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SUSY-QCD corrections to neutralino pair production in association with a jet

  • Regular Article - Theoretical Physics
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Abstract

We present the NLO SUSY-QCD corrections to the production of a pair of the lightest neutralinos plus one jet at the LHC, appearing as a monojet signature in combination with missing energy. We fully include all non-resonant diagrams, i.e. we do not assume that production and decay factorise. We derive a parameter point based on the p19MSSM which is compatible with current experimental bounds and show distributions based on missing transverse energy and jet observables. Our results are produced with the program GoSam Cullen et al. (Eur. Phys. J. C 72:1889, 2012) for automated one-loop calculations in combination with MadDipole/MadGraph for the real radiation part.

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Acknowledgements

We would like to thank Wolfgang Hollik, Jonas Lindert, Edoardo Mirabella, Davide Pagani, and the members of the GoSam collaboration for various useful discussions. We also acknowledge use of the computing resources at the Rechenzentrum Garching. The work of G.C. was supported by DFG Sonderforschungsbereich Transregio 9, Computergestützte Theoretische Teilchenphysik. We also acknowledge the support of the Research Executive Agency (REA) of the European Union under the Grant Agreement number PITN-GA-2010-264564 (LHCPhenoNet).

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Correspondence to Gudrun Heinrich.

Appendices

Appendix A: Leading order diagrams

In Figs. 13 and 14 we display the 14 diagrams contributing at leading order to the process \(u\bar{u}\to\tilde{\chi}_{1}^{0} \tilde{\chi}_{1}^{0} g\). The diagrams for the \(u g\to\tilde{\chi}_{1}^{0} \tilde{\chi}_{1}^{0} u\) subprocess can be obtained by crossing.

Fig. 13
figure 13

Tree level diagrams 1 to 8 for the process \(u\bar{u}\to \tilde{\chi}_{1}^{0} \tilde{\chi}_{1}^{0} g\)

Fig. 14
figure 14

Tree level diagrams 9 to 14 for the process \(u\bar{u}\to \tilde{\chi}_{1}^{0} \tilde{\chi}_{1}^{0} g\)

Appendix B: Gauge dependence

In this appendix we examine the gauge dependence of the diagrams that have been removed from the amplitude in the real emission part as discussed in Sect. 2.4. We show that this gauge dependence vanishes for covariant gauges and for a large class of non-covariant gauges.

The only diagrams, once omitted, that can lead to a dependence on the choice of gauge are of the type shown in Fig. 15. In this diagram there is an s-channel gluon which decays into a squark–antisquark pair. As the biggest contribution to the cross section comes from the parts of the phase space where the two squarks are on-shell, it is sufficient for our argument to consider the 2→2 proccess of squark pair production and neglect the subsequent decay of the squarks. We denote the incoming momenta of the quarks as q 1,q 2 and the outgoing momenta of the squarks as p 1,p 2. In the following we neglect overall prefactors like color factors and coupling constants as they are irrelevant for our argument. The same holds for factors of i and any minus signs. The amplitude of the 2→2 process can be written as

$$ {\mathcal{M}}\sim\bar{v}(q_1) \gamma^{\mu} D_{\mu\nu} u(q_2)\cdot\bigl(p_1^{\nu}-p_2^{\nu} \bigr), $$
(B.1)

where D μν denotes the gluon propagator, which in Feynman gauge is simply given by

$$ D_{\mu\nu}=-\frac{g_{\mu\nu}}{k^2} \quad\text{with}\ k=q_1+q_2. $$
(B.2)

Choosing the Feynman gauge and contracting the Lorentz indices expression gives

(B.3)

and after squaring and performing the fermion spin sum one obtains

(B.4)
Fig. 15
figure 15

Diagram of squark pair production via a s-channel gluon and their subsequent decay

As the gluon propagator is gauge dependent, the gauge dependence vanishes only in the sum of all contributing amplitudes.

To calculate the effect of a specific gauge to the given diagram we start with a general covariant gauge. The gluon propagator can be written as

$$ D_{\mu\nu}=-\frac{1}{k^2} \biggl(g_{\mu\nu} +(1-\lambda) \frac {k_{\mu}k_{\nu}}{k^2} \biggr), $$
(B.5)

so that, for λ=1, we recover the Feynman gauge. In the general case the presence of a term k μ k ν leads to an extra term in Eq. (B.3) of the form . It can easily be seen that this extra term vanishes if one replaces

(B.6)

and makes use of the Dirac equation for massless quarks,

(B.7)

Next, we turn to the case of non-covariant gauges. We consider the following structure for the gluon propagator:

$$ D_{\mu\nu}=-\frac{1}{k^2} \biggl(g_{\mu\nu}-\frac{n_{\mu}k_{\nu }+n_{\nu}k_{\mu}}{n\cdot k} + \frac{n^2k_{\mu}k_{\nu}}{(n\cdot k)^2} \biggr), $$
(B.8)

where n can be a time-like, space-like or light-like vector.

The third term of Eq. (B.8) vanishes with the same argument as for covariant gauges, as well as the term ∼n ν k μ .

The remaining term can be written as

(B.9)

Momentum conservation in the on-shell limit implies

$$ q_1\cdot p_1 = q_2\cdot p_2, \qquad q_2\cdot p_1=q_1\cdot p_2, $$
(B.10)

and therefore the additional factor in Eq. (B.9) is zero.

Appendix C: Renormalisation

In this appendix we outline how we perform the renormalisation of the squark mass and wavefunction.

To begin we write the renormalised self-energy as follows:

(C.1)

where A is the one-loop contribution to the self-energy. The on-shell renormalisation condition is that the renormalised one-loop self-energy is equal to the inverse of the bare propagator in the limit (up to order \(\alpha _{s}^{2}\)):

(C.2)

We expand A:

(C.3)

and then (C.2) fixes our renormalisation constants:

(C.4)

Appendix D: Expansion around real arguments

To circumvent using scalar integrals with complex incoming momentum we show how one can expand A around the real mass and end with renormalisation constants that look like the usual ‘real’ case but with complex internal masses. We follow the argument presented in [88].

(D.1)

We can therefore expand (C.4) to

(D.2)
(D.3)

Now, if we substitute these into (C.1) we obtain (substituting \(\mu^{2} = m_{\tilde{q}}^{2} - i m_{\tilde{q}}\varGamma_{\tilde{q}} \)):

(D.4)

where

(D.5)

Therefore our expansion looks like the usual real on-shell scheme, but with complex internal masses.

We now list the explicit results for the renormalisation constants. The constant for the mass counterterm is

(D.6)

and the wave function renormalisation constant is

(D.7)

where the scalar two-point function in n=4−2ϵ dimensions is denoted by \(I_{2}^{n}\), defined in the conventions of [89] as

(D.8)

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Cullen, G., Greiner, N. & Heinrich, G. SUSY-QCD corrections to neutralino pair production in association with a jet. Eur. Phys. J. C 73, 2388 (2013). https://doi.org/10.1140/epjc/s10052-013-2388-8

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