Associated Charged Higgs Boson and Squark Production in the NUHM Model

UPTEC F10 010 Examensarbete 30 hp Februari 2010

Associated charged Higgs boson and squark production in the NUHM model

Gustav Lund Abstract Associated charged Higgs boson and squark production in the NUHM model Gustav Lund

Teknisk- naturvetenskaplig fakultet UTH-enheten Conventional searches for the charged Higgs boson using its production in association with Standard Model (SM) quarks is notoriously weak in the mid-tanB range. Hoping Besöksadress: to find an alternate channel to fill this gap, the production of the charged Higgs boson Ångströmlaboratoriet Lägerhyddsvägen 1 in association with supersymmetric squarks is studied. Using Monte Carlo generators Hus 4, Plan 0 the production at the LHC is simulated within the non universal Higgs mass model (NUHM). If the six parameters of the model (m0, m1/2, A0, tanB, u, mA) induce small Postadress: masses of the stop, sbottom and charged Higgs, the production cross section can be Box 536 751 21 Uppsala of the order pb. Through scans of the input parameter the cross section is maximized, with the requirement that the stop decays directly to a neutralino - Telefon: simplifying detection, in the point (m0, m1/2, A0, tanB, u, mA) = (190, 187, -1147, 018 – 471 30 03 179, 745, 13.2) where the cross section is 559 fb.

Telefax: 018 – 471 30 00 The production is compared to the irreducible backgrounds stop, stop, t, tbar and t, tbar + 2 jets. The former poses no severe constraints and can be easily removed Hemsida: using appropriate cuts. The latter, SM background, has a cross section almost 1000 http://www.teknat.uu.se/student times larger and strong cuts must be imposed to suppress it. Neglecting hadronization and systematic effects, we show that a 5 sigma discovery is possible at 133 fb-1. In this range, mH+ = 194 GeV and tanB = 13.2, other channels have little or no prospects of detecting the charged Higgs and the studied process shows good prospects for complementing charged Higgs searches at the LHC in the mid-tanB range.

Handledare: Johan Rathsman Ämnesgranskare: Gunnar Ingelman Examinator: Tomas Nyberg ISSN: 1401-5757, UPTEC F10 010 1 Popul¨arvetenskaplig sammanfattning p˚asvenska

För100 ˚arsedan utfördes experimentell partikelfysik p˚aett radikalt annorlunda sätt änidag. Exper- imenten, som studerade till exempel elektroner och radioaktiv str˚alning och hur dessa växelverkade med materia, kunde utföras p˚aett skrivbord i ett vanligt laboratorium. Detta eftersom elektroner är stabila och finns överallt i oss och omkring oss och radioaktiv str˚alning bildas i en stadig takt som till˚ater oss att studera den i lugn och ro. De nya, tyngre och instabila partiklar som upptäcktes under seklets första hälft, till exempel elektronens storebror muonen som upptäcktes i kosmisk str˚alning, krävde dock mycket större maskiner. Detta eftersom energin till deras stora massa m˚aste tas fr˚an rörelseenergin hos andra snabbt färdandes partiklar och deras snabba sönderfall krävde att m˚anga partiklar hela tiden kunde produceras p˚anytt.

P˚amitten av 1900-talet ins˚agsbehovet av gemensamt sammarbete föratt ha r˚admed de dimensioner som krävdes av dessa nya maskiner och tolv Europeiska länder, inklusive Sverige, gick ihop och grun- dade CERN(europeiska centret förkärnforskning) i Genève. Under slutet av 2009 startades därLHC, the Large Hadron Collider, och data förväntas börja strömmain under 2010. LHC accelererar tv˚a str˚alar av protoner i motg˚aende banor i en 27 km underjordisk tunnel intill Genèvesjöni Schweiz. Hundratusentals g˚anger per sekund, närprotonstr˚alarna till˚ats korsa varann, kolliderar protoner med varandra i ljusets hastighet och enorma mängder av partiklar bildas och skjuts ut ˚atalla h˚all. Endast ett f˚atal av dessa kollisioner ärav intresse och den största utmaningen äratt identifiera dessa över det brus som bildas av alla andra partiklar.

Den teori som förnärvarande beskriver vad vi vet om partikelfysik kallas Standard Modellen och utvecklades p˚a60- och 70-talet. Den beskriver alla kända partiklar och krafter, med undantag för gravitationen, med otrolig precision. Dess enda förutsägelse som ännu inte bekräftats ärexsistensen av den s˚akallade Higgspartikeln som fysiker hoppas kunna p˚avisa med hjälp av LHC. Higgspartikeln behövs föratt teoretiskt kunna förklara varförmassiva kraftöverföringspartiklar har massa och föratt generera massa till all materia som simmar runt i en slags ”Higgs-vätska” och därmed blir olika tröga eller massiva beroende p˚ahur h˚art de binder sig till Higgs-vätskan.

Med denna sista bekräftelse tycks d˚aStandard Modellen vara komplett. Vi tror oss dock veta att den inte kan beskriva allt, förutom sin oförm˚agaatt beskriva gravitationen verkar ocks˚an˚agra av Standard Modellens masslösapartiklar faktiskt ha massa och tydliga tecken finns att den inte kommer kunna beskriva fysik vid de högre energier som LHC skall utforska. Därförhar en uppsjöav andra teorier och utvidgningar formulerats, alla med olika förutsägelser och mer eller mindre potential att kunna förklara de fenomen som hittils undg˚att alla försöktill förklaring.

Denna rapport ärförfattad inom ramen av en av dessa teorier kallad Supersymmetri. Den förutsp˚ar en ny, supersymmetrisk partner till alla existerande partiklar och dubblar s˚aledes partikelinneh˚allet i Standardmodellen. Detta leder till matematiska korrektioner till olika massor som f˚arStandard Mod- ellen att löpa amok, perfekt tar ut varandra. Dessutom ger den en lämplig kandidat till den mörka materia som kosmologer och astronomer upptäckt existerar överallt i Universum. M˚anga av dessa nya partiklar hoppas Supersymmetri-förespr˚akare hitta med hjälp av LHC de kommande ˚aren. I tillägg till att dubbla alla vanliga partiklar förutsp˚arSupersymmetri fem Higgs partiklar istället förStandard Modellens enda. Tv˚aav dessa ärelektriskt laddade och det ärmöjligheten att upptäcka dem denna rapport skall fokusera sig p˚a.

2 Inom en Supersymmetrisk modell, NUHM, som genom olika antaganden blivit starkt begränsad, un- dersöks möjligheten att observera och massbestämmaden laddade Higgs partikeln vid LHC. Ett flertal datorprogram, som implementerar alla de antaganden som gjorts, används föratt simulera produktionen av laddad Higgs vid LHC.

Vi försöker avgöra hur signalen förladdad Higgs skulle komma att se ut i detektorerna beroende p˚a hur den produceras och sönderfaller. Denna signal jämförs sedan med n˚agra av de bakgrunder som utgörs av produktionen av andra partiklar. P˚as˚avis avgörs om produktionen av laddad Higgs kan särskiljas och hur l˚ang tid det skulle ta att f˚aett statistikt säkerställt resultat. Den normala statistiska variationen av bakgrunden som vi vet existerar, kan annars tolkas som signal om mättiden ärförkort.

Resultatet äratt under vissa speciella antaganden, som starkt begränsar den redan begränsade NUHM modellen, kan den laddade Higgs partikeln detekteras och dess massa bestämmasp˚aungefärett till tv˚a˚arnärvälLHC kommit upp i sin tänkta produktionshastighet. Jämförelsevis skulle det ta avsevärt mer äntre ˚aratt göra samma sak genom att studera andra processer inom ramen förmer allmänna teorier.

3 Contents

1 Popul¨arvetenskaplig sammanfattning p˚asvenska 2

2 Introduction 5 2.1 The Standard model ...... 5 2.1.1 Gauge invariance ...... 5 2.1.2 The massive bosons ...... 6 2.1.3 Strong interaction ...... 7 2.1.4 Problems with the Standard Model ...... 8 2.2 Supersymmetry ...... 9 2.2.1 Why supersymmetry ...... 9 2.2.2 The Wess-Zumino model ...... 10 2.2.3 Soft symmetry breaking ...... 12 2.2.4 Higgs sector and the NUHM ...... 12 2.2.5 Particle content of the MSSM and NUHM ...... 13 2.3 Project outline and motivation ...... 14

3 Simulation tools 15 3.1 Spectrum calculators ...... 15 3.1.1 SoftSUSY ...... 15 3.1.2 Susyhit ...... 16 3.2 Monte Carlo generators ...... 16 3.2.1 MadGraph/MadEvent ...... 16 3.2.2 Pythia...... 16 3.3 ROOT...... 17

4 Analysis 17 4.1 Initial scans over parameter space ...... 17 4.1.1 H-t1-b1 coupling and squark masses ...... 17 4.1.2 Cross section ...... 25 4.1.3 Decays and detection ...... 25 4.2 Signal and background ...... 30 4.2.1 MSSM background ...... 32 4.2.2 SM background ...... 35 4.3 Comparison with other channels ...... 38

5 Conclusions, Outlook 39

6 Acknowledgments 39

4 2 Introduction

The Standard model (SM) of physics was developed during the 60s and 70s and is one of the greatest achievements of modern physics. Precise measurements have been done to conﬁrm its predictions to 3 the 10− level [1]. No signiﬁcant deviations have been observed between theoretical predictions and experimental results, yet we expect the SM to fail at higher energies than those probed to date. At these energies several new models await experimental data to test their new predictions.

Section 2.1 gives an introduction to the SM, the gauge principle and the Higgs mechanism. In section 2.2 the basic idea of Supersymmetry is introduced and ﬁnally the rest of the project is outlined in section 2.3. The reader is assumed to be familiar with basic Quantum ﬁeld theory. If well aquainted with supersymmetry he or she can can skip to section 2.3. Throughout this paper units of ~ = c = 1 will be used.

2.1 The Standard model 2.1.1 Gauge invariance The SM treats three of the four known forces; the strong, the weak and the electromagnetic force. The interactions are mediated by spin 1 gauge bosons; eight massless gluons, one massless photon and the three massive bosons of the weak interaction. The underlying assumption of the SM that has proved so successful is the gauge principle or the requirement that the Lagrangian should be invariant under local gauge transformations. The simplest gauge transformation we can make is within the U(1) group, where the Lagrangian for a free fermion of mass m is = ψ¯ (iγµ∂ m) ψ. (1) L µ − iθ If we make the global gauge transformation ψ ψ′ = e ψ, where θ is some arbitrary constant the partial derivative transforms like → iθ iθ ∂ ψ (∂ ψ)′ = ∂ e ψ = e ∂ ψ µ → µ µ µ and the Lagrangian under the transformation becomes

iθ µ iθ µ = e− ψ¯ (iγ ∂ m) e ψ = ψ¯ (iγ ∂ m) ψ L µ − µ − and is clearly invariant. Demanding that the transformation has to be locally gauge invariant, meaning iθ(x) the transformation can be diﬀerent in diﬀerent points of space-time i.e. ψ ψ′ = e ψ where θ(x) is no longer constant, the partial derivative transforms like →

iθ(x) iθ(x) iθ(x) ∂ ψ (∂ ψ)′ = ∂ e ψ = e ∂ ψ + e ψ∂ iθ(x), µ → µ µ µ µ the Lagrangian becomes

iθ µ iθ = e− ψ¯ (iγ ∂ + ∂ (iθ(x)) m) e ψ, L µ µ − and is no longer invariant under the transformation and is therefore said not to be locally gauge invariant. If we exchange the partial derivative with a covariant derivative, Dµ = ∂µ + gAµ where i A A′ = A ∂ θ(x), the covariant derivative transforms like µ → µ µ − g µ iθ(x) iθ(x) iθ(x) D ψ (D ψ)′ = ∂ + gA′ e ψ = e (∂ + i∂ (θ)+ gA i∂ (θ)) ψ = e D ψ, µ → µ µ µ µ µ µ − µ µ 5 rendering the Lagrangian locally gauge invariant. Plugging the covariant derivative back into the lagrangian of eq (1) it becomes

= ψ¯ (iγµD m) ψ = ψ¯ (iγµ∂ m) ψ + igψγ¯ µA ψ L µ − µ − µ where the extra term describes the interaction. To be a true propagating ﬁeld the kinetic term = 1 F µν F where F µν = ∂µAν ∂ν Aµ should also be added to the Lagrangian. Lkin − 4 µν −

In this seemingly artificial process we have been forced to introduce the gauge field Aµ and derive its tranformation properties, which as it turns out, is the relativistic description of the electromagnetic field. Using more complicated gauge groups, SU(2) U(1) and SU(3), locally gauge invariant Lagrangians can be derived and we are forced to introduce× similar fields describing the electroweak and strong interactions. Up to an overall coupling g, the terms in the Lagrangian and thus the interactions themselves, are completely defined by the gauge principle. For example the photon must be 2 µ massless since a term mγ AµA violates gauge invariance.

2.1.2 The massive bosons Since the forces are mediated by the gauge bosons a massless photon makes sense since it will generate a force with inﬁnite range, as observed. For the short-ranged weak force however, massless bosons are a problem and something in the theory must give rise to their masses - enter the Higgs mechanism. A complex scalar ﬁeld with a Lagrangian

µ = ∂ φ†∂ φ V (φ), (2) L µ − 2 2 with a potential V (φ) = µ φ†φ + h φ†φ , is rotationally symmetric and invariant under the phase transformation iθ φ (x) φ′ (x)= e φ (x) . → 2 µ2 If h> 0 and µ < 0 the potential takes the form of a Mexican hat with a minimum for φ = − | 0| 2h ≡ v . By choosing a particular solution like θ = 0 we chose a ground state and the symmetry is broken. √2 q We can instead parametrize excitations over this new ground state, chosen at random, as 1 φ (x)= (v + ϕ1(x)+ iϕ2(x)) . √2

2 2 2 Recalculating the potential we ﬁnd a term µ ϕ1, but no ϕ2 term, meaning ϕ2 is massless corresponding to excitations in the ﬂat direction of the potential.

If we add an SU(2) doublet of scalar ﬁelds, φ(x), to our theory and parametrize it in terms of four real ﬁelds H(x) and θi(x), i = 1,.., 3,

0 φ 1 σi 0 φ(x) = exp i θi(x) , (3) ≡ φ+ √2 2 H(x)+ v where σi are the Pauli matrices, we get three massless ﬁelds. Because of the symmetry no matter what ground state nature chooses for θi we can always make a rotation such that θi = 0. The locally

6 µ µ µ µ µ invariant Lagrangian = D φ†D φV (φ) with covariant derivative D φ = ∂ + igW˜ + ig′y B φ, L µ φ can now be calculated and the kinetic term becomes 2 2 µ 1 µ 2 g µ g µ D† φ D φ = ∂ H∂ H +(v + H) W †W + Z Z µ 2 µ 4 µ 8 cos2 θ µ W where the W and Z bosons have acquired masses 1 M cos θ = M = vg. Z W W 2

This expression relates the W and Z masses in terms of θW , the Weinberg angle, and has been confirmed in numerous experiments. The photon and Z fields, Aµ, Zµ can be expressed as combinations of the 3 gauge fields introduced through the covariant derivative, Bµ,Wµ , A cos θ sin θ B µ = W W µ . Z sin θ cos θ W 3 µ − W W µ Rewriting the Lagrangian in terms of these new fields the Aµ term is expressed in terms of electroweak parameters and to correctly describe quantum electro dynamics imposes the conditions

g sin θ = g′ cos θ = e, Y = Q T . W W − 3 The first expression relates the new couplings of the SU(2) SU(1) interaction to the known electromagnetic coupling e and provides the desired electroweak× unification. The second expression fixes fermion hypercharges in terms of their electric charge and weak isospin numbers. These identities have also been verified to a very high precision. The only prediction of the standard model yet to be confirmed is the existence of the Higgs boson which is the physical manifestation of the H field. In a similar way to how the Higgs mechanism generates gauge invariant mass terms for the gauge bosons, mass terms can be generated for the fermions through coupling to the higgsfield.

2.1.3 Strong interaction In order to explain the multitude of mesons and baryons observed in particle collisions, the elementary particle quark was proposed. Together, a quark and an antiquark could produce a charged or neutral meson, and in particle or antiparticle triplets they would form baryons. In this way the full spectrum of discovered particles could be explained and many new ones could be predicted. The theory was later to be known as Quantum chromodynamics (QCD) and includes 6 quarks and their respective antiquarks neatly grouped into three generations of up- and down-type

up type u c t . down − type d s b − In order not to violate the Pauli exclusion principle, a new quantum number needs to be introduced, color. Quarks come in one of three colors, red, blue or green, and any observed state should be colorless (red, blue, green or red, anti-red, etc.) so as not to introduce unobserved states. The postulate of colorlessness is also one of conﬁnement, since no free quark can be observed. Color is the source of the strong ﬁeld/interaction and since there are three colors, QCD is an SU(3) gauge theory. The free QCD Lagrangian = q¯ (iγµ∂ m ) q L f 6 µ − f f Xf

7 is invariant under global transformations in colorspace qα qα ′ = U α qβ, where the SU(3) matrices f → f β f can be written λa U = exp i θ , 2 a and λa, (a = 1, 2,..., 8) are the generators of SU(3) algebra corresponding to the Pauli matrices of SU(2). Making the Lagrangian invariant to local transformations, θa = θa(x), requires the introduction of eight gauge ﬁelds since we have eight independent gauge parameters. These are the gluons and they come in eight color combinations. The gluon has three interactions g g g g

q q g g g g

Figure 1: Feynman diagrams showing the interactions of the gluon. where the ﬁrst is gluon emission from a quark, much like the photon emission of an electron.

Because of the running coupling of QCD, which becomes smaller at higher energies, QCD has another peculiar property called asymptotic freedom. In particle collisions at high energies the quarks and gluons become free particles but as the energy decreases confinement takes over and they form Hadrons, or undergo hadronization. This is a complicated process for which a few models exist. One is the Lund string model [2] which is implemented in the Monte Carlo generator Pythia described in section 3.2.2. While bound in hadrons like the proton, quarks and gluons are referred to as partons and their distributions, or the probability to find them with a certain fraction, x, of the proton momenta are described by the parton distribution function (PDF). When a parton hadronizes it forms many hadrons which shower a detector within a cone-like structure, these showers are called jets. A gluon being emitted from a quark as in figure 1a will also give rise to a jet which, to mimimize computations can be modelled by a computational algorithm rather than actually computing all next to leading order corrections (NLO), it is then referred to as a parton shower.

2.1.4 Problems with the Standard Model In spite of its great success in describing experimental observations, physicists agree that the SM can not be a complete description of the Universe we live in and must be improved or altered. There are several reasons for expecting the SM to fail in describing experimental results at higher energies. Some of the more important ones will be brieﬂy explained in section 2.2.

Neutrino oscillations have been observed in both solar and atmospheric neutrino ﬂux and in reactor neutrino experiments such as Kamiokande in Japan. These experiments imply that neutrinos have mass contrary to the SM neutrino which only exists in one helicity. Adding right handed neutrinos to the SM is technically simple but it introduces a completely new mass scale implying that the SM is incomplete.

8 General relativity and quantum field theory have their own separate length scales at which they are usually applied. This works for most physics applications but it is clear that the current theories cannot simply be merged into one. String Theory is currently the best candidate for a Theory of Everything but it has yet to provide any predictions that can be verified experimentally. Several other theories have been constructed attempting to do the same thing and although there is yet no self consistent theory capable of such a unification, common for most of them are predictions that differ greatly from the SM and can be tested for experimentally. One such theory is Supersymmetry(SUSY) which is also an integral part of String Theory, effectively doubling the number of elementary particles, which will be discussed in greater detail below.

According to cosmological evidence dark matter and dark energy should make up approximately 95 % of the energy content of the Universe. Within the standard model there is no realistic candidate to make up this mass and so new physics is needed to explain the presence of invisible matter in the universe.

The hierarchy problem gives the most hope of new physics being at an attainable energy scale. The Higgs mass, MH , parameter of the standard model can be found by inserting the Higgs doublet 0 −µ parametrization of eq (3) into the Lagrangian in eq (2), yielding MH = q 2 for the bare mass. The 2 problem is that through radiative corrections MH gets a contribution which is proportional to the renormalization cut-oﬀ Λ2. Since Λ is the scale to which we integrate the energy and momenta we interpret it as the validity range of the SM. This is logically consistent and we could let Λ run all the way to the Planck scale M 1019 GeV. However it means that in order for the physical Higgs mass, P ≃ 2 2 M phys M 0 + Λ2 + (ln Λ), H ≃ H O to be of the order a few hundred GeV as suggested by both unitarity and experiment, the correction would have to cancel to one part in 1026 which is unsatisfactory to say the least. Since the amount of required ﬁne-tuning quickly increases with Λ one would expect Λ 1 TeV. This is exciting because it is exactly the energy range that will be probed by the LHC at CERN∼ which will reach a center of mass energy √s = 14 TeV.

2.2 Supersymmetry 2.2.1 Why supersymmetry The arguments above simply tell us to expect some new physics at the TeV scale but nothing of what this physics should be. Initially focusing on the hierarchy problem one potential solution is introduc- ing new particles whose loop corrections cancel those of the known particles. Since the corrections of bosons and fermions have different signs, it is tempting to introduce them in fermion-boson pairs. There is however no reason why this cancelation should happen to any order and particularily to all orders, unless the new particles are somehow related to the known ones. Relations like these are referred to as symmetries but all previously known symmetries related bosons and fermions to them- selvs, never to each other. A new such symmetry was discovered in the 60’s and 70’s and was called a supersymmetry. It is the most general extension of the Poincarégroup algebra into which all quantum field symmetries are grouped. Such a theory requires that for each fermion or boson, exists a boson or fermion partner exactly cancelling each others loop corrections thus providing an elegant solution to

9 the Hierachy problem.

In addition to the hierarchy problem, there are several other reasons why Supersymmetry has recieved so much attention over the last decades:

Grand unification The gauge coupling constants are in fact not constant, when processes are studied at different energy scales, they appear different in magnitude. For example the electron charge seems weakened when studied from a shorter distance, or using a probe with higher energy, since it becomes shielded by electron-positron pairs being created and annihilated according to the Heisenberg uncertainty principle. These changes can be described by the renormalization group equations(RGE) of that particular mathematical group. When the gauge couplings are evolved using SM RGEs, they do not meet at one point. If however supersymmetric RGEs are used, the gauge couplings do in fact unify incredibly well. This is true only under the assumption that supersymmetric particles, or sparticles, have masses in the range 100 GeV 10 TeV exactly as required by the hierarchy problem. The scale at which the couplings unify is− usually referred to as the GUT scale from the group of Grand Unified Theories that predict its existence. Dark matter candidate In supersymmetric theories there is a new conserved quantity called R-parity which helps to prevent proton decay which could otherwise be allowed at a disastrous rate. This conservation implies that there is one stable sparticle which could potentially make up the heap of dark matter we detect in the Universe. Graviton When generalising SUSY from a global to a local symmetry, as discussed in section 2.1.1, one is forced to introduce a massless spin 2 field. These are the properties demanded of the graviton which is the hypothetical carrier of the gravitational force. Light Higgs Precision measurements of electroweak parameters which through radiative corrections are sensitive to the Higgs mass indicate that mh 120 GeV and mh 200 GeV, whereas the SM prediction allows any value up 800 GeV.≃ The supersymmetric≤ prediction is m 135 GeV. h ≤ GUT extrapolation Almost all other extensions of the SM are effective theories and only push the need for new physics few orders of magnitude. SUSY however, can in principle be extended all the way to the GUT scale. String theory SUSY is an essential ingredient in string theory believed by many to be our best and perhaps only candidate for a theory of everything.

The simplest functional supersymmetric extension of the SM, the Minimal Supersymmetric extension of the Standard Model (MSSM) was developed in the 1980’s but the ﬁrst supersymmetric quantum ﬁeld theory was constructed by Wess and Zumino in 1974.

2.2.2 The Wess-Zumino model In all supersymmetric models including the Wess-Zumino model, all particles have a partner whose spin diﬀers by 1/2. These are together grouped into supermultiplets with fermions and bosons which should be related by transformations. In quantum ﬁeld theory such transformations are symmetries if the action S = d4x remains invariant under them according to Noether’s theorem. Invariance of the action is guaranteedL if is invariant or changes with a total derivative which disappears in the R L

10 integration through Gauss’ theorem.

Wess and Zumino introduced a supermulitplet consisting of a majorana fermion ﬁeld ψ and a complex scalar ﬁeld φ =(A + iB)/√2 with a Lagrangian

µ 2 1 µ = ∂ φ∗∂ φ m φ∗φ + ψ¯(iγ ∂ φ m)φ LWZ µ − 2 µ − 1 1 i 1 1 = (∂µA)2 + (∂µB)2 + ψ¯∂ψ m2(A2 + B2) mψψ.¯ 2 2 2 6 − 2 − 2 This Lagrangian is invariant under the transformation A A + δA etc. where → δA = iαγ¯ 5ψ, δB = αψ,¯ − δψ = Fα + iGγ α + ∂γ Aα + i∂Bα, − 5 6 5 6 δF = iα¯∂ψ, 6 δG =αγ ¯ ∂ψ. 56 Here the auxiliary ﬁelds F and G have been introduced, they are related to A and B through the equations of motion and can be removed, but are convenient for expressing variations as linear transformations. ψ B A

A A A Figure 2: Tadpole diagrams showing lowest order corrections to A.

We now have a Lagrangian invariant under supersymmetric transformations to which we can add interaction terms and calculate radiative corrections. The corrections arise from interactions caused be the terms g g g Aψψ¯ + mAB2 + mA3 (4) √2 √2 √2 in the interaction Lagrangian. To illustrate how the cancellations that save us from the hierarchy problem work, the loop corrections from the tadpole diagrams of ﬁgure 2 are simple taken from equation (3.47) in [1]. They are proportional to d4p d4p d4p Tr m 3m p m − p2 m2 − p2 m2 Z 6 − ψ Z − B Z − A d4p d4p d4p = 4m m 3m (5) ψ p2 m2 − p2 m2 − p2 m2 Z − ψ Z − B Z − A where m is the mass parameter in eq (4) and mA, mB and mφ are the masses that enter through the respective propagators. The numerical factor 3 comes from the three possible contractions of of the

11 A3 term and the 4 comes from the trace of a four-dimensional matrix. It is clear that all terms cancel if and only if mψ = mA = mB = m which is true in the basic supersymmetric model since the ﬁelds belong to the same supermultiplet. Even if only mψ = m = mA, mB, the quadratic corrections will cancel and only a logarithmic divergence will remain, allowing6 for the scalar mass to be diﬀerent from the fermion mass as will be discussed in the next section.

2.2.3 Soft symmetry breaking No matter how beautiful, SUSY cannot be an exact symmetry. Each particle would have a supersymmetric partner with the same mass which would have been detected in experiments a long time ago. So SUSY must be a broken symmetry and the question arises how it can be broken without ruining the cancellations for which it was introduced. This is referred to as breaking SUSY softly. The second d4p and third term in eq (4) give corrections proportional to 1 and 3 times m p2 m2 as seen in eq (5). − A,B So adding a term R = c(A3 3AB2) Lsoft − to the interaction Lagrangian will add corrections which cancel if mA = mB and whose quadratic corrections cancel regardless. This is a very simple example of soft symmetry breaking but it can be done in diﬀerent ways in the diﬀerent sectors of SUSY and it is not trivial how SUSY should be broken, only that it must be.

2.2.4 Higgs sector and the Non Universal Higgs mass model In the SM we introduced one complex Higgs doublet which with its four degrees of freedom gave mass to the W ±,Z bosons leaving one degree of freedom for the scalar Higgs boson. In supersymmetric theories, it turns out a second Higgs doublet is needed to prevent gauge anomalies, to give mass to both up- and down-type quarks and to ensure that the Superpotential, W , is an analytic function. Now the Higgs sector has eight degrees of freedom three of which are still needed for mass generation of the gauge bosons, but the ﬁve remaining give rise to ﬁve separate Higgs bosons h0 - a light neutral CP even scalar, similar to the SM Higgs H0 - a heavier neutral CP even scalar A0 - a neutral CP odd scalar

H± - two charged scalars. Together with the parameters needed to parametrize the new Higgs doublet come a total of 105 independent parameters related to SUSY breaking in addition to the 19 parameters of the SM in order to describe the MSSM [4]. To constrain supersymmetric models and reduce the number of free variables other physics inputs are needed that can provide additional relations between the parameters. For example, as mentioned in section 2.2.1, SUSY is not an eﬀective theory and can be extrapolated as far as the GUT scale. If we reverse this argument we can assume that certain parameters can be related to one single universal value at the GUT scale. Then all these variables can be evolved down to their pertinent scale using their RGEs.

The constrained MSSM (cMSSM) is one such model which is commonly used for phenomenological studies. Using assumptions and arguments similar to the one above it has been constrained to only

12 require four variables and one sign.

The SUSY breaking parameters

m0 - the universal scalar mass m1/2 - the universal gaugino mass

A0 - the universal trilinear coupling and the Higgs potential parameters tan β - the ratio of the vacuum expectation values of the two Higgs doublets sgn(µ) - the Higgs superpotential coupling where all universal parameters are given at the GUT scale. One assumption is that the masses of the two Higgs doublets are the same as the universal gaugino mass. Lifting this constraint one can specify the value of the Higgs mass at the GUT scale mH or separate masses mHu and mHd for the two Higgs doublets. In this paper we will use the latter approach. Using the relations

m2 m2 tan2 β 1 µ 2 = Hd − Hu m2 , | | tan2 β 1 − 2 Z − m2 = m2 + m2 + 2 µ 2 A Hd Hu | | the new parameters can instead be chosen as mA - the CP odd higgs scalar mass at the electroweak scale and the µ parameter.

This model is referred to as the non universal Higgs mass model (NUHM) which was ﬁrst presented in [5]. For our purposes it has the advantage of being able to input directly the A0 mass at the electroweak (EW) scale, which otherwise is hard to control using GUT scale values and provides more freedom on the Higgs mass scale. All work within this paper will be performed within the NUHM model.

2.2.5 Particle content of the MSSM and NUHM The principle of supersymmetry introduces one new particle for each of the SM particles, or rather one supersymmetric degree of freedom for each SM degree of freedom. In the case of the fermions - the quarks and leptons, their partners are called squarks and sleptons. But since the fermions have chirality and therefore two degrees of freedom, there should be two spin 0 sfermions for each fermion. The supersymmetric particles are indicatedq ˜ and they are given chiral indexes

˜l l L . → ˜l R From constraints on ﬂavor changing currents, the two lighter squark generations should be nearly degenerate in mass and are not given distinct names. Because of their large masses, the top and bottom partners can be well separated from the other squarks and are usually referred to as their mass eigenstates, ˜b t˜ b 1 , t 1 . → ˜b → t˜2 2

13 The superpartners of the B,W and H ﬁelds are called Bino, Wino and Higgsino and they mix to 1 produce neutralinos and charginos which are spin 2 fermions. There are four neutral neutralinos and two charged charginos 0 0 0 0 1 1 χ1, χ2, χ3, χ4, χ1, χ2 0 and in most supersymmetric models the lightest neutralino, χ1, is the lightest supersymmetric particle (LSP) which is a candidate for dark matter. Finally two higgsinos also result as a mixture of the higgsino ﬁelds and the gluons have a supersymmetric equivalent in the gluino.

2.3 Project outline and motivation The Higgs boson is probably the most sought after object in physics today. If found, it would confirm the last prediction of the SM. Simultaneously, signs for beyond the standard model are searched for in many experiments. The signal here is more ambiguous and could be one of many particles or phenomena. For example, an additional Higgs boson, like the charged Higgs, would be a clear indication of beyond the SM physics, pointing towards SUSY. For the charged Higgs boson to be found, its cross section, or probability to be produced, at LHC energies must be sufficiently high to produce enough Higgs bosons to unambiguoulsy confirm its existence and measure its mass. The charged Higgs can be produced in different processes, the dominant process is in association with t

t¯ gHtb H b −

¯b

Figure 3: Example of one Feynman diagram of charged Higgs production in association with SM quarks.

heavy quarks, pp H±tb as in ﬁgure 3. The cross section of the process scales with the square of the coupling constant→ [6]

ig 5 5 gHtb = mb tan β 1+ γ + mt cot β 1 γ , (6) 2√2mW − which has a minimum around tan β = 6 as can be seen in figure 4. Since the current limits on tan β is that it cannot be too large nor too small, 5 . tan β . 30 [7], this potentially places it near the minimum. If so it will be much harder to detect the charged Higgs through this channel and it is desirable to find another process that could plug this hole. In order to retain a large cross section this should also be a strong process involving different kinds of particles such that the coupling has a different dependence on tan β. Charged Higgs production in association with squarks is potentially such a process being similar to the production in figure 3 but q q˜. Since the minimal Supersymmetric theories do not allow direct input control over the Higgs mass,→ the process is studied within the NUHM were the Higgs mass can

14 2 2 be controlled by changing the value of mA since mH± = mA + mW . Because of the resulting large parameter space of six parameters, instead of performing a general analysis of the detection possibilities p in parameter space, we try to maximize the cross section and ﬁnd one point where charged Higgs can be detected in order to show that detection is possible. This would then suggest that the production in association with squarks could potentially be used to search for charged Higgs in the parameter range where other processes cannot.

Figure 4: Dependence of the charged Higgs to t, b-quark coupling on the parameter tan β as in eq (6). The coupling is in arbitrary units.

3 Simulation tools

Even though scientists have yet to receive any data which could confirm or reject Supersymmetry it is a self consistent and well defined theory, so are the sub models like NUHM. Given the six input parameters of the NUHM, any mass, branching ratio or cross section can be calculated. This is however in most cases a quite complicated process and the calculations get even more arduous when particle decays and interactions have to be calculated. Therefore we need to rely on several different simulation tools in order to calculate these different measurable quantities. The ones used for this paper are described in varying detail below.

3.1 Spectrum calculators 3.1.1 SoftSUSY SoftSUSY [8] is a spectrum calculator for the MSSM and some of its extensions. Provided with some basic SM masses and coupling constants, as well as the model dependent supersymmetric input parameters required, SoftSUSY calculates all SUSY parameters at the EW scale and gives the spectrum of masses. The renormalisation group equations are iterated until a convergent result is achieved. SoftSUSY also performs a number of checks to ensure that the results are consistent like checking for

15 negative square masses of non-Higgs scalars and that all the results converge properly.

3.1.2 Susyhit Using the output ﬁle of SoftSUSY, Susyhit [9] calculates the decay widths and branching ratios of all sparticles. The program consists of three independent programs: the spectrum calculator SuSpect, the higgs decay calculator HDecay and the sparticle decay calculator SDecay. The spectrum and decay information is all that is needed for an event generator to generate unweighted events or events with correct statistical probabilities. The information is shared between programs using input/output ﬁles in the les Houches format, a standard format to store process and event information [10].

3.2 Monte Carlo generators When any experiment is performed, like proton-proton collisions at the LHC, there are inevitably other processes which can give a similar or identical result in the detector, called backgrounds. Even for backgrounds that normally look very diﬀerent, they can occur much more often than the signal and therefore mimic it enough to completely overshadow it. On an event by event basis it is generally impossible to say whether it was caused by a signal or background event. To make any conclusions, experiments must be repeated many times until enough statistics have been gathered to compare the result with the expected one. To know what is the expected result, experimental particle physics today relies heavily on Monte Carlo simulations where many events are generated using random number generators. This of course requires a good knowledge of what processes give rise to backgrounds and how they should be calculated. Any uncertainties in the models or variables will make the Monte Carlo simulations less realistic. For this paper two Monte Carlo event generators are used simulating diﬀerent stages of the proton-proton collisions.

3.2.1 MadGraph/MadEvent In high energy proton-proton collisions it is actually the constituent particles of the protons, the quarks and gluons - in this context usually referred to as partons, that collide with each other to produce new particles. This is referred to as the hard interaction. Many different indistinguishable interactions can lead to the same final state and must all be summed over to calculate the total probability of producing that particular final state. The program package MadGraph/MadEvent(MGME) [11] is an event generator that produces a record of four-momenta for the hard interaction final state particles. MadGraph generates all contributing Feynman diagrams and corresponding matrix elements whereby MadEvent produces the record of unweighted events. The event record is generated in a Les Houches Event file following the les Houches standard format.

3.2.2 Pythia Since the event record from MGME consists of elementary particles, many of which are unstable or cannot exist freely (quarks), another program is needed to decay and hadronize the particles into stable ones which can be observed in detectors. This can be done using the Pythia [12, 13] program package. It allows a complete event record to be calculated to the level required. For example all decays and hadronization can be turned on or oﬀ. The end result is an event record of ﬁnal state particles and their four-momenta which can be analysed to compare what the signal and backgrounds will look like

16 in the detector.

In the analyses performed in this paper no hadronization is performed and only some particles are allowed to decay like t, H± and W ± whereas other particles are analysed as ﬁnal state particles like τ-leptons and b-quarks. This in order to facilitate the analyses and determine if detection is even possible at parton level.

3.3 ROOT ROOT [14] is a C++ based data analysis framework developed at CERN. It is a very powerful tool for handling large amounts of data to produce graphs and histograms while for example performing cuts on variables. ROOT was used to produce all plots in this paper.

4 Analysis 4.1 Initial scans over parameter space We are trying to determine if the production of charged Higgs in association with squarks can be used to detect the charged Higgs boson in any region of the NUHM parameter space. This basically requires that the production cross section is not too small compared to the background or that they are easy to separate in the detector. Since we have little or no idea as to the true values of the NUHM parameters, large scans over the parameter space are initially required. These scans are performed by randomly selecting values of the NUHM variables m0, m1/2, A0, mA, µ and tan β in the ranges indicated in table 1. Once the values of the NUHM parameters have been chosen at random, the spectrum of all

Parameter m0 [GeV] m1/2 [GeV] A0 [GeV] mA [GeV] µ [GeV] tan β min 50 50 -2000 -2000 5 1 max 2000 2000 2000 2000 600 60

Table 1: Ranges of NUHM parameter values for initial random scans. supersymmetric parameters is calculated using SoftSUSY. A number of checks are then performed to ensure that the resulting spectrum is both convergent and does not violate any experimental data. All the internal SoftSUSY checks are used to check for problems. All sparticle masses are checked against their experimental lower limits shown in table 2 and it is veriﬁed that the LSP is neutral. Finally points in the (mH± , tan β) space that have been excluded from experimental data [15] are also 2 2 excluded by making a linear cut tan β 0.123 mH± 1.9, where mH± = mA + mW , as indicated by the red line in ﬁgure 5. This is also≤ the region where− the standard channels for analysis has the p least sensitivity. The cut removes most of the black, blue and yellow regions while leaving nearly all points coloured in green.

4.1.1 H-t1-b1 coupling and squark masses Figure 6 shows the Feynman diagrams for two examples of charged Higgs production with squarks. Like all other diagrams they contain the triple vertex of a charged Higgs, an up type and a down type squark. Consequently the cross section is proportional to the corresponding coupling constant which

17 + 0 + ˜ ˜ Particle H h χ1 χ1 e˜R µ˜R τ˜1 ν˜ b1 t1 g˜ mass limit [GeV] 79.3 111 46 94 73 94 81.9 94 89 95.7 308

Table 2: Experimental lower limits at 95% conﬁdence level of sparticle masses [16].

Figure 5: Allowed(green) and forbidden regions of the (mH± , tan β) space from experimental data [15]. The red line indicates the linear cut implemented in the simulations.

will be refered to as gmn where m, n = L, R, 1, 2 indicate the index of the up and down type squark. Since squarks are scalars, two of each squark is needed to cancel the two radiative contributions of the corresponding spinor quark. In Supersymmetry all squarks therefore come in pairs and an index L, R for chiral states or 1, 2 for mass eigen states, where 1 is the lightest, is required to specify the squark. ˜ b1∗ H−

˜b1 ˜b∗ ˜ ˜ 1 t1 g11 t1 g11

˜b1

˜ H− b1∗

Figure 6: Feynman diagrams of associated charged Higgs and squark production. The coupling g11 between H±, t˜1 and ˜b1 shown in the ﬁgure, is always present.

In order for these large scans to be eﬃcient, only a small number of calculations should be performed for each point in parameter space. Since the cross section is proportional to gmn which is very simple to calculate, it is therefore a good starting point. The coupling constants can be calculated using the

18 expressions for the chiral states of up- and down-type squarks [6],

2 2 igmW md tan β + mu cot β gLL = − sin 2β √2 − m2 W igmumd gRR = (tan β + cot β) √2mW igmd gLR = (µ + Ad tan β) √2mW igmu gRL = (µ + Au cot β) , √2mW where the quark masses appear in the couplings and the tan β dependence is diﬀerent. The superpartners of the chiral states mix to produce the mass eigenstates corresponding to the propagating ﬁnal state particles that will be detected. The mixing is determined by two mixing angles θu and θd according to u cos θ sin θ u 1 = u u L u sin θ cos θ u 2 − u u R d cos θ sin θ d 1 = d d L d sin θ cos θ d 2 − d d R Finally, using the abreviations cos θu,d = cu,d and sin θu,d = su,d, the equations for mass state-couplings are g11 = cucdgLL +cusdgLR +sucdgRL +susdgRR

g12 = cu( sd)gLL +cucdgLR +su( sd)gRL +sucdgRR − − (7) g = ( s )c g +( s )s g +c c g +c s g 21 − u d LL − u d LR u d RL u d RR g = ( s )( s )g +( s )c g +c ( s )g +c c g . 22 − u − d LL − u d LR u − d RL u d RR Now all couplings can be calculated for all generations of squarks. With four couplings for each of the three generations we have twelve diﬀerent couplings to study. To be able to select the best candidate we consider the eﬀect of particle mass on the cross section. The total cross section can be written

σ = g(x1) g(x2) dx1 dx2 dσ,ˆ (8) Z where g(x1,2) are the parton distribution functions (PDF), x1,2 are the fractional momenta of the colliding partons andσ ˆ is the cross section for the hard interaction at parton level. From dimensional arguments,σ ˆ is inversely proportional to the center of mass (CM) energy,s ˆ, needed to produce the mass 2 sˆ = x1x2s (m ± + m˜ + m˜ ) ≥ H t1 b1 where s is the total centre of mass energy. Larger masses produced means more energy needed and smaller cross section for the hard process. Also, to produce larger masses the partons need to carry additional fractional momenta raising the lower limit from which the integration in eq (8) is performed, further reducing the value of the total integral. So in the production of smaller masses the cross section is larger and we will focus on the coupling constant between the squarks of lowest mass. Because of signiﬁcant mixing in the top sector, most supersymmetric models predict that the stop be the lightest squark. The cross section should therefore be the largest when the squarks produced are t1 and b1 and

19 the coupling under study will be g11 which is calculated using the ﬁrst of eqs (7).

Figure 7 shows the result of a random scan of 2.5 105 points in the parameter space. The coupling · g11 has been plotted against all six input parameters. A few conclusions can be drawn: All values in the range 2000 g 1600 are allowed although positive values are favored. − ≤ 11 ≤ Only for a few parameter ranges are any ranges of g11 excluded. An example is 1600 m1/2 2000 which excludes 2000 g 600. ≤ ≤ − ≤ 11 ≤− With the exception of µ there is little dependence on the parameters, i.e when changing a parameter the probability for a certain value of g11 remains approximately the same. Small values are allowed and favored for almost all ranges of all parameters. In general values 0 g 500 are the most probable. ≤ 11 ≤ As was explained earlier, the cross section of the hard interaction depends strongly on the masses of the produced particles. For the process to be used as a Higgs detection tool, the masses should therefore be small. Figure 8 and 9 show the stop and sbottom masses plotted against the NUHM parameters. Both masses have a lower cut-off in terms of m0 and m1/2 because of terms in the RGEs which parametrizes the squark masses at the EW scale as functions of the universal parameters at the GUT scale. They are shown in eqs (9) [18]. 1 m2 = m2+ K + K + K + ∆u ˜ + stop corrections u˜L 0 3 2 36 1 L 2 2 4 mu˜R = m0+ K3 + K1+ ∆u ˜R 9 (9) 2 2 1 ˜ m ˜ = m0+ K3+ K2+ K1+ ∆dL+ sbottom corrections dL 36 2 2 4 ˜ m ˜ = m0+ K3 + K1+ ∆dR dR 9 where K 0.15 m2 , K 0.5 m2 and K (4.5 to 6.5) m2 . Both masses also show a depen- 1 ≃ 1/2 2 ≃ 1/2 3 ≃ 1/2 dence on A in the applicable low mass (m 500 GeV) regime. 0 t1/b1 ≤ Figure 10 shows how the coupling depends on the squark masses, it is obvious that in order for the masses to be smaller than a couple of hundred GeV, the coupling must also be small, g11 500 800. Both the coupling and the masses affect the total cross section but these effects work against| |≤ each− other since a smaller mass forces a weaker coupling. When choosing the NUHM parameters to maximize the cross section we should be guided by the masses and not the coupling since they have the clearest dependence on NUHM parameters.

20 Figure 7: The coupling constant g11 plotted against the NUHM input parameters from a random scan of 2.5 105 parameter points selected in the ranges speciﬁed in table 1. Colors indicate the number of points· plotted in one bin, indicating statistically favored regions in terms of the other parameters.

21 Figure 8: The stop mass plotted against the NUHM input parameters from random scan of 2.5 105 parameter points. Colors indicate the number of points plotted in one bin, indicating statistically· favored regions in terms of the other parameters. The stop mass shows a strong dependence on the parameters m1/2 and m0.

22 Figure 9: The sbottom mass plotted against the NUHM input parameters from random scan of 2.5 105 parameter points. Colors indicate the number of points plotted in one bin, indicating statistically· favored regions in terms of the other parameters. The sbottom mass shows a strong dependence on the parameters m1/2 and m0.

23 Figure 10: In the upper plot the the coupling g11 is plotted against the sbottom mass and in the lower against the stop mass. To have small masses, mb1 < 500 GeV and mt1 < 200 GeV the coupling cannot be larger than g . 800 at least. | 11|

24 4.1.2 Cross section Using C++ and Linux scripts, a program is written which generates random values for the variables within set ranges, calculates the spectrum of variables using SoftSUSY, calculates decay widths and ratios using SusyHit and then computes the total cross section using MGME. The NUHM parameter ranges are constrained with consideration to minimizing mt1 and mb1 from previous scans and minimizing mH± by minimizing mA. A positive value on µ is imposed since it is clearly favored in order to explain corrections to the anomalous magnetic moment of the muon [17].

Initially a scan of 3 105 parameter points is run with the constraints in table 3. The calculated cross section shown against· the NUHM parameters can be seen in ﬁgure 11. As expected there is a clear dependence on the mass parameters m1/2, m0 and mA favoring small values. We see that the cross section reaches values as large as 1 pb, small compared to backgrounds that are expected to be of the order 100 1000 pb. ∼ −

Parameter m0 m1/2 A0 mA µ tan β min 150 145 -2000 100 200 7 max 800 350 -800 350 1350 28

Table 3: Constrained ranges of NUHM parameter values for random scans of total cross section. The ranges have been constrained to maximize the cross section based on arguments and previous scans in section 4.1.1

4.1.3 Decays and detection In any real experiment, the squarks and Higgs bosons produced will decay almost instantaneously and so will most of their decay products. What the detector sees are essentially leptons and showers of hadronised quarks called jets. In order to get a clear and detectable signal with neither too many undetectable particles, like neutrinos and neutralinos, nor too many jets, the decay chains of the produced particles should be investigated. The decay information can be found in the output from SusyHit as the branching ratios of the particles, these ratios decide how often a particle will decay into a certain ﬁnal state. The charged Higgs will almost always decay predominantly through two decay channels with branching ratios of about 50 - 100 % and 0 - 50 % respectively,

H+ τ +ν and H+ t¯b. → τ →

The ˜b1 also has two major decay channels which are equally populated, together making up about 60 - 80 % of the total branching ratio,

+ + ˜b∗ t˜∗H and ˜b∗ t˜∗W . 1 → 1 1 → 1

In both cases the ˜b1, being the heavier squark, will produce a second t˜1 together with a charged Higgs or a W boson. The decay products of the stop squark depend on which ﬁnal states are kinematically accessible. The possibilities for the main channel are

t˜ bχ+ and t˜ cχ0. 1 → 1 1 → 1

25 Figure 11: Scatter plot of data from scans over 3 105 parameter points of the cross section of the hard interaction, displayed against the NUHM input· parameters. The cross section displays a strong dependence on the parameters, in particular the mass parameters m1/2, m0 and A0.

26 + If mt1 mb + m then the stop will decay through the first channel with a 100 % probability. If the ≥ χ1 mass spectrum forbids this decay the stop will proceed through the latter, flavour changing neutral channel with a branching ratio of 80 - 90 %. The major difference between these two decays is that + 0 the first contains an unstable supersymmetric particle, the χ1 , whereas the second contains χ1 which is stable since it is the LSP. The decay of this unstable particle will give rise to either more jets or missing energy since its decay channels are

χ+ χ0q q¯ and χ+ χ0l+ν . (10) 1 → 1 u d 1 → 1 l To summarize, the ﬁnal state depends on the mass spectrum, and the two possibilities are

+ H 0 0 qq¯ qq¯ ¯ + H− + bbχ1χ1 , if mt1 > mb + mχ ˜ ˜ W lνl lνl 1 H−t1b1∗  + (11) →  H 0 0  +  H− ccχ¯ 1χ1, if mt1 mb + m W + ≤ χ1  It is clear that the latter is much simpler and in all coming scans the cross section will only be calculated

+ if the condition mt1 mb + m is fulfilled such that the mass spectrum forces the stop to decay ≤ χ1 into this final state. When making this demand we limit ourselves to the rather few points in which the spectrum fulfils this condition and inevitably loose generality. However the objective, as discussed in section 2.3, is to maximize the detection possibilities in one single point whether it is the most general or not. Since this new demand generally forces a low stop mass these parameter points will generally produce large cross sections as well. We also add an additional constraint, from searches by the DØ Collaboration of the Fermilab Tevatron, on the (m , m 0 ) space, which excludes the square t1 χ1 area where m 140 GeV and m 0 80 GeV. Details on this constraint are discussed in more detail t1 χ1 in [19]. ≤ ≤

Point # m0 m1/2 A0 mA µ tan β σtot [fb] mH± [GeV] 1 193 182 -1136 213 822 11.8 572 203 2 190 187 -1047 179 745 13.2 559 216 3 185 186 -955 178 672 11.7 502 194 4 182 188 -1031 202 736 11.7 466 194 5 235 184 -983 188 665 12.2 433 225

Table 4: The ﬁve points found with largest cross sections.

A new smaller scan, with the narrower parameter limits shown in table 5 and the above mentioned added constraints, is run and the results can be seen in ﬁgure 12. As expected the average cross section is much higher than before because of the new constraint on the stop mass. The maximum cross sections reach values of just under 600 fb. The parameter points with the largest cross sections are shown in table 4 and these are the points that will be considered in the signal background comparisons of section 4.2. Worth to note is that these points all lie within a very narrow section of parameter space and there is no reason to believe nature has chosen a value in this particular region. They still serve as an example of what could be possible.

27 Figure 12: The cross section for the hard interaction plotted against the NUHM parameters. The parameter ranges have been restricted compared to earlier scans and the constraint on the masses

+ mt1 mb + m has been imposed. The cross section reaches values just under 600 fb. ≤ χ1

28 Parameter m0 m1/2 A0 mA µ tan β min 180 180 -1400 175 500 6.5 max 280 235 -900 215 960 15

Table 5: Further constrained ranges of NUHM parameter values for random scans of total cross section. The ranges have been constrained to maximize the cross section based on arguments and previous scans in section 4.1.1

Figure 13: The cross section of the hard interaction is plotted in a histogram against tan β in the range 180 GeV < m ± < 215 GeV. The curve shows σ(pp H±t¯b) for tan β = 10, m ± = 200 GeV scaled H → H with the coupling constant of the H±t¯b vertex squared.

As outlined in section 2.3, if the cross section for H± is larger in association with squarks rather than SM quarks, that channel can be used for earlier detection. In ﬁgure 13 there is a histogram of the cross section with squarks from our simulation. Superimposed is the cross section for H±t¯b production which has been obtained by scaling the value of the cross section for tan β = 10 with the coupling constant g2 . In the region 6.5 tan β 15 there are several points where squark scenari o does Htb ≤ ≤

29 indeed give a higher cross section and we can conclude that in the limited parameter space window of this scan, the prospects are good for charged Higgs detection using the squark production channel.

4.2 Signal and background A few points in parameter space have now been determined that give a relatively large cross section, are not excluded by any experimental data and should give somewhat simple final states that can be identified in the detectors. To compare the signal and background, many events must be generated and analysed to see what they look like in the detector. Such an analysis can be performed in Pythia and is generally very complicated and time consuming since many types of particle interactions have to be modelled and calculated. In the subsequent analysis quarks and leptons will therefore be considered as final state particles and no hadronization or parton level interactions like parton showering will be performed when the decays are analysed in Pythia.

In Pythia particles have an exact mass representing the best experimental value. When reconstructing the invariant mass of particles the result will be a very sharp peak in the spectrum. But in reality, when a particle decays and is detected, the limitations in detector resolution will cause a smearing of this peak and in the invariant mass spectrum it will have a width which can be modelled by smearing the energy and momenta

px′ = (1+ s) px

py′ = (1+ s) py

pz′ = (1+ s) pz 2 2 E′ = (m) + p¯ , | ′| using a smearing factor s that is taken from ap gaussian distribution G(¯x, σ) with mean valuex ¯ = 0 and standard deviation 0.62 σ = + 0.022. s pT Considering eq (11), the charm quarks will be analysed as final state particles instead of the jets that would have been detected in a real detector. In coming discussion we will still refer to these quarks as jets since they will form jets even if they are not analyzed as such in our simulations. Ultimately we wish to reconstruct the invariant mass of the Higgs boson and see a peak in the resulting spectrum. Therefore it is desirable to detect all of its rest products and only events where one of the Higgs bosons proceeds though the decay H+ t¯b W +b¯b qqb¯ ¯b, → → → are saved. The other Higgs particle must then decay into τντ so as not to flood the detector with four more jets while maintaining a large branching ratio. This decay is also available for the W boson giving us more signal events. In the subsequent analysis Pythia models the decays and only saves events resulting in the final state 0 0 H−t˜ ˜b∗ ccq¯ qb¯ ¯bτν χ χ 1 1 → τ 1 1 and the detector signal will be two b-jets, four other jets, one τ-jet and missing transverse momenta or p from the neutrino and neutralinos. The missing momenta is calulated as the sum of all momentas 6 T

30 from the detected particles (j = 1, 2...n) with opposite sign such that the total momenta adds up to zero, the missing transverse momenta is therefore

2 2

pT = v px + py . 6 u    u j j u X X t    The event analysis is split up into three parts performed by three different C++ programs. The first program loads the event record using Pythia, performs the requested decays and saves the four-vectors in a ROOT-tree. The second program analyses these four-vectors. It sorts the jets by pT and calculates the invariant mass for different jet combinations. Two jets should come from the decay of a W and the best candidate pair is determined. This W boson, along with one other b-jet comes from a top-decay and the best candidate is found to enable efficient cuts in the final stage. Together with the last b-jet these four particles are the rest products of the Higgs boson and the invariant mass can be reconstructed in each event and then compared to the background. In the detector, b-jets can be identified by b-jet tagging which reduces the number of total events since the efficiency is only about 50%. The same is true for τ-jets with an efficiency of about 30%. The third and final program performs several cuts on the transverse momenta, as detailed below, before presenting the information in histograms.

The first set of cuts correspond to what the detector can be expected to resolve. θ and φ specify a particle’s direction in terms of its angle against the beam line and its azimuthal angle. The pseudorapity, η, is defined as η = 1 ln tan 1 θ and gives a nearly flat distribution from particle collisions. The trans- − 2 2 verse momenta p = sin(θ) p¯ and the angular separation between particles ∆R = (∆φ)2 + (∆η)2 T both need to be above detector| | separation thresholds and are set as follows p p 20 GeV, η 5, η 2.5, η 2.5, ∆R 0.4. T ≥ | |≤ | b|≤ | τ |≤ ≥ Essentially an event is only used if all particles are separated enough from the beam and each other in (φ,η) space. In order for b- and τ-jets to be tagged they need to pass through the silicon detector which is smaller than the surrounding detectors, forcing a more stringent cut on ηb and ητ .

To analyse the signal, the hard process of 1 105 signal events are generated using MGME in one single point of parameter space which is selected after· evaluating the background in section 4.2.1. At a cross section of 559 fb for the process and its hermitean conjugate, this number of events corresponds to 1 approximately 90 fb− of integrated luminosity or one year of data collection at LHC when it reaches design luminosity.

Knowing what kind of background we expect, and knowing what it will look like from Monte Carlo simulations, it can be subtracted from the experimental events. However, if there are very few total events the statistical uncertainties are large. Assuming any errors have a normal distribution, we can evaluate the number of standard deviations required to give an observed excess of events compared to expectations. This value measures the statistical signiﬁcance of any signal and is computed as S n = , √B generally a value n 5 is required to claim a discovery. ≥

31 4.2.1 MSSM background One irreducible background within the MSSM is the production of stop and top quarks

pp t˜ t˜∗tt,¯ → 1 1 where the stop quarks decay into charm quarks and neutralinos and the top-quarks each produce one b-quark and either two other jets or a τντ pair thus precisely mimicking the signal. Initially the cross section for this process is evaluated in the ﬁve points from table 4. Point number 2 in table 4, with parameter values m = 190 GeV, m = 187 GeV,A = 1047 GeV, m = 179 GeV, µ = 745 GeV, tan β = 13.2, 0 1/2 0 − A has the best signal-to-background ratio when taking into account branching ratios to our ﬁnal state, and this point is used for all event generation. In this point the charged Higgs mass mH± = 194 GeV

As for the signal, the hard interactions are generated using MGME. Given that the background has a 10 times larger cross section one should generate 10 times more events to get correct statistics. This can be circumvented by changing the decay settings in Pythia such that

+ W hadrons W − τν → → τ for all events. Thereby increasing the events that we are interested in by a factor 7. It is therefore sufficient to generate 1.5 105 events. These are then analysed by the same program as was the signal. The result of the comparison· can be seen in figure 14 where the signal has been plotted on top of the background. The number of events at the peak, approximately corresponding to mH± 15 GeV or 180 mINV (2j + 2b) 206.6GeV due to bin sizes, are counted and scaled to represent an± integrated ≤ 1 ≤ luminosity of 100 fb− . The significance value is computed to n = 29.

Figure 14: The invariant mass of the Higgs boson is reconstructed for the MSSM background (red) and the signal + background (blue). No cuts have been made and the signiﬁcance value in the range 180

This picture can be improved somewhat by making appropriate cuts based on the diﬀerences of the signal and background kinematics. In ﬁgure 15 we can compare the transverse momentum of all

32 detected particles for signal and background. The four quark jets have been ordered from hardest, being the one with largest pT , to softest and the b-quarks have been identified as best and worst match for reconstructing the invariant mass of a top quark together with the two quarks that have previously been matched to reconstruct the invariant mass of a W boson. Setting the cuts so as to maximize n is just a matter of trying different cuts on the data material so as to remove as many background events as possible while keeping most of the signal intact. In figure 16 the following cuts, in units of GeV, have been imposed

(pT )jet1 < 110 (pT )jet2 < 90 (pT )jet3 < 50 (pT )jet4 < 60 (p ) < 190 (p ) < 60 (p ) > 20 p > 40. T b1 T b2 T τ 6 T 1 These cuts reduce the background events, scaled to 100 fb− , from 1111 to 143, while only reducing the signal from 965 events to 572. The signiﬁcance ratio is thereby increased to n = 48.

33 Figure 15: The distribution of transverse momenta for the four jets, two b-jets, one τ-jet and pT of the signal and the MSSM background. Both curves are in the same units but they have been scaled6 to the same maximum. Only cuts representing detector resolution are imposed.

34 Figure 16: The invariant mass of the Higgs boson is reconstructed for the MSSM background(red) and the signal + background (blue), after imposing selective cuts on the tranverse momenta in ﬁgure 15, the signiﬁcance value is increased to n = 48

4.2.2 SM background

Another irreducible background is the SM production of tt¯+ 2jets which has a cross section σtot = 470 pb according to MGME. This is almost a factor 1000 larger than our signal. Generating a 103 times as many events as the 1 105 generated to simulate our signal is hardly feasible but as before the decay settings in Pythia saves· us a factor 7 and generating just over 1.1 106 events, using the on-line clusters available from MGME, we are a factor 13 short. This is not ideal· but will have to suffice. The events are fed through the same analysis as before and the result can be seen in figure 17. When 1 scaled to 100 fb− the number of background events is 153265 and as before 965 for the signal. The resulting significance value is n = 2.5.

Figure 17: The invariant mass of the charged Higgs boson is reconstructed for the SM background (red) and the signal + background (blue). The signal is drowned in the background and before imposing cuts it is diﬃcult to identify.

35 Figure 18: The distribution of transverse momenta for the four jets, two b-jets, one τ-jet and pT of the signal and the SM background. Both curves are in the same units but they have been scaled to6 the same maximum. Only cuts representing detector resolution are imposed.

36 Figure 19: Using the cuts in eq (12) the invariant Higgs mass is reconstructed for the SM background(red) and the sinal + background (blue). The signal now forms a peak around the Higgs mass mH± = 194 GeV.

Figure 20: The cuts in eq (12) used to reconstruct the Higgs mass in ﬁgure 19 are further constrained such that (pT )τ > 150 and pT > 130. The signiﬁcance value is increased to n = 15.8 at the cost of statistics. Ideally, more background6 events should be simulated to ensure the large value of n is not simply due to a lack of statistics.

Based on the comparison of the transverse momenta of the signal and background in ﬁgure 18, cuts in untis of GeV, are set as follows

(pT )jet1 < 120 (pT )jet2 < 60 (pT )jet3 < 70 (pT )jet4 < 50 (12) (p ) < 300 (p ) < 80 (p ) > 120 p > 110 T b1 T b2 T τ 6 T and the number of background events are reduced from 153265 to just 110 while the signal is reduced from 965 to 110 events. The significance value is thereby increased to n = 8.2. The resulting plot can be seen in figure 19. The last two cuts are the most restrictive, reducing the background a factor 7 and 10 respectively while only halving the signal. They can be restricted further removing much of the signal and background but significanlty increasing n. Strengthening the cuts to (pT )τ > 150 and

37 pT > 130, the background is reduced to just 5 events while the signal still consists of 37. Thereby n is 6increased to n = 15.8 and the result is shown in figure 20. Unfortunately we suffer from an apperent lack of statistics and must use caution when drawing conclusions about the results. In all simulations we have assumed b- and τ-jets can be identified or tagged with 100% efficiency. In fact these tagging efficiencies are approximately 50% and 30% respectively. This reduces our final number of detected events, both for signal and background, by 50% 50% 30% = 7.5% and our significance value is reduced by √7.5% = 27%. Using our largest value for· n · S = 0.27 n = 4.3. √ · max B true In order to get a 5σ signal we need to increase the runtime a factor (5/4.3)2 = 1.33 to acquire an 1 integrated luminosity of 133 fb− .

1 Figure 21: 5σ detection contour for H± in ATLAS at 1, 10, and 30 fb− respectively in the mH± ,tan β plane. There is a clear gap in the range 2 tan β 15 where the charged Higgs cannot be detected 1 ≤ ≤ at 30 fb− using the conventional channels. Prospects for closing this gap seem small considering how the contour grows with increased luminosity. The point analyzed in this paper, mH± = 194 GeV, tan β = 13.2 lies in the middle of mid-tan β void well outside the detectable region. For details regarding this plot we refer to [20].

4.3 Comparison with other channels

Figure 21 shows the 5σ detection prospects of charged Higgs at ATLAS within the mh-max scenario [20]. This scenario tries to maximize the radiative corrections to the light Higgs in order to get a value consistent with experimental expectations. The charged Higgs is produced in top decays for mH± 170 GeV and in gluon-bottom fusion for mH± 180 GeV. Within the study a full detector simulation≤ is performed including systematic eﬀects. The≥ intermediate tan β-range is largely 1 inaccessible and although the study only shows the discovery contour at 30 fb− the gap seems hard to

38 close using the studied channels. The point studied in this paper, mH± = 194 GeV and tan β = 13.2 lies well outside the discovery contour as indicated by the red dot in ﬁgure 21.

5 Conclusions, Outlook ˜ ˜ ¯ ¯ Having analysed the two processes pp t1t1∗tt and pp tt + 2 jets, assumed to be the largest → → 1 backgrounds, using MGME and Pythia, a 5σ discovery requires an integrated luminosity of 130 fb− assuming b- and τ-jet tagging efficiencies of 50 and 30 %. After including effects from hadronization, parton level interactions, other backgrounds and detector efficiencies, a slightly higher integrated luminosity may be required. This still compares well to what is required by other channels which hardly have any detection prospects in the mid-tan β range.

The 5σdiscovery assumes particular values of the NUHM parameters chosen to maximize the cross section and thereby the detection possibilities. More generally, if the parameters within the NUHM have values such that they force small masses for the squarks and the charged Higgs, preferably such ˜ 1 that the t1 bχ1 channel is closed, detecting the charged Higgs boson from its production in association with squarks→ can be possible in the void left by other production channels in the mid-tan β range.

If the Higgs production under study is to be used for detection a more general analysis should be made to determine how large the volume of parameter space allowing detection is. Also a full simulation should be done including hadronization and detector simulations.

In order to reach better results one could consider performing the same analysis within a more general model of the MSSM that provides better control over the t˜1, ˜b1 masses thereby allowing a more general treatment of the other variables.

6 Acknowledgments

The author would like to thank the whole THEP group for fruitful discussions and comments at our meetings. In particular Johan for his invaluable guidance over the course of this project. A special thanks to Oscar for his endless patience to questions. Also a big thanks to Glenn for his many useful tips and great company along the way and Gunnar for participating in all administrative requirements and his shown interest.

39 References

[1] H. Bear, X. Tata, Weak Scale Supersymmetry, Cambridge University Press, Cambridge, 2006

[2] B.Andersson, G. Gustafson, G. Ingelman, T. Sj¨ostrand, Parton Fragmentation and String Dy- namics, Phys. Rep. 97(2 & 3), 31-145 (1983)

[3] A. Pich, The Standard Model of Electroweak Interactions, hep-ph/0705 4264v1, 29 may 2007

[4] P.Bin´etruy, Supersymmetry - Theory, Experiment and Cosmology, Oxford University press, New York, 2006

[5] D. Matalliotakis and H. P. Nilles, Implications of non-universality of soft terms in supersymmetric grand uniﬁed theories, hep-ph/9407251, June 1994

[6] John F. Gunion, Howard E. Haber, Gordon Kane, Sally Dawson, The Higgs Hunter’s guide, Perseus Books, Massachusets, 1990

[7] O. Buchmueller, Likelihood Functions for Supersymmetric Observables in Frequentist Analyses of the CMSSM and NUHM1, hep-ph/0907.5568. July 2009

[8] B.C. Allanach, SOFTSUSY manual, Comput. Phys. Commun. 143 (2002) 305-331, hep- ph/0104145

[9] Decays of Supersymmetric Particles: the program SUSY-HIT, A. Djouadi, M.M. Muhlleitner, M. Spira, ActaPhys.Polon.B38:635-644,2007, hep-ph/0609292

[10] J. Alwall et al., A standard format for Les Houches Event Files, Comput. Phys. Comm. 176 (2007) 300-304, hep-ph/0609017

[11] J. Alwall, P. Demin, S. de Visscher, R. Frederix, M. Herquet, F. Maltoni, T. Stelzer, MadEvent, a multi-purpose event generator powered by MadGraph, 1 March 2007

[12] T. Sjöstrand, S. Mrenna and P. Skands, Pythia 6.4 physics and manual, JHEP05 (2006) 026 [13] T. Sjöstrand, S. Mrenna and P. Skands, A Brief Introduction to PYTHIA 8.1, Comput. Phys. Comm. 178 (2008) 852, arxiv:0710.3820 [14] Rene Brun and Fons Rademakers, ROOT - An Object Oriented Data Analysis Framework, Pro- ceedings AIHENP’96 Workshop, Lausanne, Sep. 1996, Nucl. Inst. & Meth. in Phys. Res. A 389 (1997) 81-86. http://root.cern.ch/ [15] D. Eriksson, F. Mahmoudi, O. St˚al, Charged Higgs bosons in minimal supersymmetry: updated constaints and experimental prospects, JHEP11 (2008) 035 [16] C. Amsler et al., Particle Data Group collaboration, Supersymmetric particle searches, Phys. Let. B667, 1 (2008) [17] F. Feroz et al., Bayesian selection of sign µ within mSUGRA in global fits including WMAP5 results, JHEP 10 (2008) 064, arxiv:0807.4512 [18] S. Martin, A Supersymmetry Primer, hep-ph/9709356, 10 Dec 2008

40 [19] DØ Collaboration, Search for 3- and 4-body decays of the scalar top quark in pp¯ collisions at √s = 1.8TeV, 10.1016/j.physletb.2003.12.001 [20] ATLAS Collaboration, Expected Performance of the ATLAS Experiment: Detector, Trigger and Physics, CERN-OPEN-2009-020 (2008) 1451-1479