Theoretical Aspects

Due to its large mass, top quarks have a very short lifetime of aboutτt≈5·10⁻²⁵s [167]. The lifetime is related to the decay width according to:

Γt= ħh τt

, or in natural units : Γt= 1 τt

,

with the reduced Planck constantħh=h/2π=6.582119514(40)·10⁻¹⁶eV·s[28]. This notation ofħhillustrates why the decay width is usually given in units of electron volts. This short lifetime of the top quark has different implications. Pursuant to the above definition, the decay width of the top quark is the largest of all SM fermions. Furthermore, as it takesτhad≈3·10⁻²⁴s to form bound state hadrons, top quarks decay before hadronisation. This hadronisation time results from the formulaτhad=ħh/ΛQCDand thus relies on the actual value of the cutoff parameterΛQCD[168]. As a consequence, bound states which comprise top quarks do not exist in nature. This is the reason why, as already mentioned in the last sections, properties of the bare top quark are experimentally accessible through the decay products of the top quark. Any hadronisation would lead to a loss of information about the underlying decay process.

The decay width of a particle can also be visualised using its mass distribution. This is highlighted in Fig. 2.9 where a Breit-Wigner function f serves to describe an idealised mass distribution of the top quark.

Figure 2.9:Illustration of the relationship between the top quark decay widthΓt within a top quark mass distribution represented by an idealised Breit-Wigner function.

The top quark mass in this idealised curve is consistent with the peak position, and its decay width can be extracted from this curve: Γt corresponds to the width of the curve at half maximum. That is why this quantity is often abbreviated as FWHM standing for “full width half maximum”, as

indicated in Fig. 2.9.

In a real measurement, the top quark decay width cannot be extracted easily from such a curve directly since the detector resolution of objects used to define corresponding observables like the mass needs to be taken into account. The resulting resolution of corresponding observable distribu-tions is about one order of magnitude larger than the underlyingΓt, making a direct measurement of this quantity so challenging. This is further explained in the next subsections. Before, theoretical calculations ofΓt are presented.

The top quark decay width is in good approximation proportional to the third power of the top quark mass and proportional to|V_{t b}|² with the CKM matrix elementV_{t b}. The calculations are usually performed for the process t→W band thus the decay widthΓ(t→W b)is calculated, which can be equated with the full Γt according to the value of|V_{t b}|. V_{t b} determines the coupling strength at theW t bvertex in combination with the universal electroweak coupling constant. The absolute value ofV_{t b}can be evaluated using a measurement of the single top quark production cross-section since it scales with|V_{t b}|²[141, 142, 144]. By the application of unitarity constraints, CKM matrix elements except forV_{t b}are used to indirectly calculate|V_{t b}|more accurately resulting in|V_{t b}|values close to one. The current measured number amounts to|V_{t b}|=0.99915±0.00005[28].

Figure 2.10:Feynman diagram for the decay of a top quark into aW bo-son and a b quark including arrows representing the four-momenta.

At leading order, the decay widthΓ(t→W b)can be calculated from the corresponding Feynman diagram, drawn in Fig. 2.10, exploiting the fact that theW boson from the top quark decay is on shell. The following assumptions need to be made to facil-itate the calculation: |V_{t b}|=1, theW boson is treated as a real on-shell particle and the mass of the bquark is neglected, i.e.

m_b=0. This results in the following four-momenta of the three participating particles: p_t = (m_t, 0, 0, 0), p_b = (P_M, 0, 0,P_M) and p_W = (E_M, 0, 0,−P_M)with the magnitude of the momen-tum in the centre-of-mass frameP_Mand the energy of theW bo-son E²_M= P_M² +m²_W with theW boson massm_W. The calcula-tion based on these three four-momenta leads to the squared spin-average matrix element:

where g_wdenotes the weak coupling constant g_w=p

4παw. Taking into account the relationship between this matrix element and the decay width for a standard decay of particleain the products

bandc, one obtains, after integrating over the full 4πsolid angle:

Γ(t→W b) = P_M

2 . 3 T O P Q U A R K D E C AY W I D T H

Replacing the momentum componentP_M in this expression results in:

Γ(t→W b) = G_F

whereg_wis expressed in terms of the Fermi coupling constantG_F=p

2g²_w/(8m²_W). This equation

Such a leading order approach constitutes only an approximate computation of the top quark decay width. In the following, first and second order QCD corrections are introduced, which have a large impact on the calculated value ofΓt. As the top quark can be treated as an almost free particle, perturbative methods are applicable to estimate those quantum corrections. The full decay width for a top quark decaying into aW boson and abquark can then be written as[169, 170]:

which includes first and second order QCD corrections described by the termsA⁽¹⁾andA⁽²⁾, respec-tively. The first term, Γ0A⁽⁰⁾, refers to the leading order calculation according to Eq. (2.2) with Γ0 =G_Fm³_t|V_{t b}|²/(8πp

2)andA⁽⁰⁾=1−3m⁴_W/m⁴_t+2m⁶_W/m⁶_t. The additional factor C_F is set to 4/3. The one-loopO(αs) correction is known in analytical form[167] and can be written using terms ofm_W/mtas: This first order correction reduces the top quark decay width by about 10%. An approximation withm_W =0 so that only the first mass-independent terms remain, results in an error inA⁽¹⁾of 22%. Including only the quadratic mass terms yields an error of 4% while the first order correction with all terms in Eq. (2.4) leads to an almost negligible error.

In order to estimate predictions for the decay width at orderO(α²_s), the factorA⁽²⁾is decomposed into terms reflecting the underlying colour structure:

A⁽²⁾=C_F²A⁽_A²⁾+C_AC_FA⁽_{N A}²⁾+C_FT n_lA⁽_l²⁾+C_FTA⁽_F²⁾.

These colour factors are defined asC_F =4/3,C_A=3 andT=1/2. The number of quark flavours is set ton_l =5 and thus includes all SM quarks except top quarks. A⁽_A²⁾denotes the Abelian contri-bution already present in QED andA⁽_{N A}²⁾the non-Abelian. A⁽_l²⁾andA⁽_F²⁾are corrections comprising a second fermion loop with either massless or massive quarks. The individual contributions of the

A⁽_i²⁾, i=A,N A,l,F, can be split into different components in terms ofm²_W/m²_t according to:

A⁽_i²⁾=A⁽_i²⁾|m_W=0+m²_W

m²_t A⁽_i²⁾|m²_W +m⁴_W

m⁴_t A⁽_i²⁾|m⁴_W +... . (2.5) The contribution A⁽²⁾_l |m_W=0 is known analytically and amounts to about 2.859 [171]. All other factors are evaluated based on the calculation of propagator-type diagrams which contribute to the self-energy of the top quarks in terms of an expansion aroundq²/m²_t =0 whereqdefines an external momentum. The limitq²→m²_t is considered by performing a Padé approximation where the resulting polynomial is written as a rational function:

[m/n](z) = a₀+a₁z+...+a_mz^m 1+b₁z+...+b_nzⁿ .

Applying this procedure to determine all terms present in Eq (2.5) results in the following solution forA⁽²⁾:

A⁽²⁾=−16.7(8) +5.4(4)m²_W m²_t + m⁴_W

m⁴_t

11.4(5.0)−7.3(1)ln m²_t m²_W

.

Taking this equation forA⁽²⁾and the analytical solutions forA⁽⁰⁾andA⁽¹⁾conforming with Eq. (2.2) and Eq. (2.4), the total top quark decay width can be evaluated using Eq. (2.3). This leads to a value ofΓt=1.33 GeV for a top quark mass ofm_t=172.5 GeV with an approximate precision of 1% [169, 170]. These values of the decay width and the mass of the top quark are exploited to generate the Monte Carlo (MC) events used in this measurement as described in Ch. 5.

In the last few years, further effort was spent on increasing the precision of the theoretical value forΓt. One of the recent calculations [172] does not only contain the dominant next-to-leading order and next-to-next-to-leading order QCD corrections, δ⁽_QCD¹⁾ andδ⁽_QCD²⁾ , but also smaller NLO electroweak correctionsδEWas well as finite bquark mass andW boson width effectsδ^b_f andδ^W_f . Hence, the corrections to the leading order approximationΓ_t⁽⁰⁾, calculated using again Eq. (2.2), can be written as:

Γt=Γ_t⁽⁰⁾·(1+δ^b_f +δ^W_f +δEW+δ⁽_QCD¹⁾ +δ_QCD⁽²⁾ ).

Decay width valuesΓ_t⁽⁰⁾andΓ_t are listed for different underlying top quark masses in Table 2.7.

For a top quark mass ofm_t=172.5 GeV, one obtains a leading order width ofΓ_t⁽⁰⁾≈1.481 GeV and, utilising the above listed corrections, a total decay width ofΓt ≈1.322 GeV[172]which is in good agreement with the prediction used in the baseline MC samples employed in this analysis. The quoted uncertainty of this second estimate amounts to about 0.8% and is derived from a variation of the renormalisation scale. Considering experimental uncertainties on the parameters which enter the formula for Γt increases the uncertainty on this evaluation up to 6%. The dominant contribution originates from the relatively large uncertainty on the CKM matrix element|V_{t b}|, using

2 . 3 T O P Q U A R K D E C AY W I D T H

m_t [GeV] δ^b_f [%] δ^W_f [%] δEW[%] δ⁽_QCD¹⁾ [%] δ_QCD⁽²⁾ [%] Γ_t⁽⁰⁾[GeV] Γt [GeV]

172.5 -0.26 -1.49 1.68 -8.58 -2.09 1.4806 1.3216

173.5 -0.26 -1.49 1.69 -8.58 -2.09 1.5109 1.3488

174.5 -0.25 -1.48 1.69 -8.58 -2.09 1.5415 1.3764

Table 2.7:Total decay width of the top quark at leading order (Γt⁽⁰⁾) and with corrections (Γt) for different top quark massesm_t. Decay width values and masses are given in GeV, the different corrections due to a finite bquark mass, a finiteW boson width and higher orders comprising NLO EW corrections as well as NLO and NNLO QCD corrections are given in percentages [172].

a conservative approach for the|V_{t b}|measurement without the CKM unitarity assumption which results in|V_{t b}|=1.021±0.032[28].