On the interpretation of Λ spin polarization measurements
OOn the interpretation of Λ spin polarization measurements Wojciech Florkowski ∗ Institute of Theoretical Physics, Jagiellonian University, PL-30-348 Krak´ow, Poland
Radoslaw Ryblewski † Institute of Nuclear Physics Polish Academy of Sciences, PL-31342 Krakow, Poland (Dated: February 8, 2021)
Abstract
The physics interpretation of the recent measurements of the spin polarization of Λ hyperonsproduced in relativistic heavy-ion collisions is discussed. We suggest that the polarization measuredin the Λ rest frame should be projected along the direction of the total angular momentum thatis first transformed to the same frame, and only then averaged over Λ’s with different momenta inthe center-of-mass frame. As this improved procedure is not expected to significantly change thepresent results regarding the global spin polarization, it may affect the estimates of the magnitudeof the polarization and its energy dependence. Such a treatment is also generally more appropriatewhenever directions in the Λ rest frame and in the center-of-mass frame are compared. Throughoutthe paper we deliver explicit expressions for various boosts, rotations, and transformations ofangular distributions, which may help to compare model predictions with the experimental results.
Keywords: spin polarization, Lambda hyperons, relativistic heavy-ion collisions, proton-proton collisions ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ h e p - ph ] F e b . INTRODUCTION For a few decades now the phenomenon of spin polarization of the Λ hyperons producedin proton-proton and heavy-ion collisions has been an intriguing topic of both experimentaland theoretical investigations [1]. For example, the longitudinal polarization of the ¯Λ hy-perons was discussed in 1980s as a possible signal of the quark-gluon plasma formation [2].However, the first heavy-ion experiments that measured the Λ spin polarization in Dubna [3]and at CERN [4] reported negative results. More recently, several theoretical predictionsof the global spin polarization signal in A+A collisions were given in Refs. [5–7]. Theseworks predicted a rather substantial experimental signal, of the order of 10%, and were notconfirmed by the STAR data of 2007 [8]. The idea of a non-vanishing global polarizationreappeared in the context of statistical physics and equilibration of spin degrees of free-dom [9–13]. The much smaller predictions of this approach [14–17] have been eventuallyconfirmed by STAR [18, 19] and independently by ALICE [20]. This has triggered a vast the-oretical interest that includes several highly debated topics: the importance of the spin-orbitcoupling [21, 22], global equilibrium with a rigid rotation [23–26], hydrodynamic [27–31] andkinetic [32–41] models of spin dynamics, anomalous hydrodynamics [42, 43], the Lagrangianformulation of hydrodynamics [44, 45], and hydrodynamic treatment of the spin tensor usingholographic techniques [46]. For recent reviews of the experimental and theoretical situationsee, for example, Refs. [47–51].As the outcome of the spin polarization experiments, one commonly cites the magni-tude of the polarization along a specific direction in the center-of-mass frame (COM). Mostpreferably, the results refer to the direction that is orthogonal either to the reaction plane(in non-central heavy-ion collisions) or to the production plane (in proton-proton collisions).In the case of heavy ions, the direction transverse to the reaction plane agrees with the di-rection of the total angular momentum of the system L (with the orientation of L oppositeto the y -axis, see Fig. 1).To determine the magnitude of the polarization in different directions, however, one firststudies distributions of various three-momentum components of protons emitted in the weakdecay Λ → p + π − , which are measured in the Λ rest frame. As the center-of-mass frameand the Λ rest frame are connected by the Lorentz transformation depending on the three-momentum of Λ, the spatial directions in these two frames are linked by a non-trivial relation.2 IG. 1: The center-of-mass (COM) frame for non-central heavy-ion collisions. In this case boththe reaction and production planes can be defined.
Consequently, any interpretation of the results obtained in the Λ rest frame requires thatan appropriate Lorentz transformation is done before one describes such results in terms ofthe COM variables.The STAR measurements [18–20] indicate that the proton distributions in the Λ restframe are not isotropic and, consequently, unambiguously lead to the conclusion about thenon-zero Λ spin polarization. In our opinion, however, the interpretation of those resultsas the hint for specific correlations between the spin direction of Λ’s and various directionsin COM (in particular, the direction of the total angular momentum L ) requires furtherclarifications because of at least two reasons. First, typically only one component of thepolarization vector is measured — the y -component in the Λ rest frame. Second, the y -direction in the Λ rest frame is different from the y -direction in COM. Consequently, acomplete understanding of the relation between the Λ spin direction and the direction ofthe total angular momentum in COM calls for a more detailed study of the effect connectedwith the boost to the Λ rest frame. 3his is especially important if one interprets the result of the Λ polarization measurementsas an analog of the Einstein-de Haas or the Barnett effect [52, 53]. In this case we suggest firstto measure the projection of the spin polarization along the angular momentum directionthat is “seen” by a Λ in its rest frame, and only then to make averaging over Λ’s withdifferent momenta in COM. Such a method guarantees that the same physical directionis used for all Λ’s. We do not expect that such a procedure may change any qualitativeconclusions about the global spin polarization but, in our opinion, it is more appropriate inorder to establish the right magnitude of the polarization and its energy dependence.In this work, we give several explicit expressions for boosts, rotations, and transformationsof angular distributions that can be useful whenever model predictions are compared withthe experimental results. In particular, we give an expression for the form of the angularmomentum in the Λ rest frame that can be used to consistently project the Λ polarizationmeasured in the Λ rest frame.The paper is organized as follows: In the next section we define the center-of-mass (COM)frame for heavy-ion and proton-proton collisions. In Sec. III we introduce the canonicalboost from COM to the Λ rest frame and introduce the transformation of the total angularthree-momentum from COM to the Λ rest frame. Yet another Λ rest frame, where the Λpolarization is aligned with the z -axis, is introduced in Sec. IV. The weak decay law for theprocess Λ → p + π − is introduced in Sec. V. Finally, in Sec. VI we discuss our main pointregarding the projection of the measured polarization along the total angular momentum inCOM. We summarize and conclude in Sec. VIII. Several useful properties of the canonicalboost and transformations of the angular distribution of protons are discussed in the twoappendices. Conventions and notation:
Throughout the paper we use natural units with (cid:126) = c = 1and the metric tensor with the signature (+ − −− ). Three- and four-vectors are definedby their components, however, for three-vectors we often use the bold font, for example, p µ = ( E, p , p , p ) = ( E, p ), where E = (cid:112) m + p denotes the particle energy. For thelength of a three-vector we use the normal font, p = | p | . Scalar products of three-vectors aredenoted by a dot, a µ b µ = a b − a · b . The unit three-vectors are denoted by a hat, p = p ˆ p .4 I. CENTER-OF-MASS (COM) FRAME
In the analyzes of spin polarization of relativistic particles, it is important to define pre-cisely the reference frames where the specific physical quantities are defined and measured.In this work, we define altogether three different reference frames that are linked by Lorentzboosts and rotations: the center-of-mass frame of the total system, COM, and two restframes of Λ’s with a given momentum in COM. The last two frames differ by a rotation.We assume that the main reference frame corresponds to the center-of-mass (COM) frameof the colliding system. In the case of non-central heavy-ion collisions, the axes of the COMframe are defined by the beam axis ( ˆ z ), the impact vector ( ˆ x ), and the direction that isperpendicular to the reaction plane ( ˆ y ) spanned by ˆ x and ˆ z , see Fig. 1. We note thatthe orientation of the three-vector describing the angular momentum L is opposite to the y axis.In the case of proton-proton collisions, the z axis corresponds to the direction of theinitial protons, the y axis is defined to be perpendicular to the plane determined by ˆ z and the momentum of the emitted Λ hyperon p Λ , i.e., to the production plane, while ˆ x isperpendicular to both ˆ y and ˆ z , see Fig. 2. The Cartesian coordinate system x, y, z is takenin the two cases to be right-handed. We note that in the case of proton-proton collisionsone may also use a rotated frame where the z axis coincides with the direction of p Λ . InFig. 2 the axes of this frame are denoted by x r , y r , and z r . III. THE Λ REST FRAME S (cid:48) ( p Λ ) In the following we define two frames where the Λ hyperon with the momentum p Λ inCOM frame is at rest. The first one is defined by the canonical boost from the COM frame.The second one differs from the first by an additional rotation that aligns the polarizationvector with the z -axis. Here we tacitly assume that the reaction plane angle in the laboratory (LAB) frame can be well measuredby calculating the event plane flow vector [54], hence, the COM frame is rotated by this angle around thebeam axis in LAB. IG. 2: The center-of-mass (COM) frame for p + p collisions. A. The canonical boost
We define the rest frame S (cid:48) ( p Λ ) of Λ’s with the COM frame three-momentum p Λ =( p , p , p ) by the canonical boost [55, 56] L µν ( − v Λ ) = E Λ m Λ − p m Λ − p m Λ − p m Λ − p m Λ αp p αp p αp p − p m Λ αp p αp p αp p − p m Λ αp p αp p αp p . (1)Here E Λ and v Λ = p Λ /E Λ are the energy and three-velocity of Λ in COM, respectively, while m Λ is the Λ mass and α ≡ / ( m Λ ( E Λ + m Λ )). We stress that the frame S (cid:48) ( p Λ ) dependson p Λ – in practice one should select an ensemble of events that include Λ’s with the COMthree-momentum in a small bin placed around a given value of p Λ . The components of thefour-vectors in S (cid:48) ( p Λ ) and COM are related by the transformation p (cid:48) µ = L µν ( − v Λ ) p ν . (2)In particular, by construction we obtain p (cid:48) µ Λ = ( m Λ , , , R Λ that brings the three-vector p Λ to the form (0 , , p Λ ),the boost L along the third axis with the velocity − v Λ , and the inverse rotation R − , namely L = R − ( φ Λ , θ Λ ) L ( − v Λ ) R Λ ( φ Λ , θ Λ ) . (3)6he rotation R Λ can be written as the product of two rotations. If we use the parametriza-tion p Λ = p Λ (sin θ Λ cos φ Λ , sin θ Λ sin φ Λ , cos θ Λ ) in COM, then R Λ = R ( θ Λ ) R ( φ Λ ), where R ( φ Λ ) = φ Λ sin φ Λ − sin φ Λ cos φ Λ
00 0 0 1 (4)and R ( θ Λ ) = θ Λ − sin θ Λ θ Λ θ Λ . (5)The boost L ( − v Λ ) is defined by the expression L ( − v Λ ) = γ Λ − γ Λ v Λ − γ Λ v Λ γ Λ , (6)where γ Λ = E Λ /m Λ is the Lorentz factor. Further useful properties of the canonical boostare discussed in Appendix A. B. Transformation of the system’s angular momentum
The crucial role in the discussion and interpretation of the spin-polarization measurementsis played by the total angular momentum of the system described by the tensor J µν . It canbe decomposed into the orbital and spin parts, J µν = L µν + S µν . In non-central heavy-ioncollisions, a substantial non-zero orbital part L µν is generated at the initial stage [10]. Oneexpects that during the system’s evolution some part of L µν is transferred to the spin part S µν , of course, with the total angular momentum J µν being conserved. The generationof a non zero spin part S µν may be reflected just by the measured spin polarization ofthe produced particles. We note that the spin part may be also generated at the very earlystages of the collision but one expects anyway that the values of S µν are negligible comparedto L µν . 7 - - - - - - - FIG. 3: Direction (streamlines) of the total angular momentum of the system transformed fromCOM to the S (cid:48) ( p Λ ) frame for Λ particles at midrapidity (ˆ v z = 0) with the momentum p Λ = 1 GeV(left panel) and p Λ = 4 GeV (right panel) for various orientations of the velocity in transverseplane. If one works in the COM frame, only the spatial components of L µν are different fromzero. They determine the orbital angular momentum of the system through the relation L k = − (cid:15) kij L ij . (7)With the standard orientation of the axes in COM, one expects that the direction of thevector L is opposite to the y axis, see Fig. 1. The components of L transform like the com-ponents of the magnetic field, since they represent spatial components of an antisymmetrictensor L µν . Hence, in the frame S (cid:48) ( p Λ ) they are given by the formula [55] L (cid:48) = γ Λ L − γ γ Λ + 1 v Λ ( v Λ · L ) . (8)From Eq. (8) we find the ratio of the lenghts of the vectors L (cid:48) and L , namely L (cid:48) L = γ Λ (cid:16) − ( v Λ · ˆ L ) (cid:17) / . (9) The conserved quantities J i corresponding to Lorentz boosts are of the form J i = ER i − tP i , where R i =(1 /E ) (cid:82) d x x i T and E = (cid:82) d x T , with T being the energy density. In the center-of-momentumframe P i = 0. Moreover, if the center-of-momentum frame is also the center-of-mass frame (strictlyspeaking, the center-of-energy for relativistic systems) then we also have R i = 0. IG. 4: Three-dimensional visualization of the vectors L (cid:48) defined by Eqs. (8) and (12). Let us note that for relativistic Λ’s the directions of L and L (cid:48) (measured in their appropri-ate reference frames) may be significantly different. In general, only for the case v Λ · L = 0they are the same. For non-relativistic systems, the second term on the right-hand sideof Eq. (8) represents a relativistic correction of the order ( v Λ /c ) and can be neglected,however, for relativistic systems the second term may be equally important as the first one.Consequently, comparisons of the measured polarization direction should refer to the di-rection of L (cid:48) rather than to the direction of L . For this purpose we introduce two unitvectors ˆ L = L L , ˆ L (cid:48) = L (cid:48) L (cid:48) . (10)Taking into account Eq. (9) we may writeˆ L (cid:48) = (cid:16) − ( v Λ · ˆ L ) (cid:17) − / (cid:18) ˆ L − γ Λ γ Λ + 1 v Λ ( v Λ · ˆ L ) (cid:19) . (11)This vector is expressed only by the three-momentum of Λ and the direction of the angularmomentum in COM. We note that with our choice of COM,ˆ L = (0 , − ,
0) (12)9e obtain ˆ L (cid:48) = (cid:0) − ( v ) (cid:1) − / γ Λ γ Λ + 1 v v , ˆ L (cid:48) = (cid:0) − ( v ) (cid:1) − / (cid:18) γ Λ γ Λ + 1 v v − (cid:19) , ˆ L (cid:48) = (cid:0) − ( v ) (cid:1) − / γ Λ γ Λ + 1 v v . (13)The visualization of those components for the case v = 0 is shown in Fig. 4.For the sake of completeness, let us consider the transformation law for the component K i = − L i that behaves like an electric-like component of L µν . As we have mentioned above, K = 0 in COM. However, after making the canonical boost and using Eq. (12) we obtain K (cid:48) = γ Λ (cid:0) v , , − v (cid:1) L (14)or, after normalization, ˆ K (cid:48) = ( v , , − v ) (cid:112) ( v ) + ( v ) . (15) IV. THE Λ REST FRAME S ∗ ( p Λ ) In the Λ rest frame S (cid:48) ( p Λ ), the Λ polarization is characterized by the polarization three-vector P (cid:48) , see Fig. 5. It can be defined by the magnitude P (cid:48) and the unit vector ˆ P (cid:48) thatspecifies the polarization direction, namely, P (cid:48) = P (cid:48) ˆ P (cid:48) . The vector ˆ P (cid:48) can be expressed bythe two angles Φ (cid:48) and Θ (cid:48) with the help of the standard parametrizationˆ P (cid:48) = (sin Θ (cid:48) cos Φ (cid:48) , sin Θ (cid:48) sin Φ (cid:48) , cos Θ (cid:48) ) . (16)In the following, it will be useful to consider also the frame where only the third componentof ˆ P (cid:48) is different from zero. This is achieved by the subsequent action of the two rotations R z (cid:48) (Φ (cid:48) ) = cos Φ (cid:48) sin Φ (cid:48) − sin Φ (cid:48) cos Φ (cid:48)
00 0 1 (17)10
IG. 5: The Λ rest frame. The momentum distribution of protons produced in the weak decayΛ → p + π − depends on cos θ ∗ , where θ ∗ is the angle between the polarization vector P (cid:48) and theproton momentum direction ˆ p (cid:48) p . and R y (cid:48) (Θ (cid:48) ) = cos Θ (cid:48) − sin Θ (cid:48) (cid:48) (cid:48) . (18)The resulting frame will be called S ∗ ( p Λ ). It is trivial to see thatˆ P ∗ = R y (cid:48) (Θ (cid:48) ) R z (cid:48) (Φ (cid:48) ) ˆ P (cid:48) = (0 , , . (19)Let us now consider the three-momentum of the proton emitted in the weak decay of Λ.Similarly to the case of the polarization vector, we express it as follows p (cid:48) p = p (cid:48) p ˆ p (cid:48) p whereˆ p (cid:48) p = (cid:0) sin θ (cid:48) p cos φ (cid:48) p , sin θ (cid:48) p sin φ (cid:48) p , cos θ (cid:48) p (cid:1) . (20)11 IG. 6: The frame S ∗ ( p Λ ) is obtained from the frame S (cid:48) ( p Λ ) by a rotation that brings P (cid:48) alongthe new z -axis. In the frame S ∗ ( p Λ ) we haveˆ p ∗ p,x = cos(Φ (cid:48) − φ (cid:48) p ) sin θ (cid:48) p cos Θ (cid:48) − cos θ (cid:48) p sin Θ (cid:48) ≡ sin θ ∗ cos φ ∗ , ˆ p ∗ p,y = − sin(Φ (cid:48) − φ (cid:48) p ) sin θ (cid:48) p ≡ sin θ ∗ sin φ ∗ , ˆ p ∗ p,z = cos(Φ (cid:48) − φ (cid:48) p ) sin θ (cid:48) p sin Θ (cid:48) + cos θ (cid:48) p cos Θ (cid:48) ≡ cos θ ∗ . (21)From the last line we find ˆ P (cid:48) · ˆ p (cid:48) p = ˆ P ∗ · ˆ p ∗ p = cos θ ∗ . (22)The angular distributions of protons emitted in the frames S (cid:48) ( p Λ ) and S ∗ ( p Λ ) satisfy anobvious constraint (cid:90) dN p d Ω (cid:48) sin θ (cid:48) p dθ (cid:48) p dφ (cid:48) p = (cid:90) dN p d Ω ∗ sin θ ∗ p dθ ∗ p dφ ∗ p , (23) This is of course a trivial result. The main reason for introducing the frame S ∗ ( p Λ ) is that we find it usefulin the following to consider the angular distributions expressed by the angles ( θ (cid:48) p , φ (cid:48) p ) or, equivalently, bythe angles ( θ ∗ , φ ∗ ). dN p /d Ω behave like scalar functions of the azimuthal and polar angles.Consequently, if the distribution dN p /d Ω ∗ is a function of cos θ ∗ only, for example, dN p d Ω ∗ = F (cos θ ∗ ) , (24)where F ( x ) is an arbitrary function of x , then dN p d Ω (cid:48) = F (cid:0) cos(Φ (cid:48) − φ (cid:48) p ) sin θ (cid:48) p sin Θ (cid:48) + cos θ (cid:48) p cos Θ (cid:48) (cid:1) . (25)In the end of this section let us note that the three-vector P (cid:48) can be interpreted as aspatial part of the four-vector [57] P (cid:48) µ = (0 , P (cid:48) ) . (26)In general, we have − ≤ P (cid:48) · P (cid:48) ≤ . (27)The case P (cid:48) · P (cid:48) = − / P (cid:48) . The values larger than − P (cid:48) · P (cid:48) = 0 means that the system is unpolarized. V. THE WEAK DECAY LAW
In the frame S ∗ ( p Λ ), the Λ weak decay Λ → p + π − is described by the following law thatdescribes the angular distribution of emitted protons dN pol p d Ω ∗ = 14 π (cid:0) α Λ P ∗ · ˆ p ∗ p (cid:1) . (28)Here α Λ = 0 .
732 is the Λ decay constant. Equation (28) implies that in S (cid:48) ( p Λ ) the protonangular distribution has the form dN pol p d Ω (cid:48) = 14 π (cid:2) α Λ P (cid:0) cos(Φ (cid:48) − φ (cid:48) p ) sin θ (cid:48) p sin Θ (cid:48) + cos θ (cid:48) p cos Θ (cid:48) (cid:1)(cid:3) . (29)The averaged values of the three momentum components in S (cid:48) ( p Λ ) can be obtained bystraightforward integration: (cid:104) ˆ p (cid:48) p,x (cid:105) = (cid:90) (cid:32) dN pol p d Ω (cid:48) (cid:33) (sin θ (cid:48) p ) cos φ (cid:48) p dθ (cid:48) p dφ (cid:48) p = 13 P (cid:48) α Λ sin Θ (cid:48) cos Φ (cid:48) , (cid:104) ˆ p (cid:48) p,y (cid:105) = (cid:90) (cid:32) dN pol p d Ω (cid:48) (cid:33) (sin θ (cid:48) p ) sin φ (cid:48) p dθ (cid:48) p dφ (cid:48) p = 13 P (cid:48) α Λ sin Θ (cid:48) sin Φ (cid:48) , (cid:104) ˆ p (cid:48) p,z (cid:105) = (cid:90) (cid:32) dN pol p d Ω (cid:48) (cid:33) sin θ (cid:48) p cos θ (cid:48) p dθ (cid:48) p dφ (cid:48) p = 13 P (cid:48) α Λ cos Θ (cid:48) . (30)13he last result indicates that the magnitude and direction of the polarization can be directlyobtained from the averaged values of the three momentum components measured in S (cid:48) ( p Λ ) P (cid:48) = P (cid:48) (sin Θ (cid:48) cos Φ (cid:48) , sin Θ (cid:48) sin Φ (cid:48) , cos Θ (cid:48) ) = 3 α Λ (cid:0) (cid:104) ˆ p (cid:48) p,x (cid:105) , (cid:104) ˆ p (cid:48) p,y (cid:105) , (cid:104) ˆ p (cid:48) p,z (cid:105) (cid:1) . (31)This expression is of course a basis for the experimental determination of polarization. VI. CORRELATION WITH TOTAL ANGULAR MOMENTUM
We have discussed above how the magnitude and direction of the spin polarization can bedetermined in the frame where Λ’s are at rest. More precisely, we have considered the restframe of Λ’s with three-momentum p Λ , which is obtained by the canonical boost from COM.A natural question at this stage appears, how the direction of the measured polarization isrelated to the axes of the COM coordinate system.Equation (31) gives the prescription how to measure three independent components of theΛ polarization in its (canonical) rest frame. Assuming that the measurement of the averages (cid:104) ˆ p (cid:48) p,x (cid:105) , (cid:104) ˆ p (cid:48) p,y (cid:105) , and (cid:104) ˆ p (cid:48) p,z (cid:105) is indeed possible, we may define the projection of the polarizationalong the direction of the total angular momentum by the expressionˆ L (cid:48) · P (cid:48) = (cid:16) − ( v Λ · ˆ L ) (cid:17) − / (cid:18) ˆ L · P (cid:48) − γ Λ γ Λ + 1 v Λ · P (cid:48) v Λ · ˆ L (cid:19) . (32)The direction represented by a unit vector ˆ L (cid:48) is the direction of the total angular momentumthat is “seen” by the spin of the decaying Λ that has three-momentum p Λ in COM. Byconstruction | ˆ L (cid:48) · P (cid:48) | ≤ P (cid:48) ≤ . In the case of spin polarization of Λ’s, such a frame does not exist, since the analyzedΛ’s have usually different momenta in COM.So far, our discussion has been concentrated on Λ’s with a given momentum in COM. Fora given colliding system, beam energy, and the centrality class, such Λ’s can be treated as Although it is typically a non-inertial rotating frame, the non-relativistic treatment allows for simpleaddition of polarizations of different particles. L · P (cid:48) or ˆ L (cid:48) · P (cid:48) from the proton distributions in S (cid:48) ( p Λ ). Theadvantage of the expression (32) compared to the estimate of just ˆ L · P (cid:48) is that the spinpolarization of each Λ, irrespectively of its three-momentum p Λ in COM, is projected onthe same physical axis corresponding to L in COM. Hence, Eq. (32) is in our opinion theproper object that can be used to study the relation between the polarization of all Λ’s with L . To do so, one has to simply average Eq. (32) over all Λ’s with different p Λ . VII. PROTON-PROTON COLLISIONS
In the end of this work, let us turn to a discussion of proton-proton collisions. If theproton-proton COM frame corresponds to the case shown in Fig. 2, where the variant witha rotation in the production plane is chosen, the canonical boost is reduced to the form (6).Then, the four-vector describing the polarization in COM is obtained by the boost L (+ v Λ )acting on the four-vector (0 , P (cid:48) ). This leads to the expression P µ = P (cid:48) ( γ Λ v Λ cos Θ (cid:48) , sin Θ (cid:48) cos Φ (cid:48) , sin Θ (cid:48) sin Φ (cid:48) , γ Λ cos Θ (cid:48) ) . (33)At first sight, the interpretation of the spin polarization measurements in proton-protoncollisions seems to be easier compared to the heavy-ion case. As the transverse componentsof P (cid:48) are not affected by the boost one may try simply to add them and average overdifferent Λ’s. This procedure, however, makes sense only if the x and z components of P (cid:48) are zero. Otherwise, the results obtained for different Λ’s depend on the boost and theoriginal transition to a rotated frame.Consequently, if the spin polarization of Λ’s has non-zero x and z components it is suitableto use the frame without the rotation. In this case we may follow the procedure discussedabove for heavy ions, with the total angular momentum direction replaced by one of theother physical directions defined in the non-rotated COM frame that can be measured (forexample, the direction perpendicular to the plane determined by the beam and the fastestproton produced). Such a procedure may be also useful in the case if more Λ’s are producedin one event. 15 III. CONCLUSIONS
In this work, we have discussed the interpretation of the recent measurements of the spinpolarization of Λ hyperons produced in relativistic heavy-ion collisions. We have emphasizedthat the appropriate interpretation of the relation between the Λ spin direction (measuredin the Λ rest frame) and the total angular momentum of the system (measured in thecenter-of-mass frame) requires that the direction of the angular momentum is boosted tothe Λ rest frame. We have given the necessary formula that, we hope, may find its practicalimplementation in the polarization measurements. In particular, this expression may beused to average the measured polarization of Λ’s with different momenta in the center-of-mass frame. Several explicit expressions for boosts and rotations have been written out,which may help to compare model predictions with the experimental results.
Acknowledgements.
We thank Y. Bondar, M. Ga´zdzicki, T. Niida, I. Selyuzhenkov,G. Stefanek, and S. Voloshin for stimulating and clarifying discussions. The workof WF and RR was supported in part by the Polish National Science Center GrantsNo. 2016/23/B/ST2/00717 and No. 2018/30/E/ST2/00432, respectively.
Appendix A: Properties of the canonical boost
Of course, the three-momentum of the Λ hyperon in its frame is zero, however, it ispossible to introduce the four-vector in COM that defines the direction of a moving Λ andhas non vanishing components in the Λ rest frame. The desired object is λ µ (cid:107) = ( λ (cid:107) , λ (cid:107) ) = (0 , ˆ p Λ ) . (A1)After the canonical boost to S (cid:48) ( p Λ ) we obtain λ (cid:48) µ (cid:107) = ( λ (cid:48) (cid:107) , λ (cid:48)(cid:107) ) = γ Λ (cid:18) − p Λ E Λ , ˆ p Λ (cid:19) . (A2)The property λ (cid:48)(cid:107) = γ Λ λ (cid:107) is usually interpreted as the conservation of the angles betweenthe three-momentum of a moving particle and the frame axes by the canonical boost, as onehas λ (cid:48)(cid:107) /λ (cid:48)(cid:107) = λ (cid:107) /λ (cid:107) . The other two important four-vectors are: λ µ ⊥ , = ( λ ⊥ , , λ ⊥ , ) = 1 (cid:112) ( p ) + ( p ) (cid:0) , − p , p , (cid:1) (A3)16nd λ µ ⊥ , = ( λ ⊥ , , λ ⊥ , ) = 1 p Λ (cid:112) ( p ) + ( p ) (cid:0) , − p p , − p p , ( p ) + ( p ) (cid:1) . (A4)The four-vectors λ µ ⊥ , and λ µ ⊥ , do not change under the canonical boost (1). The three-vector λ ⊥ , represents the rotation axis for the rotation R Λ .For example, any four-vector of the form n µ = (0 , n ) = (0 , n , n , n ) , (A5)with the normalization n µ n µ = − n · n = 1, after the canonical boost weobtain n (cid:48) µ = (cid:0) n (cid:48) , n (cid:48) (cid:1) = (cid:18) − p Λ · n m Λ , n + p Λ p Λ · n m Λ ( E Λ + m Λ ) (cid:19) , (A6)where n (cid:48) · n (cid:48) = 1 + ( p Λ · n ) /m . Thus the direction n in COM can beOne can check that λ (cid:48)(cid:107) · n (cid:48) = γ λ (cid:107) · n . (A7)Since n = 0 and P (cid:48) = 0 we obtain n · P = n (cid:48) · P (cid:48) . (A8)Hence, the four-vector n µ can be used to define the polarization direction in the way that isframe independent. Appendix B: Distribution of the proton three-momenta along arbitrary direction.
If the distribution of protons coming from the Λ decay is given by Eq. (28), their angulardistribution in S (cid:48) ( p Λ ) is obtained from Eq. (29). In this section we assume that a certainangular distribution of protons dN p /d Ω (cid:48) is known and construct the distribution of the pro-ton projected momentum along an arbitrary direction in S (cid:48) ( p Λ ). The obtained formula canbe used to determine polarization in a given direction directly from the angular distribution dN p /d Ω (cid:48) . Note that if the distribution dN p /d Ω (cid:48) is isotropic, the proton projected momentumalong any direction has a flat distribution that reflects no sign of polarization.We start with the integral of the angular distribution and rewrite as follows (we are nowin the frame S (cid:48) ( p Λ ) but for clarity of notation we skip the index prime, also the number of17rotons is normalized to one)1 = (cid:90) (cid:18) dNd Ω (cid:19) sin θ p dθ p dφ p = +1 (cid:90) − dc (cid:90) (cid:18) dNd Ω (cid:19) δ ( c ∗ − c ) sin θ p dθ p dφ p . (B1)Here δ denotes the Dirac delta function and c ∗ is the cosine of the angle between the protondirection defined by the angles θ p and φ p and an arbitrary direction defined by the angles Θand Φ, hence c ∗ = cos θ ∗ = cos Θ cos θ p + cos(Φ − φ p ) sin Θ sin θ p . (B2)By construction − ≤ c ∗ ≤ +1.The distribution of the proton three-momentum direction along the direction specifiedby the angles Θ and Φ is defined by the integral dNdc = (cid:90) (cid:18) dNd Ω (cid:19) δ ( c ∗ − c ) sin θ p dθ p dφ p . (B3)To do the integral on the right-hand side we introduce the function f ( c, Θ , Φ , θ p , φ p ) = c ∗ (Θ , Φ , θ p , φ p ) − c (B4)and use the properties of the Dirac delta function to write dNdc = π (cid:90) sin θ p dθ p π (cid:90) dNd Ω ( θ p , φ p ) (cid:20) δ ( φ p − φ + p ) | f (cid:48) ( φ + p ) | + δ ( φ p − φ − p ) | f (cid:48) ( φ − p ) | ) (cid:21) dφ p . (B5)Here f (cid:48) = sin(Φ − φ p ) sin Θ sin θ p (B6)and φ ± p = Φ ± arccos (cid:18) c − cos Θ cos θ p sin Θ sin θ p (cid:19) (B7)are the two solutions of the equation c ∗ − c = 0. As a matter of fact, this equation hassolutions only if the following condition is satisfied − ≤ c − cos Θ cos θ p sin Θ sin θ p ≤ +1 . (B8) It may happen that the solutions defined by Eq. (B7) are outside of the range (0 , π ), however, since f (cid:48) and dN/d Ω are periodic this does not lead to problems. θ p must be limited — for given values of c and Θ, only those values of θ p contribute to the integral (B5) which satisfy (B8). If weintroduce the notation c = cos θ , with 0 ≤ θ ≤ π , then the limits for the θ p integration are θ min p = max(0 , Θ − θ, θ − Θ) ≤ θ p ≤ min( π, π − θ − Θ , Θ + θ ) = θ max p . (B9)Here we assumed that the range of the angles Θ and θ p is between 0 and π . Consequently,the final result can be written as dNdc = θ max p (cid:90) θ min p (cid:20) dNd Ω ( θ p , φ + p ) 1 | f (cid:48) ( φ + p ) | + dNd Ω ( θ p , φ − p ) 1 | f (cid:48) ( φ − p ) | ) (cid:21) sin θ p dθ p . (B10)If the proton distribution is given by the weak decay law discussed above, the last formulamay be interpreted as the inverse of the transformation that leads from Eq. (28) to Eq. (29).We have checked numerically that this is indeed so.19
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