2014Fall QFT Final Examination—Hunting for the Higgs

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2014Fall QFT Final Examination—Hunting for the Higgs

Yao-Chieh Hu^{1, ∗}

1Department of Physics, National Taiwan University, Taipei 10617, Taiwan (Dated: January 18, 2015)

There are many colliders and experiences now studying the Higgs bosons, like Large Electron- Positron Collider (LEP), the Tevatron, and the Large Hadron Collider (LHC). In this report, we will focus on the experiment at the LHC, where the Higgs was found on 4 July 2012 and be temporarily confirmed on 14 March 2013 with the mass about 125GeV. The production and the detection of the Higgs will be mentioned, and also some future research for new physics at LHC.

I. INTRODUCTION

After Learning the electroweak interaction, we know that the Higgs mechanism provides a general framework to explain the spontaneously symmetry breaking with SU(2) × U(1) to U(1)and give masses to the W^± and Z gauge bosons. The high energy U(1) symmetry is called hypercharge, denoted as U (1)_Y; this U (1)_Y should not be confused with the EM gauge symmetry U (1)EM. (In SM, the SU(2) × U (1)Y is broken by the VEV of a com- plex doublet H with hypercharge 1/2 called the Higgs multiplet.) According to what we have learned, we can predict the behaviours of the Higgs boson, such as how it is produced and how to detect it by its decay mode.

II. WEINBERG-SALAM MODEL AND COUPLING CONSTANTS

In the Glashow-Weinberg-Salam theory of electroweak interactions [7,8,9], the mass terms of the fermions are constructed by the spontaneous symmetry breaking of some scalar field coupled to the fermion fields. The scalar field couples to, say, the electrons and electron neutrinos, through the Lagrangian (c = ~ = 1)

∆Le= −λeE¯L· φ eR+ h.c. (1) The coupling to the up and down quarks is similar, read- ing

∆Lq = −λdQ¯L· φ dR− λu^abQ¯Laφ^†_buR+ h.c. (2) Evaluating these Lagrangians in the unitarity gauge, in which the scalar field is parametrized by

φ(x) = 1

√2

0

v + h(x)

, (3)

with v being the vacuum expectation value of the scalar field and the real-valued perturbation h(x) as the Higgs field, one obtains the general form of the coupling of the Higgs boson to any lepton or quark,

Lf = −mff f¯

1 +h

v

. (4)

∗b99606019@ntu.edu.tw

Note that the coupling is proportional to the mass of the fermion.

The couplings of the Higgs boson to the gauge fields are through the kinetic term, (D^µφ)^†(Dµφ). The covariant derivative is

D_µφ =

∂_µ− igA^a_µτ^a− i1 2g⁰B_µ

φ, (5)

in which A^a_µ and Bµ are the gauge bosons of the SU (2) and U (1) groups, respectively, with g and g⁰ being the corresponding coupling constants. After evaluating in the unitary gauge, one obtains

LK =1

2(∂^µh) (∂µh) +

m²_WW^+µW_µ⁻+1

2m²_ZZ^µZµ

·

1 + h

v

2

, (6)

where

W_µ^±= 1

√2 A¹_µ∓ iA²_µ

(7) with mass,

mW =1

2gv, (8)

and

Zµ= 1

pg²+ g⁰² gA³_µ− g⁰Bµ

(9)

with mass,

m_Z = 1 2

pg²+ g⁰² v, (10) and one remaining massless gauge boson identified as the electromagnetic vector potential. The SM Higgs boson couplings to gauge bosons, Higgs bosons and fermions are summarized in the following Lagrangian:

L = −g_{Hf ¯}_ff f H +¯ gHHH

6 H³+gHHHH 24 H⁴ +1

2VµV^µ(gHV VH +gHHV V

2 H²)

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In standard model, it predict that the Higgs couplings to fundamental fermions are proportional to the fermion

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2 masses, and the coupling to the gauge bosons are pro-

portional to the squares of the boson masses. That is, g_{Hf ¯}_f = mf

v , g_{HV V} = 2m²_V v , gHHV V = 2m²_V

v² .

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where V = W^± or Z is the gauge boson, f is any kind of fermion, and is the vacuum expectation value of the Higgs field. With these interactions in hand, one can predict the properties of the Higgs bosons, such as the cross sections and branching ratios of various processes, and compare them with the experiments.

III. PRODUCING HIGGS AT THE LHC

LHC is a proton-antiproton collider, like Tevatron but with higher center-of-mass energy (14T eV expected) and luminosity (10 ∼ 30 f b⁻¹). They produce Higgs through four major processes : gluon fusion, weak-boson fusion, associated production with a gauge boson and associated production with top quarks. The ordering is for special purpose, it is ordered with respect to the cross-section for the production of SM Higgs.

1. Gluon fusion production mechanism

At LHC, the Higgs boson production with largest cross-section is the gluon-fusion. This producing mode is by the reaction:

gg −→ H(+X) (13)

In this mechanism, the process is mediated by the ex- change of a virtual, heavy top quark. The detailed dis- cussion is in [2]. The lowest order of the Feynman diagram is :

Moreover, the gluon-gluon fusion cross section of the Higgs particle in the SM in nexttoleading order QCD is discussed in first half of [3]

2. Vector boson fusion production mechanism

This mechanism has the second-largest cross-section at LHC. The producing mode is by two reactions:

qq −→ Hqq or q ¯q −→ Hq ¯q (14) and the lowest order of the Feynman diagram is:

And because of the coupling constant is proportional to the fermion masses, the main mode is with top-quarks.

3. WH and ZH associated production mechanism

This one is commonly called ”production in associate with a vector boson.” The producing mode is by the reaction:

q ¯q −→ HV (15)

where V = W ± or Z is the gauge boson, this is the reason for its name (associate with a vector boson). And the lowest order of the Feynman diagram is

Considering the associated productions, we have the process:

pp −→ HV + X (16)

again where V = W ± or Z is the gauge boson. This process receive contributions at next-leading-order given by QCD corrections to the Drell-Yan cross-section and electroweak correction to next-leading-order. To see more de- tails of associated production of Higgs and weak bosons, one can check[4].

4. Higgs production in association with top quarks

For the t¯tH production process, first, we note that this final state can arise form both q ¯q and gg initial

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3 states. However, at LHC, only the gg contribution is

significant[5]. The process is give:

pp −→ gg −→ Ht¯t (17)

The corresponding Feyaman diagram is:

Short Conclusion

We can summary above four processes in one diagram:

This is for SM Higgs boson production cross sections as a function of the center of mass energy,√

s, for pp colli- sions.

IV. DETECTION OF HIGGS AT LHC

A variety of search channels are pursued by the LHC collaborations, A Toroidal LHC Apparatus (ATLAS) and Compact Muon Solenoid (CMS), with the channels relative importance changing due to the branching ratios of the SM Higgs boson as functions of mH:

a. At low masses , for which m_H < 120GeV :

H −→ γγ, (18)

H −→ b¯b, (19)

H −→ τ⁺τ⁻. (20)

b. At higher masses , for which 120GeV < m_H <

150GeV :

H −→ W⁺W⁻ −→ l⁺νl⁻¯ν, (21) H −→ ZZ −→ l⁺l⁻l⁺l⁻. (22)

In short, the above five channels are the major channels for low mass SM Higgs boson searches at the LHC. We can see their mass resolution for each decay channel for m_H= 125Gev in below table:

c. At even higher masses , for which mH >

600GeV :

H −→ ZZ. (23)

So far, we have confirmed that the mass of Higgs is about 125GeV. The all decay modes are shown in the picture below

V. NEW PHYSICS IMPLIED BY LHC

Although the Higgs mechanism in Standard Model (SM) is very successful phenomenally, the electroweak symmetry breaking driven by a weakly-coupled elemen- tary scalar sector still requires a mechanism to explain the smallness of the breaking scale compared with the Planck scale. This problem lead to the extension as Minimal Supersymmetric Standard Model (MSSM).

The difference between MSSM and SM Higgs mechanism is that MSSM contains the particle spectrum of a two-Higgs doublet model, Hu and Hd, which required to ensure an anomaly-free SUSY extension of the SM and to generate mass for both up-type and down-type quarks and charged leptons. In this construction, five physical Higgs particles are left in the spectrum after the spontaneous breaking of the electroweak symmetry. They con- tain one charged Higgs pair, H^±, one CP-odd scalar, A, and two CP-even states, H and h.

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4 So far, LHC is working hard to find if there exits the

other neutron and charged Higgs which is predicted by the MSSM. Although there are no yet any evidence for the existence of these new particle, we can still find some

implication from the different result from the prediction of SM. And if we find them, it will indicate the new physics beyond the SM.

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D50, 4491 (1994).

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Schroeder, 1995.

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