# Radiative corrections to stoponium annihilation decays

###### Abstract

The lighter top squark in supersymmetry can live long enough to form hadronic bound states if it has no kinematically allowed two-body decays that conserve flavor. In this case, scalar stoponium may be observable through its diphoton decay mode at the CERN Large Hadron Collider, enabling a uniquely precise measurement of the top-squark mass. The viability of the signal depends crucially on the branching ratio to diphotons. We compute the next-to-leading order QCD radiative corrections to stoponium annihilation decays to hadrons, photons, and Higgs scalar bosons. We find that the effect of these corrections is to significantly decrease the predicted branching ratio to the important diphoton channel. We also find a greatly improved renormalization-scale dependence of the diphoton branching ratio prediction.

###### Contents

## I Introduction

The Minimal Supersymmetric Standard Model (MSSM) reviews () with conserved R-parity contains a stable lightest supersymmetric particle (LSP). If the LSP is neutral, it could be the cold dark matter required by the standard cosmology. The collider signatures of the MSSM generally involve missing energy carried away by two LSPs produced in each event. Unfortunately, this suggests that there will be no true kinematic mass peaks whose reconstruction would determine superpartner masses. In favorable models, it is possible to obtain precision measurements of superpartner mass differences and other combinations of masses at hadron colliders by finding kinematic edges from decays. However, the overall mass scale of the superpartners will be much harder to obtain precisely at the CERN Large Hadron Collider ATLASTDR (); CMSTDR ().

In models with a relatively small mass difference between the lighter top squark and the neutralino LSP , there is an exception that would allow a sharp mass peak. If the lighter top squark has no kinematically allowed two-body decays that conserve flavor, then it will form hadronic bound states. Among these is stoponium, a stop-anti-stop bound state, which can be directly produced at hadron colliders through gluon-gluon fusion. The largest production cross-section is for the () state, denoted in the following by , but other stoponium states can contribute to the signal either by prompt decays to the ground state or direct annihilation decays. This state will form if

(1.1) | |||||

(1.2) |

so that the decays
and
are both kinematically forbidden.
(Here stands for the lighter chargino mass eigenstate.)
These conditions are almost never satisfied in the MSSM parameter
space with the so-called mSUGRA boundary conditions, but they can
easily be satisfied in other motivated models. These include
“compressed supersymmetry”
models
compressed ()
in which the predicted thermal relic density of dark matter is in
agreement with that observed by WMAP
and other experiments
WMAP ()-PDG (), due to the enhanced annihilation
mediated by -channel
exchange of top squarks.^{†}^{†}†The inequalities (1.1)
and (1.2) can also be satisfied in
the stop-neutralino co-annihilation region
stopcoannihilationone ()-stopcoannihilationthree ()
of parameter space, but at the price of a much more extreme fine-tuning
of input parameters.
Another class of models consists of those that generate
the baryon excess over anti-baryons at
the electroweak scale baryo (), baryoDM (), baryonew ().
In these and other DiazCruz:2007fc () cases,
is necessarily small enough to
guarantee the formation of stoponium.

Stoponium can decay directly by the decays of one of the top-squark constituents through the 3-body process , or if that is kinematically forbidden, through the flavor-violating 2-body process and/or the 4-body process . However, the corresponding partial widths are many orders of magnitude smaller Hikasa:1987db ()-Hiller:2008wp () than the binding energy of stoponium, which will be of order a few GeV Hagiwara:1990sq (). Therefore, will decay primarily by annihilation, including the possible two-body final states , , , , , , , and . Of these, the most promising final state, both for detectability over backgrounds and reconstruction of the mass peak, is , as was first pointed out long ago by Drees and Nojiri Drees:1993yr (); Drees:1993uw () (See also refs. Nappi:1981ft ()-Fabiano:2001cw () for other work related to stoponium at colliders.) The diphoton stoponium signals for both compressed supersymmetry and supersymmetric electroweak-scale baryogenesis have recently been studied in Martin:2008sv ().

In much of the parameter space in which stoponium can form,
the two-body final state leading to hadronic jets dominates.
If so, then the leading-order
prediction for BR is nearly model-independent, and
is of order . The QCD corrections to the bound state annihilation
decays are quite significant, however, and need to be taken into account
along with corrections to the production cross-section in order to
obtain a realistic estimate of the LHC sensitivity for a given model.
In this paper, we will calculate the QCD next-to-leading order
corrections to
-wave stoponium^{‡}^{‡}‡Although we have in mind the top squark,
our results can also be applied to
any new strongly interacting fundamental scalar whose width is small
enough that it hadronizes before it decays.
decay into the , , and final states.
The decay widths into and are model-independent to
leading order, and one may also argue that they are the most important,
since usually dominates the total width, and is the
observable
signal. However, the final state may dominate in some parts
of parameter space,
particularly if stoponium is just above the
threshold for that decay.

We will proceed following the strategy (and some of the notation) used in ref. Hagiwara:1980nv () where the analogous case of quarkonium decay was studied. (The quarkonium annihilation beyond leading order was calculated earlier in ref. Barbieri:1979be (), which regulated infrared divergences and mass singularities using a gluon mass instead of dimensional regularization.) The -wave stoponium decay width is related to the low-velocity () limit of the stop-anti-stop annihilation cross-section by:

(1.3) |

where is the bound-state wavefunction at the origin. [This is often expressed instead in terms of the radial wavefunction at the origin, .] Here, is the relative velocity of the squarks in the center-of-momentum frame. (The same formula (1.3) holds for excited states with 0 angular momentum. Obtaining the decay widths of higher angular momentum stoponium states would require keeping contributions at higher order in .) For diphoton and hadronic final states, the cross-section on the right-hand side of eq. (1.3) is in turn related by the optical theorem to the imaginary part of the amplitude for through two-particle and three-particle cuts. For the final state, we find it easier to just calculate the radiative corrections to the decay directly. In both cases, we work in Feynman gauge, and regulate amplitudes using dimensional regularization in dimensions. Ultraviolet divergences are indicated separately by writing , while infrared divergences and mass singularities are indicated by and for the pole terms. The top-squark propagator is renormalized on-shell, and the QCD gauge coupling will be renormalized in the scheme.

An important issue that arises in all calculations of this type is that obtains contributions that are divergent as due to the exchange of massless gluons in diagrams of the form shown in figure 1. The relevant next-to-leading order contribution in QCD is related to the leading order contribution by

(1.4) |

where is the quadratic Casimir invariant, for . This Coulomb singularity can be absorbed into the definition of the bound state wave-function . Alternatively, since it is universal in character, it cancels when one considers branching ratio observables. This provides a useful test of the calculation.

The rest of this paper is organized as follows. In section II, we find the next-to-leading order QCD corrections to stoponium annihilations into and final states. Section III discusses the one-loop QCD corrections for the final state. In section IV, we discuss the numerical impact of these results. Section V contains some concluding remarks.

## Ii Decays to hadrons and to photons

In this section, we calculate the next-to-leading order QCD corrections to the and partial widths. We use the cut method derived from the optical theorem, which allows the direct computation of the squared amplitude and avoids having to square a matrix element involving many terms. To calculate the amplitude for an arbitrary process , draw all of the diagrams involving scattering to the desired order, then cut the diagrams through the propagators that correspond to the desired final state (for at leading order, we have the three diagrams in figure 3). These cut propagators are put on mass-shell, using the appropriate Feynman rule corresponding to external particles (see Figure 2). Then the sum of all cut diagrams, multiplied by an extra factor of and summed and averaged over spins and colors as appropriate, is denoted by and equals the squared amplitude of .

In order to get the partial widths, the contributions to must be integrated over -dimensional Lorentz-invariant phase space. The individual contribution of a single cut diagram to the cross-section is

(2.1) | |||||

In equation (2.1), the labels and are for the initial-state and for the final-state momentum four-vectors, is the relative velocity of the initial-state particles, and is the integral over -body Lorentz-invariant phase space. Since we are calculating the annihilation of a bound state, we multiply both sides by the relative velocity and set as the relative velocity goes to zero. Therefore,

(2.2) |

Adding up all of the terms from the appropriate cut diagrams gives the total cross-section multiplied by the relative velocity, , which is related to the partial width in equation (1.3).

At tree-level, the cross section for the annihilation of into a gluon-gluon final state in dimensions is

(2.3) |

with , where is the bare QCD coupling and is the regularization mass. The diagrams , , and in figure 3 contribute to this result in the ratio .

The tree-level diphoton cross section can be obtained by the replacement in the result. For this final state,

(2.4) |

where is the charge of the squark, and . Therefore, the leading-order decay rates are

(2.6) |

where we have replaced the bare coupling with the renormalized coupling , since they are equal at leading order. Taking the ratio of these partial widths eliminates the bound state wavefunction and produces the simple leading-order result

(2.7) |

The non-vanishing cut diagrams that correspond to the annihilation of
into , , and final
states at next-to-leading order are given in figure
4^{†}^{†}†Note that several diagrams not shown in the
figure vanish because the color indices of three-gluon final states must
be antisymmetric by charge conjugation invariance. This is because
has , while a final state with gluons has
, where is () for antisymmetric
(symmetric) adjoint color indices Novikov ()..

Many of these
diagrams can be cut in more than one way. In diagrams with three
cut propagators, there is either real gluon emission or the
pair-production of quarks^{‡}^{‡}‡We include some
final state contributions,
even
though these may be regarded as corrections to
final states, which we do not treat here. The light
partial widths are suppressed by small Yukawa couplings at
leading order, and
the final state is often strongly suppressed
by kinematics and couplings. However, the
contributions from 3-particle cuts in
diagrams p1, p2, and p3 in figure 4
cancel
large logarithms in the limit
of small
due to the gluon vacuum polarization (2-particle cut)
contributions from the same diagrams., and diagrams
with two cut propagators have one-loop integrals.

In diagrams with three-particle cuts, the principal difficulty is integrating the momentum fractions of the final-state particles over three-body phase space. To do this, the phase space integrals can be reduced to integrals of the form given in the Appendix of ref. Hagiwara:1980nv (). Care must be taken in evaluating diagrams with multiple distinct three-propagator cuts. Diagram d2, for example, has two cuts that are not equal.

Evaluation of the two-particle cuts involves expanding the loop integral from the virtual gluon in partial fractions to obtain a set of scalar integrals, which are well-known. A complete set of scalar integrals that occur in the calculation can be found in ref. Beenakker:1988bq () (for a complete set of divergent and many finite scalar loop integrals, see ref. Ellis:2007qk ()). In contrast with the three-particle cuts, the phase space integration is quite easy Hagiwara:1980nv (). Since the cut diagrams do not depend on the final-state momentum directions, they are proportional to their contributions to the cross-section

(2.8) |

where

(2.9) |

There is an important simplification that can be made in the massless two-particle cuts of diagrams with potential Coulomb singularities (diagrams f1, f2, f3, and f4 in figure 4). Using the identities Hagiwara:1980nv ()

(2.10) | |||||

(2.11) |

where and are the final-state gluon momentum 4-vectors, is the number of spacetime dimensions, and the integrated two-body phase space, it is easy to show that, for integrals performed in this calculation, dot products of the particle momentum 4-vectors in the numerators of loop integrals cannot contain terms linear in the relative velocity . The effect of the divergence comes only from the scalar loop integrals in the calculation, and one may set everywhere else.

In Tables 1 and 2, the contribution from each diagram to is given in the form

(2.12) |

Here,

(2.13) |

in keeping with the notation of ref. Hagiwara:1980nv (). Also, is the quadratic Casimir invariant, is the Casimir invariant of the adjoint representation, and is the index of the fundamental representation, given for by , , and . We have combined diagrams with and with in the table because the individual self-energy diagrams are not proportional to the projector .

diagram | , three particle cut | , two particle cut |
---|---|---|

a | ||

b1 | ||

b2 | ||

c1 | ||

c2 | ||

c3 | ||

d1 | ||

d2 | ||

d3 | ||

e1 | ||

e2 | ||

f1 | ||

f2 | ||

f3 | ||

f4 | ||

g1 | ||

g2 | ||

g3 | ||

g4 | ||

h | ||

i | ||

j1 | ||

j2 | ||

k | ||

l1 | ||

l2 |

diagram | , three particle cut | , two particle cut |
---|---|---|

m1 | ||

m2 | ||

m3 | ||

n1 | ||

n2 | ||

n3 | ||

o | ||

p1 | ||

p2 | ||

p3 | ||

q1 + q2 | ||

q3 + q4 |

Taking the sum of the diagrams, we find the next-to-leading order result

(2.14) | |||||

where is either 1 or 0 depending on whether or not the
four-point squark interaction in Figure 2 is
included,^{§}^{§}§In the MSSM,
. However, one can
imagine
non-supersymmetric theories with fundamental strongly interacting
scalars, in which these formulas would apply with .
is the number of light quarks, and
or 0 depending on whether or not the top quark is included in
the effective theory. In this formula, we have written

(2.17) |

The function is defined so that it parametrizes the effects of a non-zero top-quark mass. In the limit that the top quark is massless compared to the top squark we have , and when the masses are identical we have .

The one-loop order correction to the diphoton cross-section can now be found simply by dropping the diagrams that involve gluon self-coupling or real gluon emission (equivalent to setting ) as well as any vacuum polarization diagrams (which no longer involve strong couplings), then making the replacement at vertices to change gluons into photons. Following this procedure, we find

(2.18) | |||||

In the renormalization scheme, the bare coupling is written in terms of the renormalized running coupling using

(2.19) |

The gluon cross section as a function of the renormalized coupling and the renormalization scale can therefore be written as