2-body decay diagram

Transcription

1 2-body decay diagram Figure 1: The two-body cascade decay: each decay in the cascade is a two-body decay. All of the particles are assumed to be onshell, including B and C. The expressions for the kinematical endpoints in terms of the particle masses contain m B. 3-body decay diagram Figure 2: The three-body decay chain: the first decay is a two-body decay, but the second and final decay is a three-body decay. All of the particles in the diagram are assumed to be onshell, including C, but notice that B does not appear in the diagram. The expressions for the kinematical endpoints in terms of the particle masses do not contain m B. Also notice that the leptons are not labeled with a subscript because it is no longer meaningful. Abstract We consider the decay chain D qc ql nearb ± ql nearl ± far A where B, C, and D are heavy, A is unobservable, q is observable as a massless jet, and l nearl ± far are observable as a pair of oppositesign-same-flavor massless leptons. 1 We showed in [1] that, while the endpoint method can provide a unique mass spectrum for A, B, C, and D in the most popular examples, such as SPS1a, there are some spectra that have not been considered which the endpoint method would not be able to identify unambiguously. In this paper we present a generalization of the endpoint method from single variable mass distributions to 2-variable mass distributions, and show that these 2- variable distributions might provide a resolution to the ambiguity in the mass determination. 1

2 final state particle combos Figure 3: The four visible particle combinations are circled: a) ll, b) qll, c) ql near, d) ql far. We assume that ql near and ql far cannot be distinguished. 1 Setup We consider a massive particle, D, that decays to a lighter massive particle, C, and a massless quark. C then decays to two massless opposite-signsame-flavor leptons, l ± l, and an unobserved particle, A, which we assume to contribute to the missing momentum in the event. If the particle, B, is lighter than C, then it is possible that the decay chain proceeds entirely via two-body decays (see 1). Otherwise, the decay chain is considered as a two-body decay followed by a three-body decay (see 2). 2 Combinations of the quark and two leptons can be formed, namely ll, qll, ql near, and ql far (see 3), and we consider the total 4-momentum squared of each combination, (m ll ) 2, (m qll ) 2, (m qlnear ) 2, and (m qlfar ) 2, respectively. l near refers to the lepton produced from the decay of C (and so nearer to the quark in the cascade diagram), and l far refers to the lepton produced from the decay of B (and so farther from the quark in the cascade diagram). However, whereas (m ll ) 2 and (m qll ) 2 are recorded directly, we assume that there is no way to distinguish l near and l far in principle, so we record the 4-momenta squared of the two single-lepton combinations as (m ql(high) ) 2 and (m ql(low) ) 2, where (m ql(high) ) 2 is the larger of (m qlnear ) 2 and (m qlfar ) 2 and (m ql(low) ) 2 is the smaller of the two.[?] These values are recorded for each event, and it was traditionally believed that the extreme kinematically allowed values (i.e. endpoints and threshholds ) could be used to determine m D, m C, m B (when B is produced onshell), and m A.[?],[?] However, we found that this is not always the case.[1] 1 This implies a slight model dependence in that D is expected to be in the color-triplet representation of SU(3) and B is expected to carry the same lepton number as l. More exotic models would relax these conditions at the expense of introducing more exotic properties such as lepto-quarks, or lepton number violation. 2 By two-body decay, we mean that the two decay products are strictly onshell. By three-body decay, we mean that, even though there may be an intermediate particle between two of the three decay products, this intermediate particle cannot be produced onshell due to kinematical constraints, i.e. its nominal mass is larger than that of the mother particle. 2

3 region plot Figure 4: The 4 relevant regions of parameter space. The three regions inside the square are the onshell regions. the region outside the square is the offshell region. The region boundaries are independent of R C and are relevant to the single lepton combinations. We organize parameter space by introducing, as in [?], three new variables: ( ) ma 2 ( ) mb 2 ( ) mc 2 R A R B R C m B Our analysis requires that m C m D 0 < R A < 1 0 < R B R A R B < 1 0 < R C < 1 thus allowing B to be heavier than C (and even heavier than D) but not lighter than A. A fourth parameter is needed to unambiguously represent the 4 unknown masses, and we use (m D ) 2, since it factors out of all of the expressions, which makes this choice of parametrization particularly convenient. In addition to the inequalities for R A, R B, and R C already required, the remaining available parameter space can be partitioned into the following 4 regions relevant to the single lepton invariant mass distributions. Region 1 2 R A > 1/R B R B < 1 Region 2 R A < R B 2 R A < 1/R B Region 3 R A > R B Region 4 R B > 1 The numbering of the region follows that found in [?], with the addition of Region 4 not considered in [?] but in, for example, [?]. As we will show, the shape of the (m ql(high) ) 2 vs. (m ql(low) ) 2 distribution can be used to identify in which of these four regions the parameter point occurs. 2 2-Variable Distributions We have chosen to pair (m qll ) 2 with (m ll ) 2 and (m ql(high) ) 2 with (m ql(low) ) 2 in our 2-variable distributions. We use squared mass rather than mass be- 3

4 table of four predicted distributions Figure 5: Four qualitative examples of (m qll ) 2 vs. (m ll ) 2 distributions: a) clipped, upward tilted onsell example, b) corresponding offshell example, c) corresponding R A = R B example, d) downward tilted R A = R B example. cause this makes the expressions simpler (and all predicted boundaries on the (m ql(high) ) 2 vs. (m ql(low) ) 2 are straight lines). 2.1 (m qll ) 2 vs. (m ll ) 2 As in [?], we consider the dilepton pair as a quasi-particle with mass m ll. The dilepton quasi-particle itself pairs with the quark to form the invariant mass m qll. If the dilepton were a true particle (i.e. if the dilepton had a constant mass), then the only variable in the invariant mass would be the cosine of the angle between the three-momenta of the dilepton and quark cos α. 3 So, for a given m ll, there is a vertical range on the (m qll ) 2 vs. (m ll ) 2 distribution that represents the variation in cos α. However, the dilepton mass can also vary, and this is of course represented by the horizontal axis of the distribution. If there is no constraint from the dilepton invariant mass, then the entire area between the curves defined by cos α = ±1 is populated. This occurs in the three-body decay scheme, and in this case (m max ll ) 2 is determined by the point at which these two curves meet (see, for example, 5 (b)). In the two-body decay scheme, the scatter plot is further constrained by the (m max ll ) 2, and the tip of the distribution is clipped off by a vertical line. The clipping is exactly vertical because the constraint is given directly in terms of (m ll ) 2, the horizontal variable. The lower curve, defined by cos α = +1, always passes through the origin. This is so because the invariant mass of a pair of massless particles travelling in the same direction always vanishes. The shape of the curved boundaries can be identified with a hyperbola in parameter space, R A R B = constant, where the constant determines the shape of the curved boundary. If constant < R C, then the top curved boundary has an initially positive slope (see, for example, 5 (a-c)). If constant > R C, then the top curved boundary has a negative slope (see, for example, 5 (d)). The location of the parameter point along the hyperbola determines how much of the 3 This is true only if A, C, and D are strictly onshell. 4

5 area inside the curved boundaries of the 2-variable distribution is clipped off. The closer the parameter point is to the line R A = R B, the less of the area is clipped off, with the exception that there is no vertical boundary in the offshell case. Three vertices can be identified on the (m qll ) 2 vs. (m ll ) 2 distribution, and we label them as: (m (1) qll )2, qll )2, and (m (3) qll )2. (m (1) qll )2 is the (m qll ) 2 intercept of the upper curved boundary line (m (1) qll )2 max((m qll ) 2 m ll. = 0) Regardless of the region of parameter space (m (1) qll )2 = (m D ) 2 (1 R C )(1 R A R B ) This can be derived by setting x =. 0 in the expression for y + in the appendix. qll )2 and (m (3) qll )2 are defined as the maximum and minimum values of (m qll ) 2 when (m ll ) 2 = (m max ll ) 2 qll )2 max((m qll ) 2 m ll (m (3) qll )2 min((m qll ) 2 m ll. = m max ll ). = m max ll ) The expressions for these two values depend on whether R A > R B or R A < R B, and whether R B < 1 or R B > 1. When R A > R B qll )2 = (m D ) 2 (1 R A R C )(1 R B ) (m (3) qll )2 = (m D ) 2 (1 R A )(1 R B R C ) When R A < R B < 1 qll )2 = (m D ) 2 (1 R A )(1 R B R C ) (m (3) qll )2 = (m D ) 2 (1 R A R C )(1 R B ) These can be derived by setting x =. (1 R A )(1 R B )R C in the expressions for y ± in the appendix. When R B > 1 qll )2 = (m (3) qll )2 = ( 1 R A R B ) (1 R C R A R B ) This can be derived by setting x =. ( 1 ) 2 R A R B RC in the expression for either y ± in the appendix. The appropriate case can be identified by examining the (m ql(high) ) 2 vs. (m ql(low) ) 2 distribution. 5

6 table of four predicted distributions Figure 6: Four qualitative examples of (m ql(high) ) 2 vs. (m ql(low) ) 2 distributions: a) corresponding to same point as in 5 (a) in Region 3, b) corresponding to same point as in 5 (a) in Region 1, c) corresponding to same hyperbola as in 5 (a) in Region 2, d) Region 4 example generic far vs. near distribution Figure 7: The trapezoidal shape of the (m qlfar ) 2 vs. (m qlnear ) 2 distribution. The right boundary is (m qlnear ) 2 = (m max ql near ) 2. The negatively sloped top boundary line is (m qlfar ) 2 = (m max ql far ) 2 given (m qlnear ) (m ql(high) ) 2 vs. (m ql(low) ) 2 The 2-variable distributions of (m ql(high) ) 2 vs. (m ql(low) ) 2 are formed by assigning the smaller (m ql ) 2 value to the horizontal coordinate (m ql(low) ) 2 and the larger (m ql ) 2 value to the vertical coordinate (m ql(high) ) 2 for each event. These distributions exhibit features that allow the region of parameter space to be identified. They also contain information needed to extract the values of the masses m A, m B, m C, and m D. The boundary lines of these distributions can be understood by first considering the unobservable 2-variable distribution of (m qlfar ) 2 vs. (m qlnear ) 2 and then making a simple transformation to (m ql(high) ) 2 vs. (m ql(low) ) 2. The (m qlfar ) 2 vs. (m qlnear ) 2 distribution is bounded inside a trapezoidshaped area. The bottom boundary represents the minimum (m qlfar ) 2 = 0. The left boundary represents the minimum (m qlnear ) 2 = 0. The right boundary represents the maximum (m qlnear ) 2 = (m max ql near ) 2. The top boundary represents the conditional maximum (m qlfar ) 2 = (m max ql far ) 2 given (m qlnear ) 2 and has a negative slope = (1 R A ). The ratio of the (m qlfar ) 2 value at the top-right vertex to the (m qlfar ) 2 at the top-left vertex is R B. (See 7.) If the line (m qlfar ) 2 = (m qlnear ) 2 passes through the negatively sloped top boundary, and (m max ql far ) 2 < (m max ql near ) 2, then the parameter point is in Region 3. If the line (m qlfar ) 2 = (m qlnear ) 2 passes through the negatively sloped top boundary, and (m max ql far ) 2 > (m max ql near ) 2, then the parameter point is in Region 2. If the line (m qlfar ) 2 = (m qlnear ) 2 passes through the vertical 6

7 three folding steps Figure 8: a) (m qlfar ) 2 vs. (m qlnear ) 2, b) folding across (m qlfar ) 2 = (m qlnear ) 2, c) (m ql(high) ) 2 vs. (m ql(low) ) 2 three images of onshell high vs. low with points labeled Figure 9: The negatively sloped boundary line intercepts the (m ql(high) ) 2 axis at (m (1) ql )2 and obtains its maximum (m ql(low) ) 2 value at (m max ql(low) )2. The horizontal boundary line intercepts the (m ql(high) ) 2 axis at ql )2. The boundary line kink occurs at (m (3) ql )2 right boundary, then (m max ql far ) 2 > (m max ql near ) 2, and the parameter point is in Region 1. Everything below the line (m qlfar ) 2 = (m qlnear ) 2 is folded across it into the area above it. Then, the horizontal axis becomes (m ql(low) ) 2 and the vertical axis becomes (m ql(high) ) 2, by definition. (See 8.) There are a few boundary lines that appear on the resulting (m ql(high) ) 2 vs. (m ql(low) ) 2 distribution, and we have identified four points. There are of course (m max ql(low) )2 and (m max ql(high) )2. Expressions for these values in terms of the masses have already been derived (see, for example, [?]). (m max ql(high) )2 occurs at the (m ql(high) ) 2 intercept of a boundary line, and since there are two boundary lines that intercept the (m ql(high) ) 2 axis, we identify two values: (m (1) ql )2 and ql )2. One of these values will be (m max ql(high) )2, but the other value is new. We define (m (1) ql )2 to be the (m ql(high) ) 2 intercept of the negatively sloped boundary line, and we define ql )2 to be the (m ql(high) ) 2 intercept of the horizontal boundary line. Another point, (m (3) ql )2, occurs at a kink at the endpoint of either the horizontal or the vertical boundary line segment 4 and one of the negatively sloped boundary line segments. And finally, we of course have (m (max) ql(low) )2. Throughout the onshell Region of parameter space (m (1) ql )2 = (m D ) 2 (1 R A )(1 R C ) ql )2 = (m D ) 2 (1 R B )(1 R C ) 4 In Region 1 the line is vertical; in Regions (2) and (3) the line is horizontal. 7

8 (m (3) ql )2 = (m D ) 2 (1 R A )R B (1 R C ) { (md ) 2 (1 R (m max ql(low)) 2 B )(1 R C ) Region 1 = (m D ) 2 (1 R A ) (1 R (2 R A ) C) Regions 2 and 3 These values are derived in the appendix. 3 Obtaining the Masses Now that we have described the observable distributions and identified some key features, they can be used to obtain the mass spectrum. Firstly, note that, whereas an endpoint analysis requires all of the different inversion formulae for all of the different regions of parameter space to be tested for consistency with the corresponding conditions on the parameters, the (m ql(high) ) 2 vs. (m ql(low) ) 2 distribution immediately reveals the correct region of parameter space, 5 so trial-and-error is unnecessary. Also, the number of regions is reduced from 13 down to 4. Secondly, note that some of the information that is only conditionally available in an endpoint analysis is always available when using the 2-variable distributions. However, along with these benefits come more experimental complications in identifying the features on the scatter plots, especially in terms of statistical significance. This issue will be addressed in a future paper. In the following treatment, the parameters R A, R B, and R C are given in terms of ratios of the available values on the 2-variable distributions, and then m D is determined as the scale factor. Since there are many more observables than there are parameters, we have many different choices for expressing the parameters in terms of the observables. In fact, although relatively simple analytical expressions can be derived, it may be better to perform a numerical fit to make full use of all of the information and reduce the error. In this section we present one possible analytical solution for each of the four shapes of (m ql(high) ) 2 vs. (m ql(low) ) 2. 8

9 Region 3 high vs. low and generic qll vs. ll, with labels Figure 10: If the (m ql(high) ) 2 vs. (m ql(low) ) 2 distribution is bounded by a quadrilateral that exhibits a horizontal top-left boundary connecting to a negtively sloped top-right boundary, as shown here, then the parameter point is in Region Region 3 Since ql )2 may be difficult to observe, 6 we ignore it. There remain seven observable values on the 2-variable distributions (see 10). We express the ratio parameters in terms of four of these values: (m (1) ql )2, (m (3) ql )2, (m max and (m max ll ) 2. R C = R A = 2 (m(1) ql )2 (m max ql(low) )2 R B = (m(3) ql )2 (m (1) ql )2 (m max ll ) 2 (m max ll ) 2 + (m (1) ql )2 (m (3) ql )2 ql(low) )2, The scale factor, (m D ) 2, can now be easily extracted from any of the expressions by simply inserting the above three values for the ratio parameters. However, for completeness, we have derived one possible analytical expression for the scale factor directly in terms of observable values. (m D ) 2 = (m (1) qll )2 (m (1) ql )2 (m max ql(low) )2( (m max ll ) 2 + (m (1) ql )2 (m (3) ql ( )2) (1) (m ql )2 (m (3) )( ql )2 (1) (m ql )2 (m (3) ql )2 + (m (1) ql )2 (m max ql(low) )2 2(m (3) ) ql )2 (m max ql(low) )2 5 There are, in fact, some potentially problematic points, e.g. R A = 0. However, we claim that, in such cases, the signal would not pass any reasonable experimental cuts. So, if we consider this analysis in the ideal case, then these points are not really problematic, and if we consider this analysis in a realistic scenario including reasonable experimental cuts, then these points are irrelevant. 6 We have seen that the internal boundary line is difficult to discern in some PYTHIA simulations. 9

10 Region 2 high vs. low and generic qll vs. ll, with labels Figure 11: If the (m ql(high) ) 2 vs. (m ql(low) ) 2 distribution is bounded by an irregular pentagon that exhibits a negatively sloped top-left boundary connecting to a horizontal top-central boundary and then another negatively sloped top-right boundary, as shown here, then the parameter point is in Region 2. Region 1 high vs. low and generic qll vs. ll, with labels Figure 12: If the (m ql(high) ) 2 vs. (m ql(low) ) 2 distribution is bounded by a quadrilateral that exhibits a negatively sloped top boundary connecting to a vertical right boundary, as shown here, then the parameter point is in Region 1. Recall that the other three masses are now given by m C = m D R C m B = m D R B R C m A = m D R A R B R C 3.2 Region Region 1 similarly to Region 3 similarly to Region Region 4 I haven t thought about this one yet. Region 4 high vs. low and generic qll vs. ll, with labels Figure 13: If the (m ql(high) ) 2 vs. (m ql(low) ) 2 distribution is bounded by a triangle that exhibits no internal structure, as shown here, then the parameter point is in Region 4. 10

11 A Derivation of (m qll ) 2 vs. (m ll ) 2 Boundaries In any reference frame, In the rest from of C, (m qll ) 2 = (p q + p ll ) 2 = (p q ) 2 + 2p q p ll + (p ll ) 2 = 0 + 2(E q E ll p q p ll ) + (m ll ) 2 p q. = E q = (m D) 2 (m C ) 2 2m C E ll = (m C) 2 (m A ) 2 + (m ll ) 2 2m C ( p ll = (E ll ) 2 (m ll ) 2 = (mc ) 2 (m A ) 2 + (m ll ) ) 2 2 (m ll) 2m 2 C The only frame dependent quantity that remains is cos α, the cosine of the angle between p q and p ll. Since we only need the boundaries, then we only need the maximum and minimum of this quantity. The near lepton has no preferred direction, and the far lepton can always travel in the same direction as the near lepton, so the full polar range is available to the dilepton pair, and both cos α = ±1 are allowed. Define x (m ll) 2 (m D, and define y ± (m qll) 2 ) 2 (m D ) 2 as a function of x, given cos α =. 1. The curved boundaries can then be expressed in terms of y as a function of x. y ± = ( 1 (1 R 2 AR B )(1 R C ) ) + ( ) 1+R C 2R C x ± ( ) ((1 ) 1 R 2 ( ) C 2R C RA R B )R C 2(1 + RA R B )R C x + x 2 B Derivation of (m ql(high) ) 2 vs. (m ql(low) ) 2 Boundaries Show derivation for high vs. low boundaries. 11

12 References [1] K. Matchev, M. Park, PAPER #1. [2] D. J. Miller, P. Osland, A. R. Raklev, Invariant mass distributions in cascade decays, arxiv:hep-ph/ v1 27 Oct [3] B. K. Gjelsten, D. J. Miller, P. Osland, Measurement of SUSY Masses via Cascade Decays for SPS 1a, arxiv:hep-ph/ v2 7 Jan [4] Christopher G. Lester, Michael Andy Parker and Martin J. White, Three body kinematic endpoints in SUSY models with non-universal Higgs masses, JHEP10(2007)051. [5] Christopher G. Lester, Constrained invariant mass distributions in cascade decays. The shape of the m qll -threshhold and similar distributions, Phys. Lett. B 655 (2007) [6] PYTHIA 12