Lorentz Invariant Relative Velocity and Relativistic Binary Collisions

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Overall, we shall 2 1 2 2 1 2 1 2 1 2 1 2 − · · (7) emphasize how the hyperbolic nature of the relativistic velocity space manifests itself in concrete physical appli- where we used again the relativistic energy formula. If cations. the velocities are collinear, the last two terms in (7) are equal and sum to give the last term in (6). In this case the expressions (6) and (7) coincide and the flux (5) becomes II. THE DEFINITION OF INVARIANT CROSS n n F = 1 2 . (8) SECTION: THREE PROBLEMS AND A E1E2 F SOLUTION The product of the number densities is n1n2 = 4E1E2, hence In quantum field theory the invariant rate corresponds to the probability per unit time per unit volume of a F =4 . (9) single transition i f, th = dPfi and is given by F → Rfi dV dt This is the most popular form of the flux in particle

n 3 physics also reported in the review of the Particle Data d p + th = 2(2π)4δ4(P P ) j . (4) Group [O 14]. Rfi |Mfi| i − f (2π)32E j=3 j Z Y A. Why collinear velocities? Equation (4) refers to the scattering of unpolarized particles, hence the square of the amplitude is summed over the final spins and averaged over the initial spins. With If we start from Eq. (5), the assumption of collinearity is necessary because, taking the velocities along the Pi and Pf we indicate the total 4-momentum of the initial and final states, respectively. z axis for example, the expression E1E2 v1z v2z = E p E p is at least invariant under| boosts− along| In (4) we use the normalization of one–particle states | 2 1z − 1 2z | 3 3 the z axis, even if not under a general Lorentz transfor- p p′ = (2π) 2Eδ (p p′) both for bosons and fermions, whichh | i corresponds to− ‘2E particles per unit volume’ in- mation. This corresponds to the simplification of the last stead of ‘one particle per unit volume’. The densities ap- terms in Eq. (7). The final expression (9) is anyway a Lorentz scalar, pearing in the flux according to this convention are thus 3 n = 2E. We shall see that for an experimentalist or in hence also valid for noncollinear velocities as can be eas- the case of a gas of particles, the densities are something ily seen in the following way. Dividing Eq. (7) by E1E2, more concrete. the right hand side can be written as Although in the Introduction we argued that Eq. (2) is 1/2 p p 2 p p 2 p 2 p 2 not valid in the relativistic framework, the starting point 1 2 + 1 2 1 2 E − E E · E − E E of quantum field theory and particle physics textbooks is " 1 2 1 2 1 2 # nonetheless the nonrelativistic expression = [(v v )2 + (v v )2 v2v2]1/2. (10) 1 − 2 1 · 2 − 1 2 F = n n v v , (5) Using the vector identity 1 2| 1 − 2| (a b)2 = a2b2 (a b)2 (11) where velocities v1 and v2 are required to be collinear. × − · This includes as particular cases the expression of the flux in the rest frame of massive particles and in the center of momentum frame. We shall discuss in a moment this 2 Four-vectors are indicated as a = (a0, a) and the Minkowski 0 0 assumption. Let us proceed and use v = p/E to rewrite scalar product as a b = a b a b. Natural units ~ = c = kB = 1 · − · Eq. (5) as are used. 3 In many textbooks, for example [BD64],[HM84], [Bro94],[Kak93],[GR08],[Tul11] it is incorrectly stated E2p1 E1p2 that Eq. (9) is valid only for collinear velocities. The motivation n1n2 − . E1E2 is that the cross section must be invariant under boosts along the perpendicular direction in order to transform as an area

2 2 p2 transverse to the beams direction, see also [PS95],[Zee10]. Employing E = m + we ﬁnd This argument is irrelevant for the deﬁnition of the theoretical quantum cross section, and we shall see in Section V, when discussing luminosity of a collider, that collisions with real E2p1 E1p2 | − | beams are almost never collinear, thus full Lorentz invariance is 2 2 2 2 2 2 1/2 = [m p + m p 2E E p p +2p p ] . (6) necessary. 2 1 2 2 − 1 2 1 · 2 1 2 3 in (10), it follows B. ‘Who’ is the relative velocity?

2 2 F = (v1 v2) (v1 v2) v.¯ (12) It is important to remark that Møller puts the product E1E2 − − × ≡ p E1E2 in ratio with the product of densities n1n2, not with Therefore for every v1 and v2 we can write the scalar B. He does not introduce any ‘Møller velocity’ 2 2 F = n1n2 (v1 v2) (v1 v2) . (13) or call relative velocity the quantityv ¯ defined in (12). − − × The misleading identification ofv ¯ = /E1E2 with a The collinear flux (5p) is just a particular case of the in- relative velocity or a ‘Møller velocity’ isF posterior, prob- variant expression (13) when v v = 0. 1 × 2 ably suggested by the deceptive similarity of F = n1n2v¯ with the nonrelativistic expression Fnr = n1n2vr. As a matter of fact, in particle physics literature, some The Møller’s flux authors consider the quantityv ¯ as the relative velocity that generalizes the nonrelativistic expression vr, for oth- Formula (13) was proposed a long time ago in [Mø45]. ers the relative velocity is vr. In statistical physics liter- In this paper Møller notes that textbooks define (at that ature [GLvW80] and in dark matter literature [GG91],v ¯ time) the cross section in the center of momentum frame is called Møller velocity to distinguish it from the rela- with the flux given by Eq. (5) (hence not differently from tive velocity, which is considered to be given by vr in any today). His purpose is to find a general expression valid case. 4 also for non collinear velocities. The above identification is unfortunate because albeit Møller does not derive, but affirms that such an expres- the product E1E2v¯ is a scalar,v ¯ by itself is not invariant. sion is given by Eq. (13) and then prove the invariance. Even worse, for many configurations and magnitudes of In order to do that, he writes the squared root in (13) as the two velocities, vr andv ¯ take values larger than the B , with E1E2 velocity of light, both for massive and massless particles. 2 2 For example, in the center of momentum frame of two B = (E2p1 E1p2) (p1 p2) . particles with equal masses, we have vr =v ¯ = 2v and − − × ∗ The squared expressionsp under the squared root are anti- for v > 0.5 the ‘relative velocity’ is superluminal. ∗ symmetric in the indices 1 and 2. Møller thus introduces Explicitly or tacitly, in high energy physics literature is the antisymmetric tensor an accepted fact that the relative velocity of two particles can be larger than the velocity of light.5 Aµν = pµpν pν pµ, 1 2 − 1 2 In reality this is a macroscopic violation of the princi- and it is easy to see that ples of relativity. Fock [Foc64] expresses the point with the clearest words: 1 B = Aµν A −2 µν In pre-relativistic mechanics the relative r velocity of two bodies was defined as the dif- is a scalar and coincides with . The flux (13) is then F ference of their velocities. Let the velocities of rewritten as two bodies, both measured in the same frame n n F = 1 2 B. (14) of reference, be u and v respectively. Then E1E2 the velocity of the second body relative to the Since energy is the time component of the 4-momentum first used to be defined as w = v u. This and number density is the time component of the 4- definition is invariant with respect− to Galileo current transformations but not Lorentz transformations. Therefore it is not suitable in the The- J = (n,nv)= n0(γ,γv)= n0u, (15) ory of Relativity and must be replaced by where n0 is the number density in rest frame and u the another. The fact that w = v u has no 2 − 4-velocity u = γ(1, v), u = 1, ni and Ei transform in physical meaning becomes evident by exam- the same way under Lorentz transformations. The ratio ining the following example. Let the veloc- n1n2 is then a Lorentz invariant quantity. The flux is ities u and v have opposite directions and E1E2 invariant because is the product of two scalar quantities. have magnitudes near to the speed of light Anyway, neither the derivation we have given, nor the or equal to it. The ‘velocity’ w will have a Møller’s paper clarifies how to derive expression (13) from magnitude near or equal to twice the speed first principles without starting from (5) with collinear of light, which is evidently absurd. velocities.

5 Weinberg considersv ¯ as the relative velocity, called uα in 4 The invariant Møller flux was reported in some influential books [Wei95], but, in evaluating the flux in the center of momentum such as [JR55],[GW64] (and [LL75] as we shall discuss at length) frame, notes: ”However, in this frame uα is not really a physi- but then disappeared from quantum field theory and particle cal velocity; (...) for extremely relativistic particles, it can take physics literature, replaced by the ’dogma’ of collinearity. values as large as 2”. 4

In special relativity every physical velocity must satisfy In the rest frame of one particle 1 we already said that vrel = v2 . The 4-momenta are p1 = (m1, 0), 2 2 2 | | v1 + v2 + v3 < 1 (16) p = (E , p ) with scalar product p p = m E . It fol- 2 2 2 1 · 2 1 2 lows that v = p /E = E2 m2/E can be written in every inertial frame. Clearly, massless particles prop- rel 2 2 2 2 2 as6 | | − agate at the velocity of light, thus the relative velocity of p 2 2 2 two photons, or an electron and a photon is equal to 1 in (p1 p2) m1m2 every inertial frame. vrel = · − . (18) p1 p2 Strictly speaking, following Fock, the mathematical ex- p · Using (12) and p p /E E = 1 v v , in terms of pressions v1 v2 andv ¯ are not even velocities in special 1 · 2 1 2 − 1 · 2 relativity.| − | velocities Eq. (18) becomes (v v )2 (v v )2 v = 1 − 2 − 1 × 2 . (19) rel 1 v v C. Transformation of densities p − 1 · 2 From Eq. (17) and (18) the invariant flux is The product n1n2 is not Lorentz invariant. In fact ✘✘ 2 2 2 the densities in a generic frame do not coincide with the ✘p1 p2 (p1 p2) m1m2 F = n1n2 · · ✘− , proper densities because of the volume contraction, or, E1E2 ✘p1✘p2 p · from another point of view, densities are only the time which gives Eq. (8), or, in terms of velocities, component of a 4-vector. Nonetheless, the product n1n2v¯ is the physical invariant flux. This means that the true ✭ (v v )2 (v v )2 ✭✭✭✭ 1 2 1 2 relative velocity is hidden in this expression and some F = n1n2✭(1 v1 v2) − ✭−✭✭ × , − · ✭1 ✭v1 v2 cancellation takes place. We shall see that this is what p − · happens. which coincides with Eq. (13). Landau-Lifshitz7 thus provide an ab initio derivation of the Møller formula (13). A similar discussion was also given in [Ter70]. D. The Landau-Lifschitz solution We have highlighted the cancellation because is the central point. Formula (13), and its particular case (5), An answer to the above problems is given in [LL75]. arises because there is the cancellation between the factor (1 v v ) that comes from the transformation of den- Let us rephrase the somewhat cumbersome Landau– − 1 · 2 Lifschitz reasoning. sities with the same factor in the denominator of the rel- They assume that both the cross section and the relative velocity. Only a posteriori we can say that the relative velocity must be defined in the rest frame of one ativistic flux with collinear velocities is given by Eq. (5). particle, say particle 1 to fix the ideas. In this frame, by The result of the cancellation is an expression which for- 0 mally looks the nonrelativistic flux, but the relative ve- definition of rate, we have = σvreln1n2 = σv2n1n2. In another frame, in general,R the rate takes the form locity in the collinear case is

= An1n2, with A a factor reducing to σvrel in the v1 v2 R vk = | − | , (20) rest frames. We have to find the expression of A in a rel 1 v v generic frame. − 1 · 2 Since the rate is invariant, the product An n must be not v1 v2 as it is generally believed. 1 2 | − | invariant. Using n = γn0, with n0 the proper density What about the ‘Møller velocity’? It is just the nu- corresponding to the rest frames, and γ = E/m, we have merator of formula (19) or the product of the factor (1 v v ) with v . This explains why it does not have − 1 · 2 rel E1 E2 0 0 any physical meaning by itself. In the next sections we An1n2 = A n1n2. m1 m2 shall see that leaving such factor and the relative velocity explicitly in the formulas allows for a clearer understand- The masses and the proper densities n0 are numbers, thus ing of relativistic physics. we must require that AE1E2 = ci, with ci an invariant. Actually it is possible to give a much general and sim- Divide now both sides of the previous equality by the pler formulation than the Landau-Lifschitz’s one with- scalar p p , thus 1 · 2 out any assumption about reference frames. But before, E1E2 there are many interesting properties of the relative ve- A = c′ , p p i locity that is necessary to recall. 1 · 2 with c′ = c /p p another invariant. In the rest frame i i 1 · 2 of one particle E E /p p = 1 and A = c′ = σv by 6 1 2 1 · 2 i rel Formula (18) was also given in [Hag73]. assumption. Hence in a generic frame 7 They attribute Eq. (13) to Pauli in 1933, well before the ap- pearance Møller’s paper in 1945. They do not give any specific p1 p2 reference and we could not find any Pauli’s paper where such A = · σvrel. (17) E1E2 formula is written. 5

III. PROPERTIES OF THE INVARIANT In special relativity the velocity space is subject to VSR RELATIVE VELOCITY the constraint (16). The space SR is given by the points in the interior of sphere of unitV radius, in natural units. If we restore c in (19), the vector product in the numer- As it is well known, this is a Lobachevsky-Bolyai hy- ator and the scalar product in the denominator are di- perbolic space with constant negative curvature [Foc64], vided by c2. In the nonrelativistic limit c , v c [LL75]. → ∞ | i| ≪ they disappear, and vrel reduces to vr. When one (or Using Eq. (19) we can calculate the relative velocity both) velocity is c, then vrel = c, while when both are between two infinitesimally near points v and v + dv. smaller than c then vrel < c. All the physical require- This corresponds to the Riemann metric ments are satisfied. v 2 v v 2 Formula (19) can be found in some books on relativity 2 (d ) ( d ) dρH = − × , (25) but to our knowledge cannot be found in any particle (1 v2)2 − physics or quantum field theory book. Up to a minus sign where the subscript H stands for hyperbolic. As shown in the numerator and in the denominator, this is the well in [Foc64], the geodesics of are straight lines and known Einstein’s rule for the composition of velocities SR the points of the segment betweenV v and v along the already written in the 1905’s paper [Ein05]. 1 2 geodesic can be parametrized by linear relations as If one prefers to reason in terms of Lorentz transformations, the easiest way is to take S as the laboratory vλ = v1 + λ(v2 v1), (26) frame where v1 and v2 are given, and S1 the rest frame − of particle 1. S1 moves with velocity v1 with respect to with λ a continuous parameter varying in the interval S, hence the velocity v21 of particle 2 in S1, that is the [0, 1]. Using (26) in (25) we find the line element relative velocity, is found by applying a boost B(v1) to v . One gets, see for example [Foc64], [Tsa10], v¯ dλ 2 dρ = . (27) H v2 v1 v2 1 λ v2 v1 (γ1 1)(1 ·2 )v1 − v1 v21 = − − − − . (21) γ1(1 v1 v2) The length of the segment gives the distance between the − · two points of SR. Performing the integration we find By taking S2 as the rest frame of particle 2, with similar V 1 reasoning, instead we have v¯ dλ 1 1+ vrel ρH = 2 = ln . (28) v1 v2 0 1 vλ 2 1 vrel v1 v2 (γ2 1)(1 v·2 )v2 Z − − v − − − − 2 12 = . (22) 1 γ2(1 v1 v2) This is equal to tanh− (v ); hence, the relation between − · rel the relative velocity and the distance is Differently from the nonrelativistic case where v12 = v21, the vectors v12 and v21 belong to different direc- vrel = tanh ρH , (29) tions− that differ by a spatial rotation.8 What is important for scattering theory is that the magnitude of the which represents the relativistic analogous of (24). two vectors is the same and symmetrical in indices 1 and Every velocity in SR can be thought as a relative ve- 2, being equal to the relative velocity (19) locity with respect toV the origin O with magnitude given by the distance from the origin. In physics this distance v = v = v , (23) | 12| | 21| rel is called rapidity, and is indicated commonly with y or η. as can be verified with direct calculation. From (29) we have the usual relations v = tanh y, γ = cosh y, vγ = sinh y. (30) A. Metric and hyperbolic properties We now eliminate the vector product in (19) using the identity (11), In nonrelativistic physics the vectors v = (v1, v2, v3) are points of the Euclidean 3-dimensional velocity space (1 v2)(1 v2) . The relative velocity vr = v v is invariant under 1 2 nr 1 2 vrel = 1 − − 2 , (31) V | − | s − (1 v1 v2) Galileo transformations and coincides with the Euclidean − · distance between the two points of nr, V and associate the Lorentz factor to vrel, v = ρ . (24) r E 1 γr = . (32) 1 v2 − rel p 8 This is the Thomas-Wigner rotation and corresponds to the well From (31) and (32) it follows the fundamental relation known fact that the product of two non collinear boosts gives a boost times a spatial rotation. See for example [Tsa10],[Gou13]. γ = γ γ (1 v v ). (33) r 1 2 − 1 · 2 6

V In Figure 1 we consider the example of scattering SR of ultrarelativistic or massless particles with velocities v1 =v ˆ1 and v2 =v ˆ2 in the laboratory frame at rest iden- tified by the origin with velocity v3 = 0. Two vertices of the red triangle are at infinity and the sides are three relative velocities equal to 1, that is of infinite length. From formula (36) it is easy to see that the two angles at the O 9 v1 v2 ideal vertices, θ23 and θ31, are zero, while the angle at θ =θ the origin θ12 is equal to the angle cos θ =v ˆ1 vˆ2 individu- 12 · θ31 =0 v ated by the velocities in the laboratory. In hyperbolic ge- θ 23 =0 2 ometry the angles determine the sides of the triangle and v1 their sum is less than π by an amount given by the hyperbolic defect δ = π (θ12 + θ13 + θ23). The defect gives the area of the triangle− A = K2δ where K =1/√ k =1 being k = 1 the gaussian curvature.10 The area− of the red triangle− in Figure 1 is thus given by A = δ = π θ. − Figure 1. The hyperbolic plane of velocity space represented by the Beltrami-Klein interior disk model of radius 1. The B. Manifestly invariant representations blue segment is relative to two massive particles colliding collinearly. The red triangle describes the scattering of two ultrarelativistic particles colliding with angle θ in the labora- Since vrel is Lorentz invariant it can be written in terms tory. of scalar products of various 4-vectors. In terms of the 4-velocity u = γ(1, v), u2 = 1, we have

2 If θ is the angle between v and v , then Eq. (33) can be (u1 u2) 1 1 2 v = · − . (37) written as rel u u p 1 · 2

cosh yr = cosh y1 cosh y2 sinh y1 sinh y2 cos θ, (34) The 4-momentum representation is given by (18), and in − terms of the 4-current (15) we obtain which is the cosine rule of hyperbolic geometry. When 2 2 2 the velocities are collinear, this give the well known fact (J1 J2) (J1) (J2) vrel = · − . (38) that rapidities sum up y = y1 y2. p J1 J2 In Figure 1 the blue segment∓ passing for the origin · Another useful formula is found introducing the Mandel- describes the scattering of two massive particles with 2 stam variable s = (p1 + p2) , collinear velocities with magnitudes v1 and v2. Using (28), the distance between the velocities 1 and 2 is the λ(s,m2,m2) sum of the lengths ρ and ρ v = 1 2 , (39) 1O 2O rel s (m2 + m2) p− 1 2 1 1+ v 1 1+ v 1 1+ v ln rel = ln 1 + ln 2 , where we used the triangular function 2 1 vrel 2 1 v1 2 1 v2 − − − 2 2 2 2 λ(s,m1,m2) = [s (m1 + m2) ][s (m1 m2) ]. (40) which gives − − − For example the Mandelstam representation gives s as a v1 + v2 function of v through the Lorentz factor γ vrel = . (35) rel r 1+ v1v2 2 s = (m1 m2) +2m1m2(1 + γr), (41) The same result is obtained by (20) orienting the collision − axis along v1 and taking v2 in the opposite direction. The metric (25) also determines the angles in velocity space [Foc64], 9 The interior of the unit ball is the 3-dimensional extension of the Beltrami-Klein interior disk model for the hyperbolic plane where cos θij = the distance (28) corresponds to the Cayley-Klein projective distance, see for example [Bar11]. The metric (25) is non confor- (v v ) (v v ) (v v ) (v v ) i − k · j − k − i × k · j × k , mal, thus the hyperbolic angles given by (36) in general do not 2 2 2 2 coincide with the Euclidean angles between two velocities. Only (vi vk) (vi vk) (vj vk) (vj vk) − − × − − × when one of the velocities corresponds with the origin, the angle (36) p p at that vertex is equal to the Euclidean one. For considerations on the velocity space employing the conformal Beltrami-Poincar´e where angles are the vertices of the velocity triangle in- model see for example [RS04]. 10 dividuated by the points v1, v2 and v3, which must be The Thomas-Wigner angle is related to the defect and the area, taken in ciclic permutation in Eq. (36). see for example [RS04],[Bar11]. 7

which is useful for cross section calculations. Also the many ways the invariant flux (43) can be explicitly writ- Lorentz factor can be written in terms of invariants as ten. For example, using the Mandelstam variable representation one easily finds the well known expression p p J J s (m2 + m2) 1 2 1 2 1 2 F =2 λ(s,m2,m2). γr = u1 u2 = · = 0· 0 = − . (42) 1 2 · m1m2 n1n2 2m1m2 While formula (43) explicitly displays the invariance p Concluding this section we want to emphasize that property, in order to highlight the physical content it is the connection with the metric properties of the veloc- more useful write ity space given by Eq. (24) and Eq. (28) shows that the F = n1n2krvrel, (45) relative velocity is a concept that is a logical consequence of the relativity principle. The distance, hence the rel- where we have defined the hyperbolic correlation factor ative velocity, is a number that does not depend on the coordinate system or reference frame, both in the non- γr cosh yr p1 p2 kr =1 v1 v2 = = = · . relativistic and in the relativistic case. − · γ1γ2 cosh y1 cosh y2 E1E2 (46)

IV. LORENTZ INVARIANT DEFINITION OF The relative velocity (19) hence enters two times in the FLUX incident flux: explicitly as a factor and implicitly through kr that is nothing but the hyperbolic cosine rule given by Eqs. (33) and (34). The hyperbolic correlation factor k Let us now go back to the incident flux and present how r is not a scalar, only the product n n k is Lorentz invari- the Landau–Lifschitz reasoning can be made simpler. 1 2 r ant, and attains the maximum value of 2 for example in The nonrelativistic expression (2) can only by used as the case of head on collinear scattering of two photons. a limit to which the new expression reduces in the nonrelativistic limit. The product n1n2 must be replaced by 0 0 some Lorentz scalar that reduces to n1n2 = n1n2. The V. FLUX AND LUMINOSITY AT COLLIDERS number densities are the time component of the 4-current (15), hence we are led to take the scalar product We now discuss how experimentalists in high energy n1 n2 0 0 particle physics use the concepts of invariant cross section J1 J2 = n1n2(1 v1 v2)= γr = n1n2γr, · − · γ1 γ2 and flux. In collider physics it is common to consider the rate where we used Eq. (33). This is the only scalar that can integrated over the interaction volume. Using the ex- be formed with two 4-currents and presents the correct pression Eq. (45) for the flux we have nonrelativistic limit. Obviously vrel takes the place of vr. It follows that the natural definition of the relativistic dN f = σ dV n n k v = σ (t), (47) invariant flux is dt 1 2 r rel L Z F = (J1 J2)vrel. (43) which defines the instantaneous luminosity (t). The in- · L tegrated luminosity int = dt (t) gives the total num- This expression is a Lorentz scalar at sight and does L L ber of expected events Nf = σ int in a certain running not rely on the rest frame or the center of momentum R L 11 time. frame. For massless particles the velocity vector be- In storage rings like the Large Hadron Collider (LHC) comes the unitary vector in the direction of propagation the beams are constituted by bunches with n particles and when at least one massless particle is involved in the and Gaussian shape characterized by spatial dispersions scattering then vrel = 1. For two massless particles, say σx, σy, σz (root mean squared deviations) along the three two photons, the flux reads F = n1n2(1 cos θ), with θ directions. Assume further that the beams move along ˆ ˆ − the angle between k1 and k2. For collisions of a massless the z axis in opposite directions with ultrarelativistic ve- with a massive particle, F = n n (1 v cos θ). 1 2 − 2 locity v 1 and produce head-on collisions at the origin When computing cross sections in quantum field the- of the z∼= 0 as in Figure 2. The number densities are ory, the 4-momentum representation (18) is more useful. thus given by We can write 2 n x2 y2 (z±vt) ( 2 + 2 + 2 ) x − 2σx 2σy 2σz F = 4(p1 p2)vrel, (44) n ( ,t)= 3/2 e . (48) · ± (2π) σxσyσz and when at least one massless particle is involved F = The luminosity is then 4(p p ). Clearly expression (9) is only one of the 1 · 2 3 (t)= fNb d x n+(x,t)n (x,t)krvrel, (49) Lk − Z 11 A similar discussion was given in [Wea76] (where anyway the rest where we have multiplied by the revolution frequency frame is used), [Fur03] and [Can16]. f and the number of bunches Nb. The parallel symbol 8 x is the reduction factor. For example, at the LHC, the crossing angle is φ = 285 µrad and σz = 7.7 cm, σx σx z σx 16.7 µm, that give S = 0.835 [HM03]. Note that σz σz the≃ crossing angle φ in the laboratory is equal to the hyperbolic defect of the velocity triangle in velocity space of Figure 1. At this point it is necessary to make two remarks. φ The quantity A = 4πσxσy that appears in Eq. (50), φ or A/S(φ) in Eq. (52), can be considered as an effective z φ/2 φ/2 cross sectional area of the bunches transverse to the plane θ that contains the beams, but this quantity depends on the details of the machine and has nothing to do with the theoretical cross section as we argued in footnote 3. Figure 2. Schematic representation of two bunches in a The crossing angle affects the luminosity φ and the L collinear head-on collision, top figure, and collision with a number of observed events N˙ f but not the ratio, that is crossing angle φ, bottom figure. the extracted cross section σexp = N˙ f / φ, which is an invariant quantity. The theorist does notL have to care about the fact that real beams have a crossing angle indicate the collinear beams. Performing the Gaussian if the cross section σth = th/F is invariant. For this integrations in x, y, s = vt, we obtain the well known f reason the flux must be invariantR under general Lorentz result also reported by the Particle Data Group [O+14] transformations and not only under boosts along a fixed

n1n2fNb direction. The invariant cross section can be calculated (t)=2 . (50) in any frame and compared with σexp. Lk 8πσxσy The factor 2 arises because for ultrarelativistic particles VI. RELATIVE VELOCITY AS KINEMATIC v 1, vrel 1 and the velocities correlation factor for ≃ ≃ 12 VARIABLE this configuration is kr 2, thus krvrel 2. In reality at the LHC≃ beams are not collinear≃ but there is an angle θ between them. Collider physicist prefer to The relative velocity vrel and the Lorentz factor γr are use the complementary angle φ = π θ, called crossing good variables to express the invariant cross section as angle, see Figure 2. The crossing angle− is necessary to much as scalar product of 4-momenta, Mandelstam vari- confine the interaction region and to avoid unwanted col- ables or rapidity. For a general example we consider the + lisions and reduce other effects, see for example [HM03]. quantum electrodynamics processes e e− γγ, pair an- + → While at LHC the crossing angle is small, around 300 nihilation, and its inverse γγ e e−, pair creation. → µrad, at the old Intersecting Storage Ring at CERN, The total annihilation cross section, see for exam- where beams were unbunched, it was about 18◦. ple [BLP82], can be written as a function of the variable The hyperbolic correlation factor with crossing angle τ = s/4m2 as and ultrarelativistic velocities reads σ0 2 1 √τ + √τ 1 2 φ σann(τ)= 2 (τ + τ ) ln − kr 1 cos θ = 1+cos φ = 2cos . (51) 2τ (τ 1) − 2 √τ √τ 1 ≃ − 2 − − − (τ + 1)√τ√τ 1 , (54) If the beams are contained in the x, z plane with σ − − z ≫ σx, σy, see Figure 2, the calculation gives [HM03] 2 2 where σ0 = πα /m , α the fine structure constant and m the electron’s mass. Using Eq. (41) with m = m = m (t, φ)= (t)S(φ), (52) 1 2 L Lk we obtain where 1 s =2m2(1 + γ ), τ = (1 + γ ), (55) 1 r 2 r S(φ)= (53) 1 + ( σz tan φ )2 σx 2 and the cross section can be written as q σ γ2 +4γ +1 σ (γ )= 0 r r ln(γ + γ2 1) ann r 1+ γ γ2 1 r r − r r − 12 p We think that this way of understanding the factor 2 is much γr +3 more clear and physical than saying that the Møller factor is . (56) 2 2. Note that collider physicists do not use the word ‘velocity’ − γr 1# − for the Møller expression but more correctly speak of kinematic or luminosity factor, see for example [MS63],[Nap93],[Fur03], This expression is valid inp any frame. In the rest frame of [HM03]. the electron to fix the ideas, the relative velocity coincides 9 with positrons velocity v+, hence corresponding to ℓ = 1 is zero. Note that (58) has an expansion in odd powers of v ∗ vrel(v+)= v+, γr(v+)= γ(v+). (57) 2 4 6 vrel =2v (1 v + v v + ...), (66) The cross section in the rest frame of the electron is thus ∗ − ∗ ∗ − ∗ given by the same Eq. (56), with γr replaced by γ(v+). while In the center of momentum frame instead Eq. (19) 1 1 v gives = + ∗ , (67) vrel 2v 2 ∗ 2v has only two terms. In order to obtain (65) from (64) it vrel(v )= ∗ , (58) ∗ 1+ v2 is necessary to substitute (67) in the first term of (64) ∗ and to take powers of (66) keeping all the terms of the with Lorentz factor 3 3 5 same order. For example, vrel 8v 24v gives the 3 ∼ ∗ − 5∗ 2 term in v and a contribution to the order v that must 1+ v ∗ 5 5 ∗ γr(v )= ∗ . (59) be summed to vrel 32v . In this way the expansion ∗ 1 v2 ∼ ∗ − ∗ (65) is recovered. If incorrectly we used the ‘Møller velocity’v ¯ = 2v Substituting (59) into Eq. (56) we find nr ∗ ∗ in σann(vrel), we would obtain wrong cross section and 2 4 expansion in the center of mass frame. In terms of s we 1 v 3 v 1+ v 2 2 σann(v )= σ0 − ∗ − ∗ ln ∗ 2(2 v ) . havev ¯ =2 1 4m /s, which gives the relation ∗ 4v v 1 v − − ∗ ∗ − ∗ ∗ − ∗ (60) 2p 2 2 4m 2 2 2 m 4 m 6 s = 2 4m + m vr + vr ++ vr + ..., 1 v¯∗ ∼ 4 16 In this way we have obtained the known formula [BLP82] − 4 using only the relative velocity. (68) The cross section of the inverse process [BLP82] is re- where the expansion corresponds to the nonrelativistic lated to Eq. (60) by limitv ¯ vr 1. This is different from the expansion ∗ ∼ ≪ 2 of (55) σpair(v )=2v σann(v ). (61) ∗ ∗ ∗ 3 5 In order to find the expression valid in any frame we s 4m2 + m2v2 + m2v4 + m2v6 + ..., (69) r 4 r 8 r invert Eq. (59) ∼ which is the correct expansion of s to use in the cross 2 γr 1 τ 1 2m section to obtain the nonrelativistic limit. The misuse v2 = − = − =1 , (62) ∗ γr +1 τ − k1 k2 of the ‘Møller velocity’ thus can also be source of errors. · Examples in dark matter phenomenology are discussed where ki are the 4-momenta of the colliding photons. For in [Can16]. example, using τ as variable, we can write τ 1 σ (τ)=2 − σ (τ). (63) VII. RELATIVE VELOCITY AND pair τ ann RELATIVISTIC COLLISIONS IN GASES

In the nonrelativistic limit vrel vr 1, and the expansion of (56) in powers of the relative∼ ≪ velocity reads The hyperbolic correlation factor kr plays a fundamental role also in the theory of collisions in a relativistic gas. 1 3 11 We consider an ideal gas composed of two species of par- σnr (v ) σ ( v3 v5 + ...). (64) ann rel ∼ 0 v − 20 rel − 105 rel ticles with mass m1 and m2 in thermal equilibrium at rel a given temperature T and work in the frame where the The expansion in terms of v+ in the rest frame of the gas is at rest as a whole, the local rest frame or comoving electron has the same coefficients. Instead, the expansion frame. Before discussing the relativistic gas it is useful of (60) to the same order for small v is to briefly review the nonrelativistic case. ∗ In kinetic theory the number density is determined nr 1 v 6 3 26 5 by the Boltzmann one particle phase space distribu- σann(v ) σ0( + ∗ v + v + ...). (65) eq 2 ∗ ∼ 2v 2 − 5 ∗ 105 ∗ tion f = exp( E/T ), E = p /2m, by the integral ∗ n = g/(2π)3 d3p−f eq, 2ℓ 1 Both expansions follow the behavior σ ℓ aℓv − as ∝ R 3/2 it is expected for inelastic exothermic processes at low eq mT m/T P n = g e− , (70) energy. At very low velocities the cross section is dom- 2π inated by the ℓ = 0 partial wave or S-wave. Anyway the coefficients are different, in particular in (64) the one with g degrees of freedoms of the particle. 10

It is well known, see for example [LL80], that the prob- where x = m/T and Kn(x) are modified Bessel functions ability density function of the relative velocity vr such of the second kind that appear in almost all the formulas ∞ 13 that 0 dvrP (vr) = 1 is given by of relativistic statistical mechanics. It is useful to introduce the normalized momentum dis- R 3/2 v2 2 µ µ r tribution 2 T 2 P (vr)= vr e− , (71) π T 1 2 2 r p √p +m /T fp( )= 2 e− , (78) where µ = m1m2/(m1 + m2) is the reduced mass. The 4πm TK2(x) mean value ∞ dvrP (vr)vr of vr is given by 3 0 such that d pfp(p) = 1. In the rest of the Section we R will abbreviate the notation indicating fp(pi) fp,i. In 8T this way theR average value of a generic function≡ of two vr P = , (72) 3 3 h i s πµ momenta is given by d p1d p2fp,1fp,2G(p1, p2). The next step is to write the probability density func- and, being σ a nonrelativistic cross section that is func- R nr tion of vrel, let us call it (vrel). It was shown in [Can14], tion of vr, the averaged rate is then and at this point it shouldP not come as a surprise, that eq eq is the averaged value of the hyperbolic correlation factor n1 n2 = σnrvr P , j12 =1+ δ12, (73) 1 hRi j12 h i p p k = d3p d3p f f 1 · 2 = dv (v )=1, h ri 1 2 p,1 p,2 E E relP rel where δ12 is the Kronecker’s delta and the factor j12 must Z 1 2 Z0 be inserted to avoid double counting when the 1 and 2 (79) are the same specie. The so-called thermal averaged cross which determines the probability density function of vrel, section is then γ3(γ2 1) r r 2 X − K (√2X√γ + ̺) 3/2 µ v √ 1 r 2 µ ∞ 2 r γr+̺ σ v = dv v e T 2 σ v . (74) (vrel)= , (80) nr r P r r − nr r P √2 K (x ) h i rπ T 0 i 2 i Z 2 2 x1 + x2 Formula (74) is commonly used for the calculation of X = √x1x2, ̺ =Q . (81) abundances of dark matter particles that decoupled at 2x1x2 a temperature when they were nonrelativistic, see for ex- 1 Thanks to (80) the mean value dv (v )v of ample [GS91]. Instead, expressed as a function of the 0 relP rel rel 2 the relative velocity is given by the expression [Can14] relative kinetic energy E = 1/2µvr , Eq. (74) takes the R 2 2 form 2 (1 + ̺) K3(ξ) (̺ 1)K1(ξ) vrel = − − , (82) P 8 ∞ E h i ξ K2(x1)K2(x2) σnrvr P = dEEe− T σnr(E), (75) h i πµT 3 where ξ = x + x . The ultra–relativistic limit x 0 r Z0 1 2 of the mean value (82) is 1, and the fluctuations tend→ to which, for example, is used for calculating nuclear re- zero, thus the bound imposed by the velocity of light is action rates in the Sun [A+11] and rates in big bang not violated even in the statistical sense. nucleosynthesis [IMM+09]. The averaged relativistic rate is then obtained by in- n1n2 tegrating = krσvrel over the momenta with dis- R j12 A. The relativistic classical gas tribution (78). This equivalent to averaging with the distribution of the relative velocity, eq eq eq eq There are other physical problems, for example the n1 n2 n1 n2 calculation of abundances of relics that decoupled when = krσvrel = σvrel . (83) hRi j12 h i j12 h iP they were relativistic, the calculation of particle yields in ultrarelativistic ion collisions and reaction rates in astro- The relativistic thermal averaged cross section is physical plasmas where the nonrelativistic approximation 3 2 1 γr (γr 1) X dvrel − K1(√2X√γr + ̺)σvrel is not good. 0 √γr+̺ σvrel = . The relativistic generalization of the Boltz- P h i R √2 i K2(xi) mann distribution, also known as Jüttner distribu- (84) tion [GLvW80], [CK02], is Q

f eq = exp( E/T), E = p2 + m2. (76) − 13 The ﬁrst terms of the asymptotic expansions for x 1 and ≫ The average number density neq = pg d3pf eq in this x 1 useful in the nonrelativistic and ultrarelativistic are re- (2π)3 spectively≪ case is given by 2 n R −x π 4n 1 (n 1)! 2 Kn(x) e (1 + − ),Kn(x) − , eq g 2 ∼ r 2x 8x ∼ 2 x n = 4πm TK2(x), (77) (2π)3 with the latter valid for n> 0. 11

Changing variable from vrel to s using equations (39) and while a and k are numerical coeﬃcients whose value de- (41), formula (84) takes the more popular form14 pends on the particular type of interactions and on the

2 2 nature of the annihilating particles. λ(s,m1,m2) √s ∞ 2 ds K1( )σ The thermal average (85) of the cross section (89) is (m1+m2) √s T σvrel = . (85) P 2 given by [Can16] h i R 8T i mi K2(xi) When x 1 all the relativisticQ formulas reduce to the σvrel = σΛaΦk(x), (90) ≫ h iP corresponding nonrelativistic expression. The analogy 1 K2(x) K2(x) Φ (x)= 8+2k +(5+2k) 1 +3 3 . between the nonrelativistic and the relativistic case is k 16 K2(x) K2(x) 2 2 thus complete. (91)

There are two particular cases that are worth to discuss B. Thermal averaged cross section in details for their frequent use in dark matter model building and phenomenology. In the case of the Fermi-Dirac and Bose-Einstein distri- If k = 0, which is the case of s-channel annihilation 5 butions it is not possible to obtain a close expression for with pseudoscalar interaction Γa =Γb = γ , we have the distribution of the relative velocity. Quantum effects 2 2 can be considered using the expansions 1 K1 (x) K3 (x) Φ0(x)= 8+5 2 +3 2 (92) 16 K2 (x) K2 (x) ∞ F,B 1 i iE/T 2 f = = ( ) ( ) e− , (86) 1+ (x− ) x 1, (93) eE/T 1 ± ± ∼ O ≫ ∓ i=1 X which describes S-wave scattering nonrelativistic limit. which amounts to the replacement In fact, the expansion of the cross section (89) is σnrvr 4 ∼ √s ∞ ∞ ( )i+j √ijs σΛa + O(vr ) thus the thermal average is constant and K1 ± K1 , (87) temperature independent in agreement with (91). T → √ij T i=1 j=1 The value k = 4 is found in the case of s-channel X X − annihilation with scalar Γa = Γb = 1 and axial-vector in formula (85), see for example [LR02]. µ 5 Γa = Γb = γ γ couplings, and t-channel Majorana In general Eq. (85) must be integrated numerically but 5 fermion annihilation with Γa = Γb = (1 γ )/2. The effective interactions of the type function ± λ λ a b ¯ 3 K2(x) K2(x) Λ = 2 (¯χΓaχ)(ψΓbψ), (88) 1 3 L Λ Φ 4(x)= 2 + 2 (94) − 16 −K2 (x) K2 (x) give cross sections that are simple enough and the ana- 3 2 lytical integration is possible [Can16]. In Eq. (88), λ + (x− ) x 1, a,b ∼ 2x O ≫ are dimensionless coupling associated with the interactions described by a combination of Dirac matrices Γa,b. gives the pure P -wave behavior in the nonrelativistic 2 Λ is the energy scale below which the effective field the- limit. The cross section (89) behaves as σ v σ a vr + nr r ∼ Λ 4 ory is valid. The χ’s are Dirac or Majorana fields and O(v4). Using (71) with reduced mass µ = m we have the have mass m, while ψ are fermions whose mass is much r 2 average v2 = 6 , thus σ v = aσ 3 in agreement smaller than m and Λ such that can be considered mass- r P x nr r P Λ 2x with (93h). Furtheri examplesh arei given in [Can16]. less with very good approximation, mψ = 0. Under these assumptions, the cross section for χχ ψψ is given by → 2 2 2 C. Boltzmann equation a √s s k λaλb m σ = σΛ + , σΛ = , 2 √ 2 4m2 4 4π Λ4 s 4m f − (89) Let (x, p,t) be a non-equilibrium one particle phase space distribution. In order to shorten the notation we assume now that the factor g/(2π3) is included in f and indicate f f(x , p ,t). The relativistic Boltz- i ≡ i i 14 Formulas (84)-(85) were found independently by physicist work- mann equation without external forces for binary colli- ing in different areas. Probably Eq. (84), written in terms of the sions 1 + 2 3 + 4 is usually written in the invariant z √ X γ ̺ χ γ v → the variables = 2 √ r + and = r rel, was first obtained form [GLvW80], [CK02] in [Wea76] for applications to astrophysical relativistic plasmas. In the form (85) it appears in [BBLZ83] for applications in heavy- 3p ion collisions physics, and in [CHH84] for the study of a model d 2 p1 ∂f1 = (f3f4 f1f2)σ , (95) for baryogenesis. Then (85) with m1 = m2 = m was found again · E2 − F in [GG91] in studying the chemical decoupling of dark matter. Z Finally, remarking the role of the invariant relative velocity and where ∂ = (∂/∂t, ). Taking the scalar product on the its probability distribution, was again obtained in [Can14]. −∇ left-hand side, dividing both sides by E1 and remember- 12

1.0 the ultrarelativistic limit tends to 4/5, which evidently X v \ J has no particular physical meaning. Another possibility X v \ is to use the mean value of the single particle velocity rel 3 p 0.8 v = d pf | | , which is easily ﬁnd to be h iJ p E X v \ M R x (1 + x) e− x ≫ 1 8 0.6 X v r \ v J =2 2 v M = . (100) h i x K2(x) −→ h i rπx

0.4 The averaged velocities and their nonrelativistic expression are shown in Figure 3. Note that vrel and v J tend to 1 in the ultrarelativistic limit x 1h as requiredi h i by rel- ≪ 0.2 ativity principle and that the relation vr = √2 v M h i h i is recovered in the nonrelativistic limit, vrel / v J 1 h i h i ∼ √2+ (x− ). O - - 10 2 10 1 10 0 101 10 2 10 3 10 4 10 5

m T D. Rate equation

Figure 3. Mean values of relative velocity and single particle velocity given by Eqs. (99) and (100). The thermal averaged cross section appears in the integrated Boltzmann equation that gives the variation in time of the number density of a particle specie as a con-

ing that = p1 p2vrel we obtain sequence of inelastic reactions and expansion of volume. F · The system departs from chemical equilibrium until it f ∂ 1 f 3 p1 p2 f f f f reaches a new stationary state or moves from a nonequi- + v1 1 = d p2 · ( 3 4 1 2)σvrel. ∂t · ∇ E1E2 − librium state towards the equilibrium state. Z (96) An example of the former process is the departure from the equilibrium and freeze out of particle species in the The hyperbolic correlation factor k thus appears explic- r early Universe.16 itly in the collision integral.15 An example of the latter is the equilibration of The average relative velocity (82) is useful in the re- the hadronic system produced in high energy nucleus- laxation time approximation of the Boltzmann equation nucleus collisions after the formation of the quark-gluon [AW74], [CK02] plasma.17 Let us consider the case of the homogeneous and f E f eq p ∂ = ( f ), (97) isotropic standard model of cosmology governed by the · −τcoll − Friedman-Robertson-Walker metric with scale factor a where τcoll is a typical collision time. If the cross section 1 da and Hubble parameter H = a dt . The distribution func- does not vary strongly with the relative velocity, we can tion depends only on E, p , t and the Boltzmann equa- approximate σvrel σ vrel such that tion takes the form [Ber88| ]| h iP ≃ h iP 1 ∂f p ∂f τcoll eq . (98) 1 1 1 3 f f f f ≃ n σ vrel H | | = d p2kr( 3 4 1 2)σvrel, h i ∂t − E1 ∂ p − P | 1| Z For a gas composed of a single specie of mass m the mean (101) value (82) is given by the simple expression where we used (96) to express the collision integral.

4 K (2x) x ≫ 1 16 Assume that the product particles remain in equilib- v = 3 v = . (99) rel 2 r rium with zero chemical potential, then h iP x K2 (x) −→ h i rπx This can be confronted for example with the compli- f f eq eq eq eq eq eq 3 4 = f3 f4 = f1 f2 = n1 n2 fp,1fp,2, cated expression for v¯ calculated in [CK02] that in h i

15 After [GLvW80] it has become popular to use the ‘Møller veloc- 16 See for example [Ber88],[KT90],[GS91],[GG91] for the decou- ity’ such that the equation takes the form similar to the nonrela- pling of dark matter and other relics. For the baryogenesis prob- tivistic one withv ¯ replacing vr. On the other hand in [CK02] the lem see for example [KW80],[CHH84],[Lut92]. The use of chem- ‘Møller velocity’ is introduced only because simplifies the formu- ical rate equations for reaction in the expanding universe was las. In our opinion this simplification is deceptive because, as we pioneered in [AFH53],[ZOP66],[VDZ77],[LW77]. have seen, the non physical velocityv ¯ is source of confusion and 17 See for example [BBLZ83],[MSM86],[LR02] and [SK09], errors and hides the hyperbolic nature of special relativity. [SMG13] for the connection between the big and the little bang. 13 where the second equality follows from energy conserva- VIII. FINAL COMMENTS tion. Further assume that the nonequilibrium distribution of the colliding particles is proportional to the equi- The Lorentz invariant relative velocity given by for- librium one through the fugacity λ = exp(µ/T ) where µ mula (19) is practically unknown in quantum field theory is the chemical potential, and particle physics literature. We have seen that thanks to (19), the formulation of the invariance flux, cross section and luminosity becomes f f = λ λ f eqf eq = λ λ neqneqf f = n n f f . 1 2 1 2 1 2 1 2 1 2 p,1 p,2 1 2 p,1 p,2 simple and transparent. In this way the use of non physical velocity like the ‘Møller velocity’, and assumptions Integrating in p the collision integral in (101) we have about reference frames and collinearity can be avoided. 1 The cross sections for 2 2 processes are easily expressed in the rest frame of→ a particle or in the center of momentum frame in terms the invariant relative veloc- (neqneq n n ) d3p d3p k f f σv , 1 2 − 1 2 1 2 r p,1 p,2 rel ity that also allows to obtain the correct nonrelativistic Z expansion. Finally we have reviewed the statistical properties in a where the integral is nothing but σvrel . The integral relativistic gas highlighting the role the probability den- h dn1 iP of the left-hand side of (101) gives dt +3Hn1, thus sity function of the relative velocity in the determination of reaction rates, Boltzmann equation and rate equations. The relative velocity determines the metric and hy- 3 1 1 d(n1a ) 1 eq eq perbolic properties of the velocity space. This is not a 3 = (n1 n2 n1n2) σvrel . (102) ν1 a dt j12 − h iP mathematical curiosity but explains the peculiarities of the various ingredients that are necessary in formulating the theory of relativistic scattering. We have inserted the stoichiometric coefficient [Can15] on the left hand side and the statistical factor on the right hand side. In this way, when the species 1 and 2 ACKNOWLEDGMENTS are the same, the equation becomes The author acknowledges M. E. Gómez and J. Ro- 3 driguez Quintero. A. Pich, G. Rodrigo, J. Portolés are 1 d(na ) 2 2 = (neq n ) σvrel , (103) acknowledged for hospitality at IFIC in Valencia and O. a3 dt − h iP Panella for hospitality at INFN in Perugia where part of this work was done. Work supported by the MINECO with j12 = 2 cancelled by stoichiometric coefficient ν1 = grants FPA2011-23778 and FPA2014-53631, by the grant 2. For an approach to rate equations based on nonequi- MULTIDARK CSD2209-00064 of MICINN Consolider- librium thermodynamics see [Can15]. Ingenio 2010 Program, and by INFN project QU ASAP.

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