Notes on Phase Space Fall 2007, Physics 233B, Hitoshi Murayama

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1 Note on Phae Space Fall 007, Phyic 33B, Hitohi Murayama Two-Body Phae Space The two-body phae i the bai of computing higher body phae pace. We compute it in the ret frame of the two-body ytem, P p + p (, 0, 0, 0. dφ (p, p d 3 p (π 3 E d 3 p (π 3 E (π 4 δ(p µ p µ p µ d 3 p (π 3 E d 3 p (π 3 E (π 4 δ( E E δ( p + p d 3 p (πδ (π 3 m + p m + p p dp d co θ dφ (πδ (π 3 m + p m + p ( m + p m + p. ( m + p m + p. ( It take ome work to olve the delta function. The end reult i that p β ( with β (m + m Then the delta function become + (m m. (3 ( δ m + p m + p δ(p β / (p/e + (p/e. (4 Therefore, dφ (p, p d co θ dφ p π (π 3 4E E (p/e + (p/e p β /,Ei m i +p

2 β d co θ dφ (π d co θ p 4(E + E p β /,Ei m i +p dφ π. (5 It i alo ueful to remember ( E + m m, (6 ( E + m m. (7 It i worthwhile looking at two pecial cae. One i when two mae are equal m m m. Then β 4m m E. (8 Thi i nothing but β v/c; hence the notation. However, for m m, two particle have different velocitie and β cannot be interpreted a the velocity either. The other i when one of the mae vanihe, m 0. Then Thi i nothing but E / becaue E p p. β m. (9 Decompoing Phae Space into Two-Body One It i often ueful to decompoe multi-body phae pace integral into a product of two-body phae pace integral. A an example, let u decompoe a fourbody phae pace into a produce of three two-body phae pace. Thi i very ueful if one conider a production of two particle, each of which ubequently decay into two-body tate. The full four-body phae pace i given by dφ 4 4 i d 3 p i (π 3 E i (π 4 δ(p p p p 3 p 4. (0

3 We firt inert two four-momentum integral with q p + p and q 34 p 3 + p 4, d 4 q d 4 q 34 (π 4 (π 4 (π4 δ(q p p θ(q(π 0 4 δ(q 34 p 3 p 4 θ(q34. 0 ( The tep function θ(x i for x > 0 and 0 for x < 0. Even though the tep function eliminate a half of the integral over q 0 and q34, 0 the delta function enure that they are given by the um of two energie and hence it upport i in the half that i retained. In addition, we alo inert d π d 34 π πδ( q πδ( 34 q 34. ( With the delta function, we can regard a ma quared of the particle whoe four-momentum i q 34. µ Then one can perform the energy integral and find d 4 q d (π 4 π (π4 δ(q p p θ(qπδ( 0 q d π d 3 q (π 3 E (π 4 δ(q p p. (3 Here, we ued the delta function q ( + q E 0 and eliminated E q 0 with the condition by the tep function q 0 > 0. In the lat expreion, it i undertod that the four-dimenional delta function contain q q. We do the ame with q 34 a well. Putting all of the above together, we find dφ 4 4 d 3 p i d d 3 q d 34 d 3 q 34 (π 3 E i π (π 3 E π (π 3 E 34 i (π 4 δ(q p p (π 4 δ(q 34 p 3 p 4 (π 4 δ(p q q 34. (4 Now, note that the integration volume d 3 p i /(π 3 E i i Lorentz invariant. Therefore, we can carry out thee integral in any frame we wih. In particular, we take the ret frame of q to perform p and p integral. In thi frame, we denote the momenta a ˆp,, and we know that dφ (ˆp, ˆp β d 3ˆp d 3ˆp (π 3 Ê (π 3 Ê d co ˆθ (π 4 δ( ˆ E Ê Êδ(ˆ p + ˆ p d ˆφ π, (5 3

4 with β (m + m + (m m. (6 The ame i true with the p 3 and p 4 integral, where the ret frame of q 34 can be choen. (Thi different reference frame i alo denoted with the ame hatted variable. Any hatted variable that refer to and are in the ret frame of q, while thoe that refer to 3 and 4 are in the ret frame of q 34. Therefore, the whole four-body phae pace i now reduced to dφ 4 β d π d 3 q d 34 (π 3 E π β 34 d 3 q 34 (π 3 E 34 (π 4 δ(p q q 34 d co ˆθ d ˆφ d co ˆθ34 d ˆφ 34 π π. (7 On the other hand, the integral over q and q 34 can be done a if each of them i a particle of ma and 34, and are done in the ret frame of P (which i the lab frame for a ymmetric collider and we do not put a hat on the variable in thi frame. Therefore, d 3 q d 3 q 34 dφ (q, q 34 (π 4 δ(p q (π 3 E (π 3 q 34 E 34 β d co θ dφ π, (8 where β ( + 34 Putting everything together, we find d d 34 β d co θ dφ dφ 4 π π π d π β + ( 34. (9 d co ˆθ d ˆφ π β 34 d co ˆθ34 d ˆφ 34 π d 34 π dφ (q, q 34 dφ (ˆp, ˆp dφ (ˆp 3, ˆp 4. (0 3 Narrow-Width Approximation The narrow-width approximation ue the fact that the amplitude through a pole of untable particle decouple a M M(i a + bm(a + M(b h a,h b ( m a + im a Γ a ( 34 m b + im. ( bγ b 4

5 For an untable calar, it i obviou. For an untable fermion, we ue the fact that p + m h±/ u h (pū h (p if the four-momentum i on-hell p m and ha poitive energy p 0 > 0. If it i one-hell with the negative energy p 0 < 0, p + m h±/ v h ( p v h ( p. Of coure, the four-momentum i not exactly one-hell, but we ignore the off-hellne in the numerator. Then one of the pinor i regarded a a part of the production amplitude, the other of the decay amplitude. The ame idea applie to vector boon, where we ue g µν + qµ q ν m h±,0 ɛ µ h (qɛν h (q. When we integrate the quared amplitude on the four-body phae pace, we ue the two-body decompoition (o far not ummed over initial and final tate helicitie σ β dφ 4 M i d d 34 β i π π dφ (q, q 34 dφ (ˆp, ˆp dφ (ˆp 3, ˆp 4 M(i a + bm(a + M(b h a,h b ( m a + im a Γ a ( 34 m b + im bγ b d d 34 β i π π dφ (q, q 34 dφ (ˆp, ˆp dφ (ˆp 3, ˆp 4 h a,h b M(i a + bm(a + M(b β i [( m a + m aγ a][( 34 m b + m b Γ b ] dφ (q, q 34 dφ (ˆp, ˆp dφ (ˆp 3, ˆp 4 m a Γ a m b Γ b M(i a + bm(a + M(b h a,h b ( In the lat tep, we aumed that the amplitude depend very little on, 34 within their width. When the detailed ditribution are not of interet, the interference term among different helicitie of the decaying particle ocillate over the phae pace and drop out. Therefore, the total cro ection reduce to σ β dφ (q, q 34 dφ (ˆp, ˆp dφ (ˆp 3, ˆp 4 i 5

6 M(i a + bm(a + M(b m a Γ a m b Γ b h a,h b β dφ (q, q 34 M(i a + b B(a + B(b i h a,h b (3 In the lat tep, we ued the definition of the partial width Γ(a + dφ (ˆp, ˆp M(a + (4 m a and the branching fraction B(a + 4 Three-Body Phae Space Γ(a + Γ a. (5 Three-body decay are often important. Example include µ e ν e ν µ, ω 3π, H W W. In thee example, no two-particle combination hit a pole of an untable particle. Nonethele the phae pace can be worked out the ame way we did earlier. Uing the two-body phae pace decompoition jut once, we find dφ 3 d3 π dφ (p, q 3 dφ (p, p 3 d3 π d co θ dφ π β ( m, 3 d co ˆθ 3 d ˆφ 3 π β 3 ( m 3, m 3 3.(6 Often we are intereted in a decay of unpolarized particle (pin averaged, and the ditribution i iotropic. Then the overall rotation of the ytem (co θ, φ integral can be dropped. It i convenient to define the polar angle ˆθ 3 relative to the direction p. Then the azimuthal dependence on ˆφ 3 i alo trivial a it correpond to the overall rotation of the ytem around the axi p. The variable 3 and co ˆθ 3 are often rewritten with the energy fraction x,,3. (Or, more traditional variable are m and m 3 3 called Dalitz variable. Firt, x E + m / 3. (7 6

7 Second, uing the aumed iotropy, we can make q 3 point along the poitive z-direction, p ( + m 3, 0, 0, β, (8 q 3 ( m + 3, 0, 0, β. (9 Therefore, the boot factor from the ret frame of q 3 to the center-ofmomentum frame i given by γ E ( 3 3 m 3 + 3, (30 γβ β. (3 3 The four-momentum of the particle in the ret frame of q 3 i ( ˆp 3 + m 3 m 3 3, β 3 in ˆθ 3, 0, β 3 co ˆθ 3. (3 The energy of the particle in the center-of-momentum frame i then [ ( ] + γβ β 3 co ˆθ 3 and hence E x 3 γ + m 3 m 3 3, (33 ( ] 3 [γ + m m 3 + γβ 3 β 3 co ˆθ 3. (34 3 Let u pecifically tudy the cae of male particle m,,3 0. Performing all the integral for overall rotation, we have dφ 3 d3 π ( The energie of the particle and are 3 d co ˆθ3. (35 x 3, (36 [ ( 3 x + ( ] 3 co 3 ˆθ 3 [( + ( co ˆθ ] 3 ( x + x co ˆθ 3. (37 7

8 Therefore, the three-body phae pace become imply dφ 3 dx 3 dx. (38 0 x Obviouly, x + x + x 3 and the phae pace i a triangle on thi plane in the three-dimenional pace. Note that the matrix element M in general depend on x i in a nontrivial way. The expreion there i only the phae pace. For intance, in the decay H W W W lν l, the virtual W propagator prefer m lνl to be a cloe a poible to m W, giving a non-trivial x W dependence. With finite mae, the edge and vertice of the triangle get rounded. For detail, there i a PDG review article. 5 Decaying Particle in the Lab Frame It i important to undertand the kinematic of the decay product of a particle of ma M decaying in flight. For a iotropic two-body decay, we tart with the imple two-body phae pace. To bring it to the lab frame, we boot the four-momentum vector from the ret frame p M ( + m M m M, β in ˆθ, 0, β co ˆθ. (39 Here, we ued our liberty to chooe the origin of the azimuth uch that the y-component of the vector vanihe. To go to the lab frame, we boot it with γ E M, β γ. (40 The energy of the particle in the lab frame i then [ ( E M γ + m M m M + γβ βco ˆθ ]. (4 The important point here i the linear relationhip between E and co ˆθ. In particular, for the iotropic decay, the ditribution i flat in co ˆθ, and therefore we obtain a flat ditribution in E for the range ( ( + m M m M β β E E 8 + m M m M + β β. (4

9 In particular for the male cae m m 0, it i particularly imple, β E E + β. (43 It i ueful to define the energy fraction x E /E, and we find a flat ditribution in β x +β. For an iotropic three-body decay into male particle, we focu on the particle (ay electron in the decay τ e ν e ν τ whoe phae pace i dφ 3 M d co dˆx ˆθ ˆx 3 0 The four-momentum of the particle in the ret frame i In the lab frame, it energy i. (44 ˆp M ˆx (, in ˆθ, 0, co ˆθ. (45 E Ex M γˆx ( + β co ˆθ. (46 For implicity, we conider the cae β. Then we can tick in a delta function and obtain dφ 3 M dˆx ˆx 3 0 M de 3 0 M de d co ˆθ dˆx ˆx de δ( M γˆx ( + co ˆθ E d co ˆθ δ(co ˆθ + E /(Mγˆx Mγˆx / dˆx 3 γ E /Mγ ( M de 3 E γ Mγ dx ( x. (47 M 3 Thi i a imple downward triangular ditribution. 9