Background dependence of dimuon asymmetry in $\bar p p$ interactions at $\sqrt{s} = 1.96$ TeV

The D0 Collaboration has reported an anomalous charge asymmetry in the production of same-sign muon pairs at the Fermilab Tevatron. The magnitude of this effect depends on the subtraction of several backgrounds, the most notable of which is due to kaons being misidentified as muons either through decays in flight or punch-through. The present authors suggested a check on such backgrounds consisting of a tight restriction on the muon impact parameter $b$, to confirm that this excess was indeed due to $B_{(s)}$ meson decays. The D0 Collaboration has performed a related check applying transverse impact parameter (IP) restrictions, whose implications are discussed. We study background asymmetry predictions for events involving two muons with IP bounds which are complementary to each other. These predictions may be used in future measurements of the net charge asymmetry from $B_{(s)}$ decays.


I Introduction
Last year the D0 Collaboration reported a charge asymmetry of about 40 times the standard model value in the production of same-sign muon pairs at the Fermilab Tevatron [1,2]. The magnitude of the effect depended on the subtraction of several backgrounds, notably muons due to misidentified or decaying kaons. The cross section for K − on matter is greater than that for K + on matter, so that K + decays in flight pose a greater source of muons than K − decays in flight. We suggested that a tight restriction on the muon impact parameter b could test whether this asymmetry was indeed due to B meson decays [3]. In an updated version of their analysis, employing a larger data sample, the D0 Collaboration has implemented this suggestion, confirming their claim for a larger-than-predicted charge asymmetry and providing partial separation of B 0 and B s contributions to the charge asymmetry [4]. The present paper is devoted to a discussion of their results and their relation to our suggestion.
In Section II we use numbers of dimuon events corresponding to different choices of transverse impact parameter (IP) [4] to estimate the effective value of a related parameter b > IP. In Section III we show that for maximum values of IP < (50, 80, 120) µm, scaling the kaon background by a factor of ≃2/3 leads to a vanishing signal for the dimuon charge asymmetry, while the net inclusive muon charge asymmetry is shifted away from zero.
For minimum values of IP > (50, 80, 120) µm, with much smaller kaon background, rescaling of that background does not alter the dimuon and inclusive muon asymmetries significantly. Section IV discusses predictions of asymmetries for two muons involving a minimum IP for one and a maximum IP for the other. We conclude in Section V.

II IMPACT PARAMETERS IP AND b
In their latest study of the like-sign dimuon charge asymmetry at the Tevatron [4], the D0 Collaboration investigated the dependence of this asymmetry on the transverse impact parameter IP, a quantity equivalent to what we called b ⊥ [3]. We advocated excluding events with an impact parameter b greater than a chosen value b 0 . If the average impact parameter in b quark decays is b , the remaining fraction of dimuons from pairs of hadrons containing b quarks will be [ Based on a study by the CDF Collaboration [5], we estimated b = 350 µm. For b 0 = 100 µm, a value we advocated in Ref. [3], this quantity is then 0.062. Such a sample, thus, should be highly depleted of dimuons from b decays.
Requiring events to involve dimuons with a certain maximum IP (≡ b ⊥ ) is less stringent than requiring them both to have the same maximum value of b, since b > IP. The two are related by where b and ψ were defined in Ref. [3]. It is difficult to impose stringent requirements on b [6,7], which is why the D0 Collaboration chose instead to restrict the parameter IP. With their most stringent constraint, we find their sample of dimuons from b decays to be reduced by a factor of 6. The D0 Collaboration studies samples with both muons having IP either less than or greater than the values 50, 80, and 120 µm. We have been provided [6] with the effect these cuts have on sample sizes. From this information we are able to extract effective values of b 0 which indeed exceed IP, by factors of 2.7 to 3. 6. In what follows we present details of the calculation.
The numbers of dimuon events both of whose muons have IP above 50, 80, and 120 µm are shown in Table I, while the numbers of events with IP less than 50, 80, and 120 µm are shown in Table II. The number of dimuon events due to b decays is where the quantities F SS are given in Tables XXI and XXII of Ref. [4], while R = 94% [6] is the fraction of the "SS" sample coming from b hadron decays. Here we use the superscripts >> and << to indicate that both muons have IP either above or below the given value. The subscript "SS" [1,2,4] refers to muons both arising from decays of particles at short distances from the interaction point. Muons produced by particles traveling long distances before decaying in the detector are labeled "L".
The total sample of dimuons due to b hadron decays is N µµ,b = N << µµ,b + N >> µµ,b + N <> µµ,b , where the last term is the contribution from events in which one muon has IP greater than the indicated value and the other has IP less than the indicated value. We can calculate this term from the first two: We should get the same value of N µµ,b for each value of IP; the values we obtain from the data provided to us are (3.81±0.17, 3.94±0.16, 3.94±0.18) ×10 6 events for IP = (50,80,120) µm. These are within statistical errors of one another.
The effective values of b 0 may now be calculated from the fractions of dimuon events due to b hadron decays with IP greater than a given amount: For the most stringent constraint, taking events with IP < 50 µm for both muons, the D0 collaboration is left with N << µµ,b /N µµ,b = 0.164 ± 0.015, or a sample of about 1/6 the size of that employed without imposing bounds on IP values. We note that while the asymmetry derived in the latter case, A b sl = −0.787 ± 0.172 ± 0.093 is 3.9σ away from zero [4], a considerably larger asymmetry obtained for IP < 50 µm, A b sl = Table III: Contributions of background sources to charge asymmetry in like-sign dimuon sample for nominal analysis of Ref. [4].
.694, is nonzero at 2.9σ. The reason for the larger measured asymmetry for muons with low IP may be that it is driven more by B s decays than by B 0 decays [3].

III RESCALING THE KAON BACKGROUND
In this Section we examine whether rescaling the kaon charge asymmetry background from the value determined in Ref. [4] can lead to a vanishing charge asymmetry for same-sign muon pairs originating from B (s) decays. We will also discuss the implication of such rescaling on the inclusive muon charge asymmetry.
Using the notations of Ref. [4], the measured like-sign dimuon asymmetry A is related to the charge asymmetry from B (s) decays A b sl through Here F SS C b + F SL c b is an effective dilution factor depending on fractions of dimuon events with two short (S) muons and with one short and one long (L) muon, while F x and A x (x = K, π, p) are fractions and asymmetries of muons produced by kaons, pions and protons, respectively. The term represents a contribution from muon track reconstruction asymmetry. We begin by quoting in Table III some relevant entries from Table XII of Ref. [4] for like-sign dimuon events, which are the main source of statistical weight supporting the claim for a charge asymmetry A b sl . In Table III the dominant contribution to A bkg is from the charge asymmetry in kaon tracks. The final result for the dimuon charge asymmetry is given by (A−A bkg ) divided by an effective dilution factor, Thus a reduction of the kaon background by a factor of 0.56 to 0.356 would be required to achieve a vanishing value of A − A bkg or of A b sl .    Ref. [4] considered a set of variations on their nominal analysis entailing the restrictions IP < (50, 80, 120) µm and IP > (50, 80, 120) µm. We are grateful to the D0 Collaboration for sharing the respective entries in Table III corresponding to each of these criteria [8]. The results shown in Tables IV and V were obtained by summing products such as F K A K over muon p T bins.
For comparison, we quote in Tables VI and VII the dominant contributions F K A K and (2 − F bkg )∆ using averaged values of A K and F x from Tables VII and XXI, XXII in Ref. [4], respectively. We note the insignificant differences between values of background asymmetries in Tables IV and VI and between those in Table V and VII. Thus, in the next section, where we discuss charge asymmetries in dimuon events with other IP constraints, we will use averaged values of fractions and asymmetries rather than summing their products over muon p T bins.
If we wish to rescale F K A K by a common factor λ, we may parametrize with the values of Table  Table VI: Dominant contributions of background sources to charge asymmetry in likesign dimuon sample with the restriction IP < (50, 80, 120) µm. We use average values of A K and F x .

VIII.
The effect of rescaling the contribution of F K A K may be seen in Figs. 1 -4. For no constraint on IP and for maximum values of IP < (50, 80, 120) µm, a common choice of λ ≃ 2/3 shifts derived values of A − A bkg to within 1σ of zero. With the exception of IP > 120 µm, it is always possible to choose a value of λ > 0.66 such that A − A bg is within 2σ of zero. However, for IP > 120 µm, with λ = 1 the value of A−A bkg = −0.402±0.152 is 2.6σ from zero, and quite insensitive to rescaling of F K A K .
Effects of rescaling the kaon asymmetry background may also be studied in the inclusive muon asymmetry a, which is less sensitive than A to a nonzero value of A b sl .  (7) for rescaling contribution of kaon background. The nominal parameters [4] involve only a very weak restriction on impact parameter (IP).
+0.356 ± 0.059 0.633 ± 0.031 IP < 50 µm +0.893 ± 0.108 1.421 ± 0.066 IP < 80 µm +0.892 ± 0.081 1.203 ± 0.053 IP < 120 µm +0.773 ± 0.082 1.047 ± 0.051 IP > 50 µm −0.054 ± 0.087 0.205 ± 0.060 IP > 80 µm −0.144 ± 0.101 0.104 ± 0.074 IP > 120 µm −0.289 ± 0.126 0.113 ± 0.087    The measured values of a − a bkg for all IP constraints are consistent with zero [4,8]. This is expected because the contribution of A b sl to this difference is proportional to c b which is much smaller than C b . (See Eq. (5) above and Eq. (7) in Ref. [4] with values of C b and c b given in this reference.) We checked that rescaling the kaon background contribution to a by λ ≃ 2/3 leaves values of a − a bkg consistent with zero within 1σ for IP > 80, 120 µm and within less than 3σ for IP > 50 µm. However, for IP < 50, 80, 120 µm this scaling moves a − a bkg significantly (6σ) away from zero. Thus we conclude that a solution A b sl = 0 cannot be obtained consistently for all IP constraints by introducing a common rescaling factor for the background kaon asymmetry.
So far D0 has analyzed dimuon events in which the two muons have IP either less or greater than the values 50, 80 and 120 µm. Other events, which have not yet been studied in the D0 sample, involve one muon with IP less than 50, 80 or 120 µm and a second muon having IP greater than the same value. The numbers of events in the three classes, N << µµ , N >> µµ and N <> µµ , are listed in Table IX for the three IP values. This Table  also quotes the calculated total number of dimuon events N total µµ for the three IP values. The total numbers are in reasonable agreement with each other. Their deviation by 7% relative to the number 6.019 × 10 6 quoted in Ref. [4] for events with no IP constraint may be due to a second order effect [6]. Noting that about half of the total like-sign dimuon events have not yet been studied experimentally (N <> µµ /N total µµ ∼ 1/2), we wish to discuss their expected charge asymmetries. For this discussion we need the fraction of background events F <> x (x = K, π, p). Using one has Thus, fraction of background events from kaons, pions and protons in the class <> under consideration may be calculated from corresponding fractions given in Tables XXI and  XXII of Ref. [4] for the two classes << and >>. These fractions are listed in Table X for IP= 50, 80, 120 µm. The three calculated fractions of kaons for events of the class <> are equal within 1σ to the fraction measured by D0 for events with no restrictions on IP [4] , F K × 10 2 = 13.78 ± 0.38. We note that kaon fractions F << K are about twice larger than the fractions F <> K , while F >> K are about twice smaller than these fractions. × 10 −2 0.29 ± 0.10 0.14 ± 0.10 0.13 ± 0.11 Results of asymmetries for background sources are given in Table XI. The calculated background asymmetries, for samples involving one muon with IP larger than 50, 80 or 120 µm and a second muon with IP smaller than the same value, should be compared with corresponding future results by D0. They may be used to extract the net charge asymmetry for B (s) decays A b sl from the raw asymmetry A.

V CONCLUSIONS
We have examined the relation between the impact parameter (IP) selections performed in Ref. [4] and those we suggested in Ref. [3]. We find that the minimum transverse IP of (50, 80, 120) µm considered in Ref. [4] is equivalent to a value of the parameter b 0 which exceeds IP by factors of 2.7 to 3.6. For the most stringent criterion, taking events with IP < 50 µm for both muons the sample of dimuons from B (s) decays is reduced to about 1/6 the size of the sample involving no bounds on IP. Fractions of backgrounds from kaons and pions in the former sample are each about twice as large as in the latter sample. In spite of the considerably smaller sample of signal events with IP < 50 µm and the larger background, the statistical significance of A − A bkg = (−0.527 ± 0.127)% measured for IP < 50 µm is 4.1σ, the same as measured for dimuons with no IP constraint, Tables III and IV.) We asked whether a rescaling of the parameter F K A K describing kaon background asymmetry could lead to annulment of the claimed charge asymmetry for same-sign muon pairs from B (s) decays. For selections of maximum impact parameters IP < (50, 80, 120) µm, a rescaling by a factor of λ ≃ 2/3 led to reduction of the net charge asymmetry to within 1σ of zero. However, for minimum impact parameters IP > (50, 80, 120) µm, with greatly reduced kaon backgrounds, this was not so, and for IP > 120 µm, no positive choice of λ reduced the net asymmetry to less than 2σ from zero. In contrast, while introducing a rescaling factor λ ≃ 2/3 in inclusive muon samples is consistent with A b sl = 0 for IP > 50, 80, 120 µm, it leads to a nonzero asymmetry for IP < 50, 80, 120 µm.
We calculated background asymmetries for dimuon samples in which one muon has a maximum IP while the other has a minimum IP, for the three cases IP = 50, 80, 120 µm. These calculated asymmetries, expected to be confirmed in future studies by D0, may be used for measuring A b sl in these samples. The fraction of dimuons from kaons was seen to decrease by imposing a minimim value for the impact parameter IP. For instance, the background asymmetry from kaons was reduced from F K A K = (0.633 ± 0.031)% with no IP restriction to F K A K = (0.205 ± 0.060)% for IP > 50 µm. The corresponding number of like-sign dimuon events decreased by about a factor 1/4 (see Table IX), and the significance of a nonzero A − A bkg went down from 4.1σ to 2.4σ (see Table V). In future studies of the same-sign dimuon charge asymmetry we thus advocate emphasis on reduction of kaon background by choosing a minimum value of impact parameter, even if at the cost of statistics. Such studies could, in principle, be performed by other collaborations such as CDF at the Fermilab Tevatron and LHCb at the CERN Large Hadron Collider.