-Mills theory at large N , where higher spin particles correspond
to approximately stable resonances (glueballs) with various masses and spins. Another is
HJEP06(218)54
tree-level scattering amplitudes of fundamental strings. In this case, higher spin particles
describe excitations of a fundamental string.
The S-matrix bootstrap aims to classify all such theories of weakly interacting higher
spin particles (WIHS) based on general principles. Two-to-two scattering in such theories is
described by a meromorphic function A(s, t). It packs together in a neat way the spectrum
of particles mi and a set of three-point couplings cijk.
Unitarity and causality, together with the presence of particles of spin higher than
two, results in a set of highly non-trivial constraints on the amplitude. At the present
moment only one solution to the problem is known explicitly — the celebrated Veneziano
formula [2] (and its cousins [3, 4]). This amplitude has several non-generic properties such
as exact linearity of the Regge trajectories, an exact degeneracy of the spectrum and soft
high energy, fixed angle behavior [5].1
Without further assumptions, (i.e. the existence
of massless spin two particle in the spectrum), these properties are not expected to hold.
In other known theories of WIHS, such as confining gauge theories in the large N limit,
the high energy behavior for real scattering angles is power-like [8] and no degeneracy at
all is expected in the spectrum [9–11]. Correspondingly, the details of A(s, t) may differ
significantly from one WIHS to another.
In [12] a systematic study of WIHS theories was initiated by studying the scattering
amplitudes at large energies and imaginary scattering angles. This kinematical regime is
controlled by the fastest spinning particles and was shown to be universal [12]. Namely, in
any WIHS the s, t ≫ 1 asymptotic of the amplitude A(s, t) coincides with the limit of the
Veneziano amplitude
s,t≫1
lim log A(s, t) = α′ [(s + t) log(s + t) − s log s − t log t] + . . . .
(1.1)
The analysis of [12] assumed that there is no accumulation point in the spectrum and
that there is a separation of scales in the large s, t limit. Correspondingly, the result (1.1)
shows that any WIHS theory satisfying these assumptions is a theory of strings. Specifically,
its spectrum contains an infinite set of asymptotically linear Regge trajectories.
To put this bootstrap program on a systematic computational path, we must
understand what the possible corrections to (1.1) are. Universality can only arise for the first
1In [6, 7] an ansatz for the scattering amplitude was analyzed for which the leading Regge trajectory
is piece-wise linear and flat for negative t. It consists of an infinite sum of the Veneziano amplitudes with
different slopes approaching zero, all sharing the same mass spectrum.
– 2 –
few terms in the large s, t expansion. Beyond the universal terms, more assumptions will
be needed to fix the amplitude and the corresponding class of WIHS theories.
The aim of this paper is to study the universal co (...truncated)