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arXiv:hep-ph/0512181v1 14 Dec 2005

Precision measurement of the top quark mass from Mb? distribution in t → b ?ν decays ?

M. L. Nekrasov

Institute for High Energy Physics, 142284 Protvino, Russia

Abstract The method of the top quark mass measurement from Mb? distribution in t → b ?ν decays can be considerably improved if applying the technique of moments and proceeding to moments of high degree. At the LHC this allows one to reduce the systematic and statistical errors of the top mass by more than a factor of two.

The precision measurement of the top quark mass is necessary for testing of the SM and/or selecting the probable scenario of its extension. The current accuracy of the top mass, attained at Tevatron, is ?Mt = 2.9 GeV, and a further progress is expected mainly at LHC and a future International linear collider. At the LHC the accuracy of measurement of the top mass is anticipated at the level of 1– 2 GeV and in view of large statistics the main di?culties are expected from systematic errors. An analysis of [1] shows that the most promising method from the point of view of optimization of the errors is based on an investigation of the invariant-mass distribution of J/ψ + ? system produced in decays t → bW → b?ν with b-jet formation and the J/ψ being a member of the b-jet. The evaluation made by Monte-Carlo modelling gives approximately 0.7 GeV for systematic error of the top mass measured by this method and of about 1 GeV for statistical error for 4 years of LHC operation [2]. In this paper we discuss an opportunity to improve this result by means of optimization of the data handling within the same experiment.

Talk given at 12th Lomonosov Conference on Elementary Particle Physics, Moscow, Russia, August 25-31, 2005.

?

1

So, following [2] we consider the processes ? q q , gg → tt → bW bW → b?ν bq1 q2 ? {b-jet + ?} + {3 jets} , ? (1)

with the b-jet and the isolated lepton ? = {e, ?} coming from one t quark, and the remaining three jets coming from another t quark. In the experiment the above states are registered and the invariant-mass distribution of J/ψ + ? system is measured. The MJ/ψ ? distributions can then be compared with a template of shapes parametrized by the top mass and thus Mt can be ?tted [2]. One could note, however, that the invariant-mass distribution of the b+? system inevitably emerges at a certain stage of this analysis. Therefore on the equal rights one can carry out the ?tting in terms of the data converted to the form of Mb? distribution. In what follows we adhere to the latter option and hence consider that (as if) a distribution F (q) = σ ?1 dσ/dq is measured, where σ is the cross-section of the process (1) and q = Mb? is the reconstructed invariant mass of the b + ? system. By proceeding in this manner, we simulate the data under the following suppositions. First we suppose that there is a satisfactory method for extracting signal from the data. (Actually this means the existence of a satisfactory model for background processes that survive after setting of kinematic cuts [2].) Secondly we describe the signal in the Born approximation, identifying the b-jet with the b quark. Finally, on the basis of the results of [2], we disregard the e?ects of the ?nite width of the top quark. The latter assumption means that σ ?1 dσ/dq is equal to Γ?1 dΓb?ν /dq, where Γb?ν b?ν is a partial width of the decay t → b?ν. (Thus the distribution F becomes process-independent.) At Fig. 1(a) the distribution F (q) is represented for various values of the top mass. We see from the ?gure that in the absolute value the distribution is most sensitive to Mt in a region located between the maximum of the distribution and a large-q tail where F (q) is almost vanishing. So in this region of q-variable one could expect the highest accuracy for the Mt determination. A practical way to enhance the role of this region is to proceed to high moments over the distribution, qn =

M 0

dq q n F (q) .

(2)

Here M is a technical parameter ?xed close to Mt , and the normalization of F (q) is adjusted so as to satisfy the equality 1 = 1. From Fig. 1(b) it is 2

obvious that with increasing n the distribution q n F (q) is concentrating more and more in the above intermediate region, increasing thus the sensitivity of q n relative to Mt . So, by basing on the above observation, we de?ne the experimentally measured top-quark mass as a solution to the equation qn

exp

= qn .

(3)

Here the moment in the l.h.s. is determined on the basis of the experimental data, and that in the r.h.s. on the basis of the theoretical distribution, which depends on the parameter Mt . Let, for a given n, a solution to (3) be Mt = Mt(n) . Then the error of the solution is ?Mt(n) = ? q n

exp

d qn dMt

Mt =Mt(n)

.

(4)

Our aim is to estimate ?Mt(n) and ?nd an optimal value of n which would minimize ?Mt(n) . Since in view of (2) the derivative d q n /dMt is known, the problem is reduced to the estimation of the error ? q n exp or its components, the statistical and systematic experimental errors. The statistical error can be estimated, in almost model-independent way, by means of the formula [3] ?stat q n

exp

=

1 2n q . N

(5)

Here N is the representative sample of events, and q 2n is a theoretically calculated moment. For the consistent determination of the systematic error the application of a proper MC event generator is needed. The analysis [2] made with the aid of the PYTHIA and HERVIG events generators showed that the major uncertainties in the MJ/ψ ? distribution were caused by (i) the uncertainties in the b-quark fragmentation and (ii) the background processes. It is obvious that at solving the inverse problem, the reconstructing the Mb? distribution from the MJ/ψ ? one, the errors should be of the same origin. With this in mind we consider ?rst the error resulting from the uncertainty in the b-quark fragmentation. For brevity we call this the type I error. At the level of the Mb? distribution it appears as an uncertainty ?q in the q variable. Let us suppose that ?q is su?ciently small, and let us neglect nonlinear e?ects. Then we have ?sys I q n

exp

=

M 0

dq [q n F (q)]′ ?q .

(6)

3

Here the prime means the derivative with respect to q. The ?q is determined 1 by the relation ?q = rq with r is a coe?cient, r ? 2 ?Eb /Eb , where Eb is the energy of the b quark in the laboratory frame and ?Eb is its error. In fact the mentioned relation for ?q follows from the observation that q 2 actually is a doubled scalar product of 4-momenta of the b quark and of the lepton ? whose momentum is considered precisely determined. The systematic error arising after subtraction of the background processes— we call it the type II error—appears in the absolute value of the distribution function. Denote the corresponding error by δF , then we have ?sys II q n

exp

=

M 0

dq q n δF (q) .

(7)

The function δF (q) should be vanishing at the boundaries of the phase space. We also assume that δF only once changes the sign when q runs the values. The simplest function satisfying to these requirements is a polynomial, δF = h q(q?M/2)(q?M) with parameter h describing the amplitude of the error. Now we turn to numerical results. We assign the following values for the parameters with global meaning: MW = 80.4 GeV, ΓW = 2.1 GeV, Mt = M = 175 GeV. The parameters N, r, h take the values depending on the conditions of observation. Since in [2] the MJ/ψ distribution was determined at N = 4000 (with kinematic cuts and for 4 years of LHC operation), we set this value for N, as well. The parameters r and h may be ?xed by basing on the properties of the MJ/ψ ? distribution. We omit here the respective discussion and show only the result: r ? 0.005, h ? 1.7 × 10?10 GeV?4 [3]. Note that the same estimation for r follows from the above mentioned 1 formula r ? 2 ?Eb /Eb when taking into consideration the 1%-precision of the determination of the energy of the b jets expected at LHC [1]. Now, as we know the parameters, we can calculate the errors of the top mass. At Fig. 2(a) we show the behavior of ?stat Mt(n) versus degree n of the moments. Fig. 2(b) shows the behavior of the summed-quadrature systematic error ?sys Mt(n) . The dashed lines in both ?gures represent the errors obtained with usage of e?ective moments determined by way of introducing a cut-o? Λn instead of M in formulas (2), (6) and (7); the aim of introducing the cut-o? is to isolate the tail of q n F (q) with an unphysical peak arising at large n, see Fig. 1(b). Having the errors ?stat Mt(n) and ?sys Mt(n) , it is easily shown that near n = 15 the total error reaches the minimum ?Mt(15) = 0.48(0.42) GeV, with ?stat Mt(15) = 0.41(0.40) GeV and ?sys Mt(15) = 0.24(0.08) GeV. Here we put into the brackets the errors 4

0.012

0.008

170 GeV 175 GeV 180 GeV

( )

a

n=1

5

15 40

(b)

0.004

0.000 0 50 100 150 GeV] 100 125 150 GeV] 175

Figure 1: The distribution F (q) = Γ?1 dΓ/dq, q ≡ Mb? , at Mt = 170, 175, 180 GeV (a),

and the shape of q n F (q) at n = 1, 5, 15, 40 (Mt = 175 GeV), arbitrary normalization (b).

1.5 ( ) 1.0 1.0 1.5

a

(b)

0.5

0.5

0.0 10 20 30 40

t(n)

0.0 50 10 20 30 40 50

Figure 2: The

?stat M

(a) and

?sys M

t(n)

(b) in GeV depending on n.

calculated by the e?ective-moment method. It can be shown by the naive counting method [3] that the theoretical errors will not spoil the above accuracy provided that the Mb? distribution will be calculated within the one-loop electroweak and two-loop QCD precision. In summary, we discussed the potentialities inherent in the technique of moments and demonstrated that at the LHC the proceeding to moments of high degree allows one to reduce the total error of Mt to the level of about 500 MeV.

References

[1] M. Beneke et al. (conveners), A.Ahmadov et al., “Top quark physics”, CERN 2000-004 [hep-ph/0003033]. [2] A.Kharchilava, Phys.Lett B 476, 73 (2000). [3] M.L.Nekrasov, Eur.Phys.J. C 44, 233 (2005).

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