A microscopic description for polarization in particle scatterings
AA microscopic description for polarization in particle scatterings
Jun-jie Zhang, Ren-hong Fang, Qun Wang, and Xin-Nian Wang
2, 3 Department of Modern Physics, University of Scienceand Technology of China, Hefei, Anhui 230026, China Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics,Central China Normal University, Wuhan, Hubei 430079, China Nuclear Science Division, MS 70R0319,Lawrence Berkeley National Laboratory, Berkeley, California 94720
We propose a microscopic description for the polarization from the first principlethrough the spin-orbit coupling in particle collisions. The model is different fromprevious ones based on local equilibrium assumptions for the spin degree of freedom.It is based on scatterings of particles as wave packets, an effective method to dealwith particle scatterings at specified impact parameters. The polarization is thenthe consequence of particle collisions in a non-equilibrium state of spins. The spin-vorticity coupling naturally emerges from the spin-orbit one encoded in polarizedscattering amplitudes of collisional integrals when one assumes local equilibrium inmomentum but not in spin.
I. INTRODUCTION
A very large orbital angular momentum (OAM) can be created in peripheral heavy ioncollisions [9, 10, 24, 45, 46, 58, 60]. Such a huge OAM can be transferred to the hot and densematter produced in collisions and make particles with spins polarized along the direction ofOAM [24, 37, 45, 60]. Recently the STAR collaboration has measured the global polarizationof Λ and ¯Λ for the first time in Au+Au collisions at √ s NN = 7 . − GeV [1, 2, 48]. Theglobal polarization is the net polarization of local ones in an event which is aligned in thedirection of the event plane. The results show that the magnitude of the global Λ and ¯Λ polarization is of the order a few percent and decreases with collisional energies. Thedifference between the global polarization of Λ and ¯Λ may possibly indicates the effect fromthe strong magnetic field formed in high energy heavy ion collisions. a r X i v : . [ nu c l - t h ] A p r Several theoretical models have been developed to study the global polarization. If thespin degree of freedom is thermalized, one can construct the statistic-hydro model by includ-ing the spin-vorticity coupling S µν ω µν into the thermal distribution function [5–7, 20–22].Here S µν is the spin tensor, ω µν = − (1 / ∂ µ β ν − ∂ ν β µ ) is the thermal vorticity, the macro-scopic analog of the local OAM, and β µ ≡ βu µ is the thermal velocity with β = 1 /T beingthe inverse of the temperature and u µ being the fluid velocity. It turns out that the averagespin or polarization is proportional to the thermal vorticity if the spin-vorticity coupling isweak.Similar to the statistic-hydro model, another approach to the global polarization assuminglocal equilibrium is the the Wigner function (WF) formalism. The WF formalism for spin-1/2 fermions [11, 16, 19, 32, 55, 59, 65] has recently been revived to study the chiral magneticeffect (CME) [23, 41, 43, 57] (for reviews, see, e.g., Ref. [36, 41, 42]) and chiral vortical effect(CVE) [3, 17, 26, 34, 51, 56] for massless fermions [12, 26–30, 33, 35]. The Wigner functionsfor spin-1/2 fermions are × matrices. The axial vector component gives the spin phasespace distribution of fermions near thermal equilibrium [18, 25, 31, 61]. It can be shownthat when the thermal vorticity is small, the spin polarization of fermions from the WF isproportional to the thermal vorticity vector. So the WF can also be applied to the study ofthe global polarization of hyperons.In order to describe the STAR data on the global Λ / ¯Λ polarization, the hydrodynamicor transport models have been used to calculate the vorticity fields in heavy ion collisions[4, 13–15, 38, 39, 54]. Then the polarization of Λ / ¯Λ can be obtained from vorticity fieldsat the freezeout when the Λ / ¯Λ hyperons are decoupled from the rest of the hot and densematter [40, 44, 52, 64].All these models are based on the assumption that the spin degree of freedom has reachedlocal equilbrium. But this assumption is not justified. The recent disagreement betweensome theoretical models and data on the longitudinal polarization indicates that the spinsmight not be in local equilibrium [8, 48, 63]. Although one model of the chiral kinetictheory can explain the sign of the data [53], it cannot reproduce the magnitude of thedata. If the spins are not in local equilbrium, how is the polarization generated in particlecollisions? This is also related to the role of the spin-orbit coupling which is regarded as themicroscopic mechanism for the global polarization. In one particle scattering such as a 2-to-2scattering at fixed impact parameter the effect of spin-orbit coupling in the polarized crosssection is obvious [24, 45], but how does the spin-vorticity coupling naturally emerge fromthe spin-orbit one? It is far from easy and obvious as it involves the treatment of particlescatterings at different space-ime points in a system of particles in randomly distributedmomentum. To the best of our knowledge, this problem has not been seriously investigateddue to such a difficulty. In this paper we will construct a microscopic model for the globalpolarization based on the spin-orbit coupling. We will show that the spin-vorticity couplingnaturally emerges from scatterings of particles at different space-time points incorporatingpolarized scattering amplitudes with the spin-orbit coupling. This provides a microscopicmechanism for the global polarization from the first principle through particle collisions innon-equilibrium.The paper is organized as follows. In Section II we will introduce scatterings of twowave packets for spin-0 particles. The wave packet method is necessary to describe particlescatterings at different space-time points. In Section III we will study collisions of spin-0particles as wave packets which take place at different space-time in a multi-particle sys-tem. In Section IV we will derive the polarization rate for spin-1/2 particles from particlecollisions. As an example, we will apply in Section V the formalism to derive the quarkpolarization rate in a quark-gluon plasma in local equilibrium in momentum. In Section VIwe will discuss the numerical method to calculate the quark polarization rate, a challengingtask to deal with collision integrals in very high dimensions. We will present the numericalresults in Section VII. Finally we will give a summary of the work and an outlook for futurestudies.Throughout the paper we use natural units (cid:126) = c = k B = 1 . The convention for themetric tensor is g µν = diag(+1 , − , − , − . We also use the notation a µ b µ ≡ a · b for thescalar product of two four-vectors a µ , b µ and a · b for the corresponding scalar product oftwo spatial vectors a , b . The direction of a three-vector a is denoted as ˆ a . Sometimes wedenote the components of a three-vector by indices (1 , , or ( x, y, z ) . II. SCATTERINGS OF WAVE PACKETS FOR SPIN-0 PARTICLES
In this section we will consider the scattering process A + B → · · · + n , wherethe incident particles A and B in the remote past are localized in some region and canbe described by wave packets. The details of this section can be found in the textbookby Peskin and Schroeder [49]. The purpose of this section is to give an idea of how thewave packets displaced by an impact parameter are treated in the scattering process, and toprovide the basis for the discussion in the next section. We work in the frame in which thecentral momenta of two wave packets are collinear or in the same direction which we denoteas the longitudinal direction. We assume that the wave packet B is displaced by an impactparameter vector b in the transverse direction, so the in state can be written as | φ A φ B (cid:105) in = ˆ d k A (2 π ) d k B (2 π ) φ A ( k A ) φ B ( k B ) e − i k B · b √ E A E B | k A k B (cid:105) in . (1)Here we see that the incident particles are treated as two wave packets | φ A (cid:105) and | φ B (cid:105) definedin Appendix A. The definition of the single particle states | k A (cid:105) and | k B (cid:105) can also be found inAppendix A. As we have mentioned that the amplitudes φ i ( k i ) center at p i = (0 , , p iz ) for i = A, B . We assume that the out state is a pure momentum state | p p · · · p n (cid:105) out in the farfuture. This is physically reasonable as long as the detectors of final-state particles mainlymeasure momentum or they do not resolve positions at the level of de Broglie wavelengths.Taking into account the normalization factors for the in-state and out-state, the scatteringprobability is given by P ( AB → · · · n ) = (cid:88) p (cid:88) p · · · (cid:88) p n | out (cid:104) p p · · · p n | φ A φ B (cid:105) in | (cid:81) nf =1 (cid:104) p f | p f (cid:105)(cid:104) φ A | φ A (cid:105)(cid:104) φ B | φ B (cid:105) = (cid:32) n (cid:89) f =1 ˆ Ω d p f (2 π ) (cid:33) | out (cid:104) p p · · · p n | φ A φ B (cid:105) in | (cid:81) nf =1 (2 E f Ω)= (cid:32) n (cid:89) f =1 ˆ d p f (2 π ) E f (cid:33) | out (cid:104) p p · · · p n | φ A φ B (cid:105) in | , (2)where the normalization of single particle states and wave packets is given in Appendix A.Since P ( AB → · · · n ) depends on the impact parameter b , we can rewrite it as P ( b ) .This probability gives the differential cross section at the impact parameter b , dσd b = P ( b ) . (3)The total cross section is then an integral over the impact parameter σ = ˆ d b P ( b )= (cid:32) n (cid:89) f =1 ˆ d p f (2 π ) E f (cid:33) (cid:89) i = A,B ˆ d k i (2 π ) φ i ( k i ) √ E i ˆ d k (cid:48) i (2 π ) φ ∗ i ( k (cid:48) i ) (cid:112) E (cid:48) i × ˆ d be i ( k (cid:48) B − k B ) · b ( out (cid:104){ p f }|{ k i }(cid:105) in ) ( out (cid:104){ p f }|{ k (cid:48) i }(cid:105) in ) ∗ = (cid:32) n (cid:89) f =1 ˆ d p f (2 π ) E f (cid:33) (cid:32) (cid:89) i = A,B ˆ d k i (2 π ) φ i ( k i ) √ E ki ˆ d k (cid:48) i (2 π ) φ ∗ i ( k (cid:48) i ) (cid:112) E (cid:48) ki (cid:33) (2 π ) δ (2) (cid:0) k (cid:48) B, ⊥ − k B, ⊥ (cid:1) × (2 π ) δ (4) ( k (cid:48) A + k (cid:48) B − n (cid:88) f =1 p f )(2 π ) δ (4) ( k A + k B − n (cid:88) f =1 p f ) ×M ( { k A , k B } → { p , p , · · · , p n } ) M ∗ ( { k (cid:48) A , k (cid:48) B } → { p , p , · · · , p n } ) , (4)where E ki = (cid:112) | k i | + m i , E (cid:48) ki = (cid:112) | k (cid:48) i | + m i with i = A, B , k B, ⊥ denotes the trans-verse part of the momentum, M denotes the invariant amplitude of the scattering pro-cess. We can integrate out six delta functions involving k (cid:48) A and k (cid:48) B , i.e. δ (2) (cid:0) k (cid:48) B, ⊥ − k B, ⊥ (cid:1) and δ (4) (cid:16) k (cid:48) A + k (cid:48) B − (cid:80) nf =1 p f (cid:17) . By integrating over k (cid:48) B, ⊥ to remove δ (2) (cid:0) k (cid:48) B, ⊥ − k B, ⊥ (cid:1) ,we can replace k (cid:48) B, ⊥ by k B, ⊥ in the integrand. By integrating over k (cid:48) A, ⊥ to remove δ (2) (cid:16) k (cid:48) A, ⊥ + k (cid:48) B, ⊥ − (cid:80) nf =1 p f, ⊥ (cid:17) , we can replace k (cid:48) A, ⊥ by − k B, ⊥ + (cid:80) nf =1 k f, ⊥ in the inte-grand. Then we can integrate over k (cid:48) B,z to remove δ ( k (cid:48) A,z + k (cid:48) B,z − p ,z − p ,z ) , in which k (cid:48) B,z is replaced by (cid:80) nf =1 p f,z − k (cid:48) A,z . The last variable that can be integrated over is k (cid:48) A,z in thedelta function for the energy conservation δ ( E (cid:48) A + E (cid:48) B − E p − E p ) . We can solve k (cid:48) A,z as theroot of the equation E (cid:48) A + E (cid:48) B = E p + E p . Note that E (cid:48) A and E (cid:48) B are given by E (cid:48) A = (cid:118)(cid:117)(cid:117)(cid:116) ( − k B, ⊥ + n (cid:88) f =1 k f, ⊥ ) + k (cid:48) A,z + m A ,E (cid:48) B = (cid:118)(cid:117)(cid:117)(cid:116) k B, ⊥ + ( n (cid:88) f =1 p f,z − k (cid:48) A,z ) + m B . (5)The delta function can be rewritten as δ (cid:32) E (cid:48) A + E (cid:48) B − n (cid:88) f =1 E f (cid:33) = (cid:88) j (cid:12)(cid:12)(cid:12)(cid:12) k (cid:48) A,z,j E (cid:48) A − k (cid:48) B,z,j E (cid:48) B (cid:12)(cid:12)(cid:12)(cid:12) − δ ( k (cid:48) A,z − k (cid:48) A,z,j ) , (6)where k (cid:48) A,z,j are the roots of the equation E (cid:48) A + E (cid:48) B = E p + E p .If we assume that the incident wave packets are narrow in momentum and centered atmomenta p A and p B , i.e. φ i ( k i ) are close to delta functions δ ( k i − p i ) , we can approximate ( E (cid:48) kA , k (cid:48) A ) ≈ ( E kA , k A ) ≈ ( E A , p A ) and ( E (cid:48) B , k (cid:48) B ) ≈ ( E kB , k B ) ≈ ( E B , p B ) . We can alsoapproximate v i = p i,z /E i ≈ k (cid:48) i,z /E (cid:48) i with i = A, B . Then we obtain σ ≈ (cid:32) n (cid:89) f =1 ˆ d p f (2 π ) E f (cid:33) ˆ d k A (2 π ) | φ A ( k A ) | E A ˆ d k B (2 π ) | φ B ( k B ) | E B | v A − v B | − × (2 π ) δ ( p A + p B − n (cid:88) f =1 p f ) |M ( { p i } → { p f } ) | = 14 E A E B | v A − v B | (cid:32) n (cid:89) f =1 ˆ d p f (2 π ) E f (cid:33) × (2 π ) δ ( p A + p B − n (cid:88) f =1 p f ) |M ( { p i } → { p f } ) | . (7)Here we have used the normalization condition for the wave amplitude (A9). We notethat the above formula is derived in the frame in which incident particles are collinear inmomemtum. We can boost the frame to the center-of-mass frame of the incident particlesand the cross section is invariant.If the number densities of A and B in coordinate space are n A and n B respectively, thecollision rate, i.e. the number of scatterings per unit time and unit volume is given by R = n A n B | v A − v B | σ = n A n B E A E B E A E B | v A − v B | σ, (8)where we have rewritten the rate in a Lorentz invariant way by making use of the fact that E A E B | v A − v B | , n A /E A and n B /E B are Lorentz invariant along the collision axis. III. COLLISION RATE FOR SPIN-0 PARTICLES IN A MULTI-PARTICLESYSTEM
In this section we will derive the collision rate in a system of spin-0 particles of multi-species. We will generalize the result of the previous section by treating the incident particlesas wave packets. The emphasis is put on the collision of two particles at two different space-time points.We will frequently use two frames in this and the next section: the lab frame and thecenter-of-mass system (CMS) of one specific collision. In the lab frame, the movementof one species of particles follows their phase space distribution f ( x, p ) . There are manycollisions taking place in the system. Figure 1 shows one collision of two incident particlesat x A = ( t A , x A ) and x B = ( t B , x B ) in the lab frame and CMS. We see that p A and p B arenot aligned in the same direction in the lab frame. When boosted to the CMS of this collisionwith the boost velocity determined by v bst = ( p A + p B ) / ( E A + E B ) , we have p c,A + p c,B = 0 as shown in the right panel of Fig. 1, see Appendix C for more details of such a Lorentztransformation. Hereafter we denote the quantities in the CMS by the index ’c’. There is aninherent problem in the collision of incident particles located at different space-time points:the collision time is not well defined. If we assume that the collision takes place at the sametime in the lab frame, i.e. t A = t B , after being boosted to the CMS, the time will be mis-matched, i.e. t c,A (cid:54) = t c,B , since x A and x B are different. The reverse statement is also true:if t c,A = t c,B then t A (cid:54) = t B due to x c,A (cid:54) = x c,B . Such an ambiguity in the collision time cannotbe avoided but can be constrained by the requirement that the difference ∆ t c = t c,A − t c,B should not be large, otherwise the incident particles are irrelevant or the collision is un-causalin the CMS. In the calculation of this paper, we will put a simple constraint ∆ t c = 0 . Inthe right panel of Fig. 1, we also see that the impact parameter b is given by the distanceof x c,A and x c,B in the transverse direction which is perpendicular to p c,A or p c,B . In thelongitudinal direction or the direction of p c,A or p c,B , two space points are also different ingeneral, i.e. ˆ p c,A · x c,A (cid:54) = ˆ p c,A · x c,B . In the calculation we also require that the distancebetween two space points in the longitudinal direction, ∆ x c,L = ˆ p c,A · ( x c,A − x c,B ) , shouldnot be large, otherwise the incident particles as wave packets lose coherence and cannotinteract in the CMS. In the calculation, we will also put a simple constraint ∆ x c,L = 0 . TheCMS constraint ∆ t c = 0 and ∆ x c,L = 0 is equivalent to the condition ∆ t = v bst · ∆ x and ( v A − v B ) · ∆ x = 0 in the lab frame, see Appendix C for the derivation. Figure 1: A collision or scattering in the Lab frame (left) and center-of-mass frame (right).
Since we will work in the CMS of incident particles in each collision, for notationalsimplicity, we will suppress the index ’c’ (standing for the CMS) of all quantities in the restpart of this section. So all quantities are implied in the CMS if not explicitly stated here.We know that the momentum integral of the distribution function gives the numberdensity in the coordinate space. Similar to Eq. (8), the collision rate in correspondingmomentum and space-time intervals can be written as R AB → = d p A (2 π ) d p B (2 π ) f A ( x A , p A ) f B ( x B , p B ) | v A − v B | ∆ σ, (9)where v A = | p A | /E A and v B = −| p B | /E B are the longitudinal velocities with p A = − p B in the CMS, f A and f B are the phase space distributions for the incident particle A and B respectively, and ∆ σ denotes the infinitesimal element of the cross section given by ∆ σ = 1 C AB d x A d x B δ (∆ t ) δ (∆ x L ) × d p (2 π ) E d p (2 π ) E E A )(2 E B ) K. (10)Here we have assumed that the scattering takes place at the same time and the same longi-tudinal position in the CMS, so we put two delta functions to implement these constraints.The constant C AB is to make ∆ σ have the right dimension of the cross section and will bedefined later. In Eq. (10) K is given by K = (2 E A )(2 E B ) | out (cid:104) p p | φ A ( x A , p A ) φ B ( x B , p B ) (cid:105) in | = 4 E A E B (2 π ) G G ˆ d k A d k B d k (cid:48) A d k (cid:48) B × φ A ( k A − p A ) φ B ( k B − p B ) φ ∗ A ( k (cid:48) A − p A ) φ ∗ B ( k (cid:48) B − p B ) (cid:112) E A,k E B,k E A,k (cid:48) E B,k (cid:48) × exp ( − i k A · x A − i k B · x B + i k (cid:48) A · x A + i k (cid:48) B · x B ) × (2 π ) δ (4) ( k (cid:48) A + k (cid:48) B − p − p )(2 π ) δ (4) ( k A + k B − p − p ) ×M ( { k A , k B } → { p , p } ) M ∗ ( { k (cid:48) A , k (cid:48) B } → { p , p } ) , (11)where φ i ( k i − p i ) and φ i ( k (cid:48) i − p i ) for i = A, B denote the incident wave packet amplitudescentered at p i , E i,k = (cid:112) | k i | + m i , E i,k (cid:48) = (cid:112) | k (cid:48) i | + m i and E i = (cid:112) | p i | + m i are energiesfor i = A, B . In Eq. (11) G i ( i = 1 , denote distribution factors depending on particletypes in the final state, we have G i = 1 for the Boltzmann particles and G i = 1 ± f i ( p i ) forbosons (upper sign) and fermions (lower sign). Note that f i ( p i ) can be in any other formin non-equilibrium cases. In (11) we have taken the following form for | φ i ( x i , p i ) (cid:105) in with i = A, B , | φ i ( x i , p i ) (cid:105) in = ˆ d k i (2 π ) (cid:112) E i,k φ i ( k i − p i ) e − i k i · x i | k i (cid:105) in . (12)Here we take the Gaussian form for the wave packet amplitude φ i ( k i − p i ) as in (A10), φ i ( k i − p i ) = (8 π ) / α / i exp (cid:20) − ( k i − p i ) α i (cid:21) , (13)where α i denote the width parameters of the wave packet A or B . For simplicity we will setequal width for two incident particles (even for different species), α A = α B = α .We can also make the approximation of narrow wave packets, so we have | k i | ≈ | k (cid:48) i | ≈ | p i | for i = A, B and then (cid:113) E A,k E (cid:48) A,k ≈ E A and (cid:113) E B,k E (cid:48) B,k ≈ E B , and the energy factors in(11) drop out. By taking the integral over x A and x B and then the integral over on-shellmomenta p A , p B , p and p , we obtain the scattering or collision rate per unit volume, R AB → = ˆ d p A (2 π ) E A d p B (2 π ) E B d p (2 π ) E d p (2 π ) E × C AB ˆ d x A d x B δ (∆ t ) δ (∆ x L ) × f A ( x A , p A ) f B ( x B , p B ) G G | v A − v B | K. (14)0Now we use new variables to replace x A and x B , X = 12 ( x A + x B ) ,y = x A − x B . (15)We can rewrite the integral over x A and x B in Eq. (14) as I = ˆ d x A d x B δ (∆ t ) δ (∆ x L ) f A ( x A , p A ) f B ( x B , p B ) × exp ( − i k A · x A − i k B · x B + i k (cid:48) A · x A + i k (cid:48) B · x B ) ≈ ˆ d Xd b f A (cid:16) X + y T , p A (cid:17) f B (cid:16) X − y T , p B (cid:17) × exp [ i ( k (cid:48) A − k A ) · b ] , (16)where we have used k A + k B − k (cid:48) A − k (cid:48) B = 0 and − k A + k B + k (cid:48) A − k (cid:48) B = 2( k (cid:48) A − k A ) implied bytwo delta functions in Eq. (11). In Eq. (16) we have integrated over y = ∆ t = t A − t B and y L = ∆ x L = ˆ p A · ( x A − x B ) to remove two detla functions, then we are left with the integralover the transverse part y µT = (0 , b ) with b being in the transverse direction. Because wework in the CMS in which all kinematic variables depend on the incident momenta in thelab frame, the impact parameter b in the CMS depends on ( x A , x B ) as well as ( p A , p B ) inthe lab frame through a boost velocity.Now we define the constant C AB in (9,14) as C AB ≡ ´ d X = t X Ω int so that the finalresults have the right dimension. Here t X and Ω int are the local time and space volume forthe interaction respectively. Note that C − AB ´ d X ( · · · ) plays the role of the average over X or (cid:104) ( · · · ) (cid:105) X . If we take the limit t X Ω int → , we obtain the local rate per unit volume fromEq. (14), d N AB → dX = 1(2 π ) ˆ d p A (2 π ) E A d p B (2 π ) E B d p (2 π ) E d p (2 π ) E ×| v A − v B | G G ˆ d k A d k B d k (cid:48) A d k (cid:48) B × φ A ( k A − p A ) φ B ( k B − p B ) φ ∗ A ( k (cid:48) A − p A ) φ ∗ B ( k (cid:48) B − p B ) × δ (4) ( k (cid:48) A + k (cid:48) B − p − p ) δ (4) ( k A + k B − p − p ) ×M ( { k A , k B } → { p , p } ) M ∗ ( { k (cid:48) A , k (cid:48) B } → { p , p } ) × ˆ d b f A (cid:16) X + y T , p A (cid:17) f B (cid:16) X − y T , p B (cid:17) exp [ i ( k (cid:48) A − k A ) · b ] , (17)where N AB → is the number of scatterings. We emphasize again that all quantities in Eq.(17) are defined in the CMS of two incident particles (we have suppressed the index ’c’).1 IV. POLARIZATION RATE FOR SPIN-1/2 PARTICLES FROM COLLISIONS
In this section we will generalize the previous section for spin-0 particles to spin-1/2ones. Our purpose is to derive the polarization rate from collisions in a system of particlesof multi-species. We assume that particle distributions in phase space are independent ofspin states, so the spin dependence comes only from scatterings of particles carrying thespin degree of freedom.As a simple example to illustrate the idea of the polarization arising from collisions, weconsider a fluid with the three-vector fluid velocity in the z direction v z that depends on x , which we denote as v z ( x ) . We assume dv z ( x ) /dx > . In the comoving frame of anyfluid cell in the range [ x − ∆ x/ , x + ∆ x/ where ∆ x is a small distance, the fluid velocityat x ± ∆ x/ is ± ( dv z ( x ) /dx )∆ x , forming a rotation or local orbital angular momentum(OAM) pointing to the − y direction. Due to the spin-orbit coupling, the scattering of twounpolarized particles with velocity ± ( dv z ( x ) /dx )∆ x and impact parameter ∆ x will polarizethe particles in the final state along the direction of the local OAM. It has been provedthat the polarization cross section is proportional to s · n c , where s is the spin quantization(polarization) direction and n c = ˆ b c × ˆ p c is the direction of the reaction plane (the localOAM) in the CMS of the scattering, where ˆ b c and ˆ p c are the direction of the impactparameter and the incident momentum respectively. This is what happens in one scattering.In a thermal system with collective motion, there are many scatterings whose reaction planespoint to almost random directions, but in average the direction of the reaction plane pointsto that of the local rotation or vorticity. To calculate the polarization in a thermal systemwith collective motion, we have to take a convolution of distribution functions and polarizedscattering amplitudes similar to (17).In this section we will distinguish quantities in the CMS and lab frame, i.e. we will resumethe subscript ’c’ for all CMS quantities, while quantities in the lab frame do not have thesubscript ’c’.Now we consider a scattering process A + B → where the incident and outgoingparticles are in the spin state labeled by s A , s B , s and s ( s i = ± / , i = A, B, , )respectively. The quantization direction of the spin state is chosen to be along the directionof the reaction plane in the CMS of the scattering. The polarization rate per unit volume2for particle 2 in the final state is given by d P AB → ( X ) dX = 1(2 π ) ˆ d p c,A (2 π ) E c,A d p c,B (2 π ) E c,B d p c, (2 π ) E c, d p c, (2 π ) E c, ×| v c,A − v c,B | G G ˆ d k c,A d k c,B d k (cid:48) c,A d k (cid:48) c,B × φ A ( k c,A − p c,A ) φ B ( k c,B − p c,B ) φ ∗ A ( k (cid:48) c,A − p c,A ) φ ∗ B ( k (cid:48) c,B − p c,B ) × δ (4) ( k (cid:48) c,A + k (cid:48) c,B − p c, − p c, ) δ (4) ( k c,A + k c,B − p c, − p c, ) × ˆ d b c f A (cid:16) X c + y c,T , p c,A (cid:17) f B (cid:16) X c − y c,T , p c,B (cid:17) exp (cid:2) i ( k (cid:48) c,A − k c,A ) · b c (cid:3) × (cid:88) s A ,s B ,s ,s s n c M ( { s A , k c,A ; s B , k c,B } → { s , p c, ; s , p c, } ) ×M ∗ (cid:0) { s A , k (cid:48) c,A ; s B , k (cid:48) c,B } → { s , p c, ; s , p c, } (cid:1) , (18)where P AB → denotes the polarization vector and n c = ˆ b c × ˆ p c,A is the direction of thereaction plane in the CMS of the scattering which is also the quantization direction of thespin. In the second to the last line of Eq. (18), the summation of s M ( · · · , s ) M ∗ ( · · · , s ) over s = ± / gives the polarized amplitude squared for particle 2 in the final state, andthe factor 2 arises from the normalization convention for the polarization that makes it inthe range [ − , instead of [ − / , / . Equation (18) is one of our main results. V. QUARK/ANTIQUARK POLARIZATION RATE IN A QUARK-GLUONPLASMA OF LOCAL EQUILIBRIUM IN MOMENTUM
In this section we will calculate the quark/antiquark polarization rate from all 2-to-2parton (quark or gluon) collisions in a quark-gluon plasma (QGP) of local equilibrium inmomentum but not in spin. We assume that the QGP is a multi-component fluid with thesame fluid velocity u ( x ) as a function of space-time for all partons. The partons in a fluid cellfollow a thermal distribution in momentum in its comoving frame with the local temperature T ( x ) . We assume that the phase space distribution f ( x, p ) depends on x µ = ( t, x ) throughthe fluid velocity u µ ( x ) in the form f ( x, p ) = f [ β ( x ) p · u ( x )] where p µ = ( E p , p ) is an on-shellfour-momentum of the parton and β ( x ) ≡ /T ( x ) .We consider the scattering, A + B → , where A and B denote two incident partonsin the wave packet form localized at x A and x B respectively, and ’1’ and ’2’ denote twooutgoing partons in momentum states. In order to calculate the polarization rate from thecollision of two wave packets displaced by an impact parameter by Eq. (18), we must work3in the CMS of the incident partons for each collision. Note that many collisions take placein the system at different space-time, the CMS of each collision depends on the momentaof incident partons which vary from collision to collision. In one collision, the phase spacedistributions for incident partons (denoted as i = A, B ) can be written in the form f i ( x c , p c ) = f i [ β ( x c ) p c · u c ( x c )]= f i [ β ( x ) p · u ( x )]= f i ( x, p ) , (19)where x, p are the space-time and momentum in the lab frame respectively, while x c , p c aretheir corresponding values in the CMS of A and B in this collision which depend on p A and p B in the heat bath (lab frame) through the boost velocity, and u µc ( x c ) denotes the fluidvelocity in the CMS as a function of the space-time in the CMS. A. Polarization rate
We now apply Eq. (18) to 2-to-2 parton scatterings. For simplicity we assume thatthe phase space distributions of incident partons follow the Boltzmann distribution, i.e. f ( x, p ) = exp[ − β ( x ) p · u ( x )] , so we have G G = 1 in (18). Also we assume that y c,T issmall compared with X c so that we can make an expansion in y c,T for the distributions, thedetails are given in Appendix B. The relevant contribution in the linear or first order in y c,T involves the term y µc,T [ ∂ ( βu c,ρ ) /∂X µc ] p ρc,A which can be rewritten as y µc,T p ρc,A ∂ ( βu ρ ) ∂X µc = − L µρ ( c ) ω ( c ) µρ + 14 y { µc,T p ρ } c,A (cid:20) ∂ ( βu c,ρ ) ∂X µc + ∂ ( βu c,µ ) ∂X ρc (cid:21) , (20)where L µρ ( c ) ≡ y [ µc,T p ρ ] c,A is the OAM tensor, ω ( c ) µρ ≡ − (1 / ∂ X c µ ( βu c,ρ ) − ∂ X c ρ ( βu c,µ )] is thethermal vorticity tensor, and y { µc,T p ρ } c,A ≡ y µc,T p ρc,A + y ρc,T p µc,A , all in the CMS. The derivation ofEq. (20) is given in Eq. (B2). Note that the OAM-vorticity coupling L µρ ( c ) ω ( c ) µρ shows up inthe y c,T expansion, which can be converted to the spin-vorticity coupling through polarizedparton scattering amplitudes encoding the spin-orbit coupling effect, as we will show shortly.The second term in Eq. (20) invloves the symmetric part of the thermal velocity derivativesin space-time, which is assumed to vanish in thermal equilibrium for the spin, known as theKilling condition [5–7, 22]. In this paper, however, we do not assume the thermal equilibriumfor the spin degree of freedom, so we keep this symmetric term in the calculation.4Keeping the first order term in the y c,T expansion and neglecting the zeroth order termwhich is irrelevant, Eq. (18) can be simplified as d P AB → ( X ) dX = − π ) ˆ d p A (2 π ) E A d p B (2 π ) E B d p c, (2 π ) E c, d p c, (2 π ) E c, ×| v c,A − v c,B | ˆ d k c,A d k c,B d k (cid:48) c,A d k (cid:48) c,B × φ A ( k c,A − p c,A ) φ B ( k c,B − p c,B ) φ ∗ A ( k (cid:48) c,A − p c,A ) φ ∗ B ( k (cid:48) c,B − p c,B ) × δ (4) ( k (cid:48) c,A + k (cid:48) c,B − p c, − p c, ) δ (4) ( k c,A + k c,B − p c, − p c, ) × ˆ d b c exp (cid:2) i ( k (cid:48) c,A − k c,A ) · b c (cid:3) b c,j [Λ − ] νj ∂ ( βu ρ ) ∂X ν × [ p ρA − p ρB ] f A ( X, p A ) f B ( X, p B ) ∆ I AB → M n c , (21)where we have used d p c,i /E c,i = d p i /E i for i = A, B , the Lorentz transformation matrixis defined by ∂X ν /∂X µc = [Λ − ] νµ = Λ νµ , the minus sign in the right-hand side comes from df i ( X, p i ) /d ( βu · p i ) for i = A, B , and ∆ I AB → M is defined by ∆ I AB → M = (cid:88) s A ,s B ,s ,s (cid:88) color s M ( { s A , k c,A ; s B , k c,B } → { s , p c, ; s , p c, } ) ×M ∗ (cid:0) { s A , k (cid:48) c,A ; s B , k (cid:48) c,B } → { s , p c, ; s , p c, } (cid:1) , (22)where the factor 2 arises from the normalization convention for the polarization. Note thatin the above formula there is a sum over color degrees of freedom of all incident and outgoingpartons. We may write ∆ I AB → M n c as ∆ I AB → M n c = ∆ I AB → M (ˆ b c × ˆ p c,A )= i (ˆ b c · I c ) e c,i (cid:15) ikh ˆ b c,k ˆ p hc,A = i e c,i (cid:15) ikh ˆ p hc,A I c,l ˆ b c,l ˆ b c,k , (23)where e c,i ( i = x, y, z ) are the basis vectors in the CMS, and ∆ I AB → M can be put into theform i ˆ b c · I c , in this way we can single out the direction ˆ b c out of ∆ I AB → M , see Eq. (40) foran example of what I c looks like.Substituting Eq. (23) into Eq. (21), completing the integration over b c , and removingdelta functions by integration, we obtain5 d P AB → ( X ) dX = π (2 π ) ∂ ( βu ρ ) ∂X ν ˆ d p A (2 π ) E A d p B (2 π ) E B ×| v c,A − v c,B | [Λ − ] νj e c,i (cid:15) ikh ˆ p hc,A × f A ( X, p A ) f B ( X, p B ) ( p ρA − p ρB ) × ˆ d p c, (2 π ) E c, d p c, (2 π ) E c, d k Tc,A d k (cid:48) Tc,A × (cid:88) j ,j =1 , | Ja( k Lc,A ( j )) | · | Ja( k (cid:48) Lc,A ( j )) |× φ A ( k c,A − p c,A ) φ B ( k c,B − p c,B ) φ ∗ A ( k (cid:48) c,A − p c,A ) φ ∗ B ( k (cid:48) c,B − p c,B ) × I c,l a (cid:2) Q Ljkl (cid:0) − J ( w ) + w J ( w ) + w J ( w ) (cid:1) + Q Tjkl (2 − J ( w ) − w J ( w )) (cid:3) . (24)Here we have used Q Ljkl = a l a j a k a ,Q Tjkl = 1 a (cid:0) a a k δ lj + a a l δ jk + a a j δ lk − a l a j a k (cid:1) , (25)with a ≡ k (cid:48) c,A − k c,A and a = | a | , w = ab with b being the upper limit or cutoff of b c , J i for i = 0 , , are Bessel functions, k c,B = p c, + p c, − k c,A , k (cid:48) c,B = p c, + p c, − k (cid:48) c,A , Ja( k Lc,A ) and Ja( k (cid:48) Lc,A ) are Jacobians for the longitudinal momenta k Lc,A and k (cid:48) Lc,A and are given by
Ja( k Lc,A ) = k Lc,A (cid:18) E c,A + 1 E c,B (cid:19) − E c,B ( p Lc, + p Lc, ) , Ja( k (cid:48) Lc,A ) = k (cid:48) Lc,A (cid:32) E (cid:48) c,A + 1 E (cid:48) c,B (cid:33) − E (cid:48) c,B ( p Lc, + p Lc, ) , (26)and k Lc,A ( j ) and k (cid:48) Lc,A ( j ) with j , j = 1 , are two roots of the energy conservation equation E c,A + E c,B − E c, − E c, = 0 and E (cid:48) c,A + E (cid:48) c,B − E c, − E c, = 0 respectively. In (24) and(25) Latin indices label spatial components in the the CMS. The derivation of (24) is givenin Appendix D.In a system of gluons and quarks with multi-flavors, there are many 2-to-2 parton scat-terings with at least one quark in the final state. The quark polarization rate for a specificflavor reads d P q ( X ) dX = (cid:88) A,B, { q a , ¯ q a ,g } d P AB → q ( X ) dX , (27)where d P AB → q ( X ) /dX is given by Eq. (24), and 2-to-2 parton scatterings are listed inTable I. The antiquark polarization rate can be similarly obtained.6 B. Polarized amplitudes for quarks/antiquarks in 2-to-2 parton scatterings
In this subsection we will derive the polarized amplitudes for quarks in 2-to-2 partonscatterings. The Feynman diagrams of all 2-to-2 parton scatterings at the tree level withat least one quark in the final state are shown in Table I. For anti-quark polarization, wecan make particle-antiparticle transformation in all processes listed in Table I, for example, q a q b → q a q b becomes ¯ q a ¯ q b → ¯ q a ¯ q b , ¯ q a q b → ¯ q a q b becomes q a ¯ q b → q a ¯ q b , gg → ¯ q a q a becomes gg → q a ¯ q a , etc.. In this subsection, we discuss polarized amplitudes for quarks, those forantiquarks can be easily obtained.In order to obtain the quark polarization, we sum over the spin states of all partonsin the scattering except one quark in the final state. For simplicity of the calculation, weassume that the quark masses are equal for all flavors and the external gluon is massless.We introduce a small mass in the gluon propagator in the t-channel to regulate the possibledivergence.In this subsection, all variables are defined in the CMS, for notational simplicity we willsuppress the subscript ’c’, for example, p A actually means p cA .7 Table I: The Feynman diagrams of all 2-to-2 parton scatterings at the tree level with at least onequark in the final state. We calculate the polarization of the quark (the second parton) in the finalstate. Here a and b denote the quark flavor, s i = ± / ( i = A, B, , ) denote the spin states, k i ( i = A, B, , ) denote the momenta, q, q , q , q denote the momenta in propagators. The processesfor antiquark polarization can be obtained by making a particle-antiparticle transformation. q a q b → q a q b ¯ q a q b → ¯ q a q b ¯ q a q a → ¯ q a q a q a q a → q a q a gg → ¯ q a q a gq a → gq a ¯ q a q a → ¯ q b q b We take the quark-quark scattering q a q b → q a q b with a (cid:54) = b (different flavor) as an example8to demonstrate how to derive the polarized scattering amplitude which depends on the spinstate of the quark in the final state. The Feynman diagram of this process is shown in TableI. The spin-momentum configurations are shown in the diagram. We can then write downthe corresponding amplitudes following the Feynman rule I = − i M ( { s A , k A ; s B , k B } → { s , p ; s , p } )= ig s t cji t clk q [¯ u ( s , p ) γ µ u ( s A , k A )][¯ u ( s , p ) γ µ u ( s B , k B )] ,I = − i M ( { s A , k (cid:48) A ; s B , k (cid:48) B } → { s , p ; s , p } )= ig s t dji t dlk q (cid:48) [¯ u ( s , p ) γ ν u ( s A , k (cid:48) A )][¯ u ( s , p ) γ ν u ( s B , k (cid:48) B )] . (28)where g s is the strong coupling constant, i, j, k, l = 1 , , denote the fundamental colors ofquarks, c, d = 1 , · · · , denote the adjoint colors of gluons, t c and t d are generators of SU ( N c ) in fundamental representation satisfying [ t a , t b ] = if abc t c , q = k A − p , and q (cid:48) = k (cid:48) A − p . Weobtain the product I I ∗ as I q a q b → q a q b M ( s ) = (cid:88) s A ,s B ,s (cid:88) i,j,k,l M ( { s A , k A ; s B , k B } → { s , p ; s , p } ) ×M ∗ ( { s A , k (cid:48) A ; s B , k (cid:48) B } → { s , p ; s , p } )= C q a q b → q a q b g s m q q (cid:48) × Tr (cid:104) ( p · γ + m ) γ µ Λ / ( − k A )( γ + 1)Λ − / ( − k (cid:48) A ) γ ν (cid:105) × Tr (cid:104) Π( s , n )( p · γ + m ) γ µ Λ / ( − k B )( γ + 1)Λ − / ( − k (cid:48) B ) γ ν (cid:105) . (29)In Eq. (29) we have used the notation p · γ ≡ p ρ γ ρ , a sum over all spins except s and overall colors of quarks and gluons have been taken, and C q a q b → q a q b is the color factor for thisprocess given in Table II. In the last two lines of Eq. (29), Λ / and Λ − / are the Lorentztransformation matrices for spinors defined in Eq. (E10), Π( s , n ) = (1 + s γ n σ γ σ ) / isthe spin projector where n σ = (0 , n ) is the spin quantization four-vector in the CMS with n = ˆ b × ˆ p A , and we have applied Eq. (E13) and Eq. (E18). From Eq. (29), we obtain the9difference of I q a q b → q a q b M between the spin state s = 1 / and s = − / for q b , ∆ I q a q b → q a q b M = I q a q b → q a q b M ( s = 1 / − I q a q b → q a q b M ( s = − / C q a q b → q a q b g s m q q (cid:48) × Tr (cid:104) ( p · γ + m ) γ µ Λ / ( − k A )( γ + 1)Λ − / ( − k (cid:48) A ) γ ν (cid:105) × Tr (cid:104) γ ( n · γ )( p · γ + m ) γ µ Λ / ( − k B )( γ + 1)Λ − / ( − k (cid:48) B ) γ ν (cid:105) . (30)The expansion of ∆ I q a q b → q a q b M gives about 200 terms. In accordance with Eq. (E10), Λ / ( p ) depends on the repidity η p and the momentum direction ˆ p , where η p is related to theenergy-momentum by E p = m cosh( η p ) and | p | = m sinh( η p ) . So the contracted trace partof ∆ I q a q b → q a q b M can be expressed as a function of (ˆ k A , ˆ k (cid:48) A , ˆ k B , ˆ k (cid:48) B ) and ( η kA , η (cid:48) kA , η kB , η (cid:48) kB ) . Table II: Color factors for all 2-to-2 processes with at least one final quark. The constants whichappear in color factors are: d F = N c , d A = N c − , C F = ( N c − / (2 N c ) , and C A = 3 with N c = 3 .color factors Color factors in scattering processes d F C F /d A C q a q b → q a q b , C ¯ q a q b → ¯ q a q b , C (1)¯ q a q a → ¯ q a q a , C (1) q a q a → q a q a , C ¯ q a q a → ¯ q b q b d F C F C (1) gg → ¯ q a q a , C (3) gq a → gq a ( C F − C A / d F C F C (2)¯ q a q a → ¯ q a q a , C (2) q a q a → q a q a , C (2) gg → ¯ q a q a , C (4) gq a → gq a d A C A C (2) gq a → gq a , C (3) gg → ¯ q a q a d F C F C A C (1) gq a → gq a , C (4) gg → ¯ q a q a The polarized amplitudes for quarks in all 2-to-2 parton scatterings listed in Table I aregiven in Appendix F, which results in more than 5000 terms. Here we give an estimate ofhow many terms there are in each process: ∆ I gg → ¯ q a q a M gives 136 terms, ∆ I gq a → gq a M gives 2442terms, ∆ I ¯ q a q a → ¯ q a q a M gives 874 terms, ∆ I ¯ q a q a → ¯ q b q b M gives 40 terms, ∆ I ¯ q a q b → ¯ q a q b M gives 210 terms, ∆ I q a q b → q a q b M gives 210 terms, ∆ I q a q a → q a q a M gives 1156 terms. It is hard to see the physicsbehind such huge number of terms unless we make an appropriate approximation. C. Evaluation of polarized amplitudes for quarks/antiquarks
The evaluation of contracted traces of quark polarized amplitudes are very complicated.This has been done with the help of FeynCalc [47, 50]. There are about terms in the0expansion of contracted traces for 2-to-2 parton scatterings.In this subsection, all variables are defined in the CMS, for notational simplicity we willsuppress the subscript ’c’ if not explicitly specified, for example, p A actually means p cA .In order to show the physics in the midst of the huge number of terms, we have tomake an appropriate approximation. As we know that the incident particles are treated aswave packets in order to describe scatterings displaced by impact parameters. A realisticapproximation is that the wave packets are assumed to be narrow, i.e. the width is muchsmaller than the center momenta of the wave packet in Eq. (13). In the extreme case thatthe width of the wave packet is zero, we recover the normal scattering of plane waves. Sincethe positions of incident particles can be anywhere in plane waves, in average the relativeOAM of two incident particles is zero, leading to the vanishing polarization of final stateparticles. This fact can be verified by setting ˆ k A = ˆ k (cid:48) A = ˆ p A , ˆ k B = ˆ k (cid:48) B = − ˆ p A , p = − p ,η A = η B = η (cid:48) A = η (cid:48) B , (31)in the trace part in Eq. (30), then we have ∆ I q a q b → q a q b M = 0 .The above result is of the zeroth order, now we turn to the first order in the deviation frommomenta in (31). We expand (ˆ k A , ˆ k (cid:48) A , ˆ k B , ˆ k (cid:48) B ) about their central values (ˆ p A , ˆ p A , − ˆ p A , − ˆ p A ) and ( η kA , η (cid:48) kA , η kB , η (cid:48) kB ) about their central values ( η pA , η pA , η pA , η pA ) to the first order in thedifferences, ˆ k A → ˆ p A + ∆ A , ˆ k B → − ˆ p A + ∆ B , ˆ k (cid:48) A → ˆ p A + ∆ (cid:48) A , ˆ k (cid:48) B → − ˆ p A + ∆ (cid:48) B ,η kA = η pA + ∆ η kA ,η (cid:48) kA = η pA + ∆ η (cid:48) kA ,η kB = η pA + ∆ η kB ,η (cid:48) kB = η pA + ∆ η (cid:48) kB , (32)where the first order quantities are denoted with ∆ (for example, ∆ A , ∆ η kA ). We also1expand ( E , p , E , p ) at ( E , p , E , − p ) , E → E + ∆ , E → E + ∆ , p → p + ∆ , p → − p + ∆ . (33)The delta functions in Eq. (21) lead to k A + k B = k (cid:48) A + k (cid:48) B = p + p . (34)So ∆ in (33) can be determined by ∆ = 12 ( k A + k B ) , (35)and p determined by p = 12 ( p − p ) . (36)Note that once p and ∆ are given, E , ∆ , ∆ satisfy ( E + ∆ ) = ( p + ∆ ) + m , ( E + ∆ ) = ( − p + ∆ ) + m . (37)So we have a freedom to choose the value of E . Then we use (32) and (33) in the contractedtrace part in Eq. (30) and expand it to the first order in ∆ -quantities. Still, the final resulthas many terms but all terms of ∆ , ∆ and ∆ cancel out.In order to further simplify the contracted trace part in Eq. (30), we use the propertythat the first order contributions do not have terms of ∆ , ∆ , ∆ by setting p = p , p = − p , (38)which leads to k A + k B = k (cid:48) A + k (cid:48) B = 0 and then ˆ k A = − ˆ k B , ˆ k (cid:48) A = − ˆ k (cid:48) B ,η kA = η kB , η (cid:48) kA = η (cid:48) kB . (39)Using (38) and (39) in the contracted trace part in Eq. (30) for q a q b → q a q b , we obtain a2shorter series of 31 terms Tr a q b → q a q b = 16 i ( n × p ) · ˆ k A × (cid:104) c A s A c (cid:48) A s (cid:48) A p · ˆ k (cid:48) A + 7 E s A c (cid:48) A s (cid:48) A ˆ k A · ˆ k (cid:48) A + 2 mc A s A c (cid:48) A − mc A s A s (cid:48) A + 4 E c A s A c (cid:48) A + E c A s A s (cid:48) A − s A s (cid:48) A p · ˆ k A (cid:105) +16 i ( n × p ) · ˆ k (cid:48) A (cid:104) mc A s A s (cid:48) A ˆ k A · ˆ k (cid:48) A − E c A s A s (cid:48) A ˆ k A · ˆ k (cid:48) A − mc A c (cid:48) A s (cid:48) A − E c A c (cid:48) A s (cid:48) A − c A s A c (cid:48) A s (cid:48) A p · ˆ k A − ms A c (cid:48) A s (cid:48) A − E s A c (cid:48) A s (cid:48) A − s A s (cid:48) A p · ˆ k (cid:48) A + 2 s A s (cid:48) A ( p · ˆ k A )(ˆ k A · ˆ k (cid:48) A ) (cid:105) +16 i ( n × ˆ k A ) · ˆ k (cid:48) A (cid:104) ms A c (cid:48) A s (cid:48) A p · ˆ k A + 8 m c A s A c (cid:48) A s (cid:48) A +4 E ms A s (cid:48) A ˆ k A · ˆ k (cid:48) A − s A s (cid:48) A ( p · p )(ˆ k A · ˆ k (cid:48) A ) − E c A s A s (cid:48) A p · ˆ k (cid:48) A − E s A c (cid:48) A s (cid:48) A p · ˆ k A − c A s A c (cid:48) A s (cid:48) A p · p − E c A s A c (cid:48) A s (cid:48) A (cid:3) +16 i ( p × ˆ k A ) · ˆ k (cid:48) A (cid:104) s A s (cid:48) A ( p · ˆ k A )( n · ˆ k (cid:48) A ) − s A s (cid:48) A ( n · ˆ k A )( p · ˆ k (cid:48) A )+ s A s (cid:48) A ( n · p )(ˆ k A · ˆ k (cid:48) A ) + 4 mc A s A s (cid:48) A n · ˆ k (cid:48) A + E s A c (cid:48) A s (cid:48) RA n · ˆ k A +3 E c A s A s (cid:48) A n · ˆ k (cid:48) A + 3 c A s A c (cid:48) A s (cid:48) A n · p (cid:105) , (40)where we denote the contracted trace part for q a q b → q a q b as Tr a q b → q a q b , c A ≡ cosh( η kA / , c (cid:48) A ≡ cosh( η (cid:48) kA / , s A ≡ sinh( η (cid:48) kA / , and s (cid:48) A ≡ sinh( η (cid:48) kA / . We see in (40) that there arefour typical terms proportional to ( n × p ) · ˆ k A , ( n × p ) · ˆ k (cid:48) A , ( n × ˆ k A ) · ˆ k (cid:48) A and ( p × ˆ k A ) · ˆ k (cid:48) A ,in which the first three terms are from the spin-orbit coupling and the last one correspondsto the non-coplanar part of p , ˆ k A and ˆ k (cid:48) A . We will show in the next section that (40) is agood approximation for the contracted trace part to the exact result.It can be proved that ∆ I AB → M for all 2-to-2 parton scatterings in Table I have the samestructure as in (40) for q a q b → q a q b under the approximation in (38,39).Note that ∆ I AB → M depends linearly on the direction of the scattering plane n = ˆ b × ˆ p A ,we can write the contracted trace part in the form of ˆ b · I , as is done in Eq. (23). We takethe term ( n × p ) · ˆ k A in (40) as an example, which can be rewritten as [(ˆ b × ˆ p A ) × p ] · ˆ k A = ˆ b · [(ˆ p A · ˆ k A ) p − (ˆ p A · p )ˆ k A ] . (41)Therefore I contains the term inside the square brackets on the right-hand side of Eq. (41).Another example is the term proportional to ( p × ˆ k A ) · ˆ k (cid:48) A , we see that all terms have factors3of the form n · V ( V = ˆ k A , ˆ k (cid:48) A , p ) inside the square brackets, these terms can be rewrittenas n · V = ˆ b · (ˆ p A × V ) , so I contains the term ˆ p A × V . VI. NUMERICAL METHOD TO CALCULATE QUARK/ANTIQUARKPOLARIZATION RATE
In this section we will calculate the polarization rate for quarks in a QGP from Eq. (24).Here we assume a local equilibrium in particle momentum but not in spin. We will considertwo cases: the approximation as in (38,39) and the exact result without any appoximation.The main parameters are set to following values: the quark mass m q = 0 . GeV for quarksof all flavors ( u, d, s, ¯ u, ¯ d, ¯ s ), the gluon mass m g = 0 for the external gluon, the internalgluon mass (Debye screening mass) m g = m D = 0 . GeV in gluon propagators in the t andu channel to regulate the possible divergence, the width α = 0 . GeV of the Gaussian wavepacket, and the temperature T = 0 . GeV.Although the 2-to-2 processes for anti-qaurk polarization are different from those forquarks, it can be shown that the polarization rate for anti-quarks is the same as that forquarks, because all 2-to-2 scatterings for anti-quark polarization can be obtained from thosein Table I by making a particle-antiparticle transformation. In the following we discussonly the quark polarization. The same discussion can also be applied to the antiquarkpolarization.The local polarization rate in Eq. (24) for quarks involves a 16-dimensional integration,which is a major challenge in the numerical calculation. In the Monte Carlo integration,the number of sample points grows exponentially with the dimension, so even a very roughcalculation in high dimensions would need huge number of sample points.To overcome this difficulty, we split the integration into two parts: a 10-dimension (10D)integration over ( p c, , p c, , k Tc,A , k (cid:48) Tc,A ) and a 6-dimension (6D) integration over ( p A , p B ) . Wecarry out the 10D integration and store the result as a function of p c,A (and p c,B = − p c,A ).Then we carry out the 6D integration using the pre-calculated 10D integral.The 10D integral, the last five lines of Eq. (24), depends on p c,A and p c,B = − p c,A whichappear in the wave packet function φ A and φ B respectively. So we denote the 10D integralas Θ jk ( p c,A ) , from Eq. (27) the polarization rate per unit volume for one quark flavor can4be rewritten as d P q ( X ) dX = π (2 π ) ∂ ( βu ρ ) ∂X ν (cid:88) A,B, ˆ d p A (2 π ) E A d p B (2 π ) E B ×| v c,A − v c,B | [Λ − ] νj e c,i (cid:15) ikh ˆ p hc,A × f A ( X, p A ) f B ( X, p B ) ( p ρA − p ρB ) Θ jk ( p c,A ) ≡ ∂ ( βu ρ ) ∂X ν W ρν , (42)where the second equality defines W ρν and the sum of A, B, is over all 2-to-2 processes inTable I. A. The 10D integration
The 10D integral Θ jk ( p ( z ) c,A ) is calculated in the CMS by assuming p ( z ) c,A = (0 , , | p c,A | ) and p ( z ) c,B = (0 , , −| p c,A | ) , where | p c,A | is determined by the momenta of two incident particlesin the lab frame as in Eq. (C1). We can obtain Θ jk ( p c,A ) by carrying out the rotationoperation on the tensor Θ jk ( p ( z ) c,A ) in accordance with the rotation matrix from p ( z ) c,A to p c,A .For the Monte Carlo integration we have to sample k Tc,A , k (cid:48) Tc,A , p c, , and p c, . First wesample k Tc,A and k (cid:48) Tc,A , where the main contribution comes from the Gaussian distribution(13). Here we draw samples of k Tc,A = ( k c,A,x , k c,A,y , and k (cid:48) Tc,A = ( k (cid:48) c,A,x , k (cid:48) c,A,y , insidethe σ ( σ = α/ √ ) region of the Gaussian distribution around the center point p ( z ) c,A . Thelongitudinal momentum k c,A,z and k (cid:48) c,A,z can be determined by the energy conservation once p c, and p c, are given.Then we sample p c, and p c, . In order to increase the efficiency of the sampling, weshould determine the range of p c, and p c, . We can first determine the ranges of lengths | p c, | and | p c, | by a numerical search. Then we determine the ranges of directions ˆ p c, and ˆ p c, . For a given ˆ p c, , which can be randomly chosen, we find that the largest value of θ ≡ arccos( − ˆp c, · ˆp c, ) between ˆ p c, and − ˆ p c, occurs when | k c,A | = | k c,B | = | p c, | = | p c, | = (cid:113) p c,A + (3 σ ) . (43)Hence we obtain the range of θ as θ ≡ arccos( − ˆp c, · ˆp c, ) ∈ , π − arccos σ (cid:113) p c,A + (3 σ ) . (44)5The azimuthal angle ϕ of ˆ p c, around − ˆ p c, is in the range [0 , π ] .With the given values of p c, and p c, , the values of k c,A,z and k (cid:48) c,A,z can be obtained bysolving Eq. (D9). Then k c,B and k (cid:48) c,B can be determined by k c,B = p c, + p c, − k c,A and k (cid:48) c,B = p c, + p c, − k (cid:48) c,A respectively.The 10D integral is done by ZMCintegral-3.0, a Monte Carlo integration package, thatwe have newly developed and runs on multi-GPUs [62]. The ZMCintegral package is ableto evaluate sample points within a couple of hours depending on the complexity of theintegrand. For our integrand with all 2-to-2 processes for quarks of all flavors and gluons,it takes about 5 hours on one Tesla v100 card. We scan the values of | p c,A | from 0.1 to 2.2GeV and those of b from 0.1 to 3.5 fm, then we store the integration results of Θ jk ( p ( z ) c,A ) forlater use. It takes a couple of days to finish the calculation. We find that when | p c,A | > . GeV, the 10D integral is almost zero. This is due to the fact that if α (cid:28) | p c,A | , the incidentwave packets can be almost regarded as plane waves which give vanishing polarization. B. The 6D integration
Now we carry out the remaining 6D integration over p A and p B in (42). As we havementioned in Section V that we assume partons with p µA = ( E A , p A ) and p µB = ( E B , p B ) in the lab frame follow the Boltzmann distribution, f i ( X, p i ) = exp[ − β ( X ) p i · u ( X )] for i = A, B .The energy-momentum p µc,A = ( E c,A , p c,A ) and p µc,B = ( E c,B , p c,B ) in the CMS of two scat-tering particles are given by Eq. (C1), where the boost velocity and the Lorentz contractionfactor are given by Eq. (C2) and (C3) respectively. The impact parameter b c in the CMSis given by Eq. (C7).In the preceding subsection, we calculated the 10D integral Θ jk ( p ( z ) c,A ) where p ( z ) c,A is inthe z direction. We have to transform the tensor Θ jk ( p ( z ) c,A ) to Θ jk ( p c,A ) so that p ( z ) c,A isrotated to the real direction of p c,A determined by Eq. (C1). The rotation matrix R ij is defined by p c,A,i = R ij p ( z ) c,A,j , with which we define the transformation for the tensor Θ jk ( p c,A ) = R jj (cid:48) R kk (cid:48) Θ j (cid:48) k (cid:48) ( p ( z ) c,A ) .Our numerical results show that the tensor W ρν has the form W ρν = W (cid:15) ρνj e j , (45)6where we see that ρ and ν should be spatial indices or W ν = W ρ = . The form of(45) will be verified in the numerical results in Section VII. Then from (42) we obtain thepolarization rate per unit volume for one quark flavor d P q ( X ) dX = (cid:15) jρν ∂ ( βu ρ ) ∂X ν W e j = 2 (cid:15) jkl ω kl W e j = 2 W ∇ X × ( β u ) , (46)where ω ρν = − (1 / ∂ Xρ ( βu ν ) − ∂ Xν ( βu ρ )] , and for spatial indices we have the 3D form ω kl = (1 / ∇ Xk ( β u l ) − ∇ Xl ( β u k )] with u being the spatial part of the four-velocity u ρ . VII. NUMERICAL RESULTS
In this section we will present our numerical results. The approximation in (38,39) isinspired by the first order contribution in the narrow wave packet approximation. In orderto see how effective the approximation is, we compare in Fig. 2 the results of the 10Dintegral Θ jk ( p ( z ) c,A ) for the scattering processes q (¯ q ) + q → q (¯ q ) + q and g + q → g + q in twocases: with and without the approximation. Here the process q (¯ q ) + q → q (¯ q ) + q standsfor a sum over 5 different processes in Table I. Note that we do not show the results for g + g → q + ¯ q for which all elements of Θ jk ( p ( z ) c,A ) are almost zero in contrast to processes withat least one incident quark. We see in the figure that the results with the approximation arein agreement with the exact ones in 20% precision. In the figure we see that all elements of Θ( p ( z ) c,A ) fluctuate around zero for | p ( z ) c,A | = 0 , which leads to vanishing polarization. When | p ( z ) c,A | is non-vanishing, the off-diagonal elements of Θ( p ( z ) c,A ) are still zero within errors, butall diagonal elements take positive values which are almost equal to each other.7 Figure 2: Comparison of the results of the symmetric tensor Θ jk ( p ( z ) c,A ) for q (¯ q ) + q → q (¯ q ) + q and g + q → g + q in two cases: (1) with the approximation in (38,39) and (2) exact calculation of theintegral without any approximation. The results for g + g → q + ¯ q are not shown because they arenegligibly small (almost zero). Here we choose b = 0 . fm and | p ( z ) c,A | = 0 , . , . , . , . GeV.The solid symbols are the exact results without any approximation, while the dashed symbols arethe results with approximation in (38,39). The unit of Θ jk ( p ( z ) c,A ) is GeV − . We then work out the rest 6D integral and obtain W ρν in Eq. (45). In the 6D integrationwe have to determine the maximum value of | p A | and | p B | or the integration range of | p A | and | p B | . In Fig. 3, as an example, we show the dependence of W y on | p A | max = | p B | max for q (¯ q ) + q → q (¯ q ) + q , where we choose b = 2 . fm, z = 0 fm and T = 0 . GeV. We see inthe figure that the value of W y is very stable when | p A | max = | p B | max > T .8 Figure 3: The dependence of the results of W y on the integral ranges | p A | max = | p B | max for q (¯ q ) + q → q (¯ q ) + q . We choose b = 2 . fm, z = 0 fm, T = 0 . GeV.
The numerical results for W ρν show the structure of (45). We can write W ρν in anexplicit matrix form W ρν = W e z − W e y − W e z W e x W e y − W e x (47)As an example, we show in Fig. 4 the results for all components of W as functions of thecutoff b for the quark polarization. We see in the figure that W x and W z are two or threeorders of magnitude smaller than the positive values of W y , which gives the polarizationin the y direction. As we can see in the figure that W y increases with the cutoff b . Thereason for such a rising behavior is due to the Taylor expansion of f A ( x c,A , p c,A ) f B ( x c,B , p c,B ) to the linear order in y c,T = (0 , b c ) as in App. B. There should exist an upper limit for b above which the coherence of the incident wave packets is broken and the results are notphysical. Such an upper limit can be set to be the order of the hydrodynamical length scale ∼ /∂ µX u ν and should be larger than the interaction length scale /m D .It can be proved that W for the anti-quark polarization is the same as that for thequark one. The numerical results show that the magnitude of all element W ρν are equal sowe denote it as W .9 Figure 4: Results for W x , W y and W z as functions of the cutoff b in fm. There are largefluctuations in W x and W z above b = 1 . fm due to the strong oscillation of Bessel functions. VIII. DISCUSSIONS
We have constructed a microscopic model for the global polarization from particle scat-terings in a many body system. The core of the idea is the scattering of particles as wavepackets so that the orbital angular momentum is present in scatterings and can be convertedto spin polarization. As an illustrative example, we have calculated the quark/antiquarkpolarization in a QGP. The quarks and gluons are assumed to obey the Boltzmann distri-bution which simplifies the heavy numerical calculation. There is no essential difficulty totreat quarks and gluons as fermions and bosons respectively.To simplify the calculation, we also assume that the quark distributions are the samefor all flavors and spin states. As a consequence, the inverse processes that one polarizedquark is scattered by a parton to two final state partons as wave packets are absent. Sothe relaxation of polarization cannot be described without inverse processes and polarizeddistributions. We will extend our model by including the inverse processes in the future.
IX. SUMMARY AND CONCLUSIONS
The global polarization in heavy ion collisions arises from scattering processes of partonsor hadrons with spin-orbit couplings. However it is hard to implement this microscopic pic-ture consistently to describe particle scatterings at specified impact parameters in a thermalmedium with a shear flow. On the other hand the statistic-hydro model or Wigner functionmethod are widely used to calculate the global polarization in heavy ion collisions. Thesemodels are based on the assumption that the spin degrees of freedom have reached a localequilibrium. So there should be a spin-vorticity coupling term in the distribution function togive the global polarization proportional to the vorticity when it is small. However it is un-known if particle spins are really in a local equilibrium. In this paper we aim to construct amicroscopic model for the global polarization from particle collisions without the assumptionof local equilibrium for spins. The polarization effect is incorporated into particle scatteringsat specified impact parameters with spin-orbit couplings encoded. The spin-vorticity cou-pling naturally emerges from particle collisions if we assume a local equilibrium in particlemomenta instead of particle spins. This provides a microscopic mechanism for the globalpolarization from the first principle through particle collisions in non-equilibrium.1As an illustrative example, we have calculated the quark polarization rate per unit volumefrom all 2-to-2 parton (quark or gluon) scatterings in a locally thermalized quark-gluonplasma in momentum. Although the processes for anti-quark polarization are different fromthose for quarks, it can be shown that the polarization rate for anti-quarks is the same asthat for quarks because they are connected by the charge conjugate transformation. Thisis consistent with the fact that the rotation does not distinguish particles and antiparticles.The spin-orbit coupling is hidden in the polarized scattering amplitude at specified impactparameters. The polarization rate involves an integral of 16 dimensions, which is far beyondthe capability of the current numerical algorithm. We have developed a new Monte-Carlointegration algorithm ZMCintegral on multi-GPUs to make such a heavy task feasible. Wehave shown that the polarization rate per unit volume is proportional to the vorticity asthe result of particle scatterings, a non-equilibrium senario for the global polarization. Sowe can see in this example how the spin-vorticity coupling emerges naturally from particlescatterings.
Acknowledgments
QW thanks F. Becattini and M. Lisa for insightful discussions. QW is supported in partby the National Natural Science Foundation of China (NSFC) under Grant No. 11535012and No. 11890713, the 973 program under Grant No. 2015CB856902, and the Key ResearchProgram of the Chinese Academy of Sciences under the Grant No. XDPB09. XNW issupported in part by the National Natural Science Foundation of China (NSFC) under GrantNo. 11890714 and No. 11861131009, and by the Director, Office of Energy Research, Officeof High Energy and Nuclear Physics, Division of Nuclear Physics, of the U.S. Departmentof Energy under Contract Nos. DE- AC02-05CH11231.
Appendix A: Single particle state as a wave packet in relativistic quantum mechanics
In this appendix, we will give definitions and conventions for the single particle state incoordinate and momentum space and those for the wave packet.2
1. Single particle state in coordinate and momentum space
For simplicity we first consider the single particle state of spin-0 particles, then we gen-eralize it to spin-1/2 particles.A position eigenstate is denoted as | x (cid:105) and satisfies following orthogonality and complete-ness conditions (cid:104) x (cid:48) | x (cid:105) = δ (3) ( x (cid:48) − x ) , ˆ d x | x (cid:105) (cid:104) x | . (A1)The normalization of the state | x (cid:105) is then (cid:104) x | x (cid:105) = δ (3) ( x − x ) = ˆ d p (2 π ) = 1Ω (cid:88) p , (A2)where Ω is the space volume.A momentum eigenstate is denoted as | p (cid:105) and satisfies following orthogonality and com-pleteness conditions (cid:104) p (cid:48) | p (cid:105) = 2 E p (2 π ) δ (3) ( p − p (cid:48) ) , ˆ d p (2 π ) E p | p (cid:105) (cid:104) p | , (A3)where E p = (cid:112) | p | + m is the energy of the particle. Note that (cid:104) p (cid:48) | p (cid:105) is Lorentz invariant.The normalization of | p (cid:105) is then (cid:104) p | p (cid:105) = 2 E p (2 π ) δ (3) ( p − p ) = 2 E p Ω . (A4)From Eq. (A1) and (A3) we can define the inner product (cid:104) x | p (cid:105) as (cid:104) x | p (cid:105) = (cid:112) E p e i p · x . (A5)With the above relation we can check δ (3) ( x − x (cid:48) ) = (cid:104) x (cid:48) | x (cid:105) = ˆ d p (2 π ) E p (cid:104) x (cid:48) | p (cid:105) (cid:104) p | x (cid:105) = ˆ d p (2 π ) e i p · ( x (cid:48) − x ) , (A6)where we have inserted the completeness relation in (A3). We can express | x (cid:105) in terms of | p (cid:105) and vice versa,3 | x (cid:105) = ˆ d p (2 π ) E p | p (cid:105) (cid:104) p | x (cid:105) = ˆ d p (2 π ) (cid:112) E p e − i p · x | p (cid:105) , | p (cid:105) = ˆ d x | x (cid:105) (cid:104) x | p (cid:105) = (cid:112) E p ˆ d xe i p · x | x (cid:105) . (A7)
2. Single particle state as a wavepacket
In the real world a particle is always localized in some finite region, so its state can berepresented by a wavepacket | φ (cid:105) which is a superposition of plane wave states, | φ (cid:105) = ˆ d k (2 π ) √ E k φ ( k ) | k (cid:105) , (A8)and φ ( k ) is the amplitude and can be normalized to unity, (cid:104) φ | φ (cid:105) = ˆ d k (2 π ) | φ ( k ) | = 1 . (A9)The energy dimension of | φ (cid:105) is . A typical form for φ ( p ) satisfying Eq. (A9) is the Gaussianwavepacket φ ( p − p ) = (8 π ) / α / exp (cid:20) − ( p − p ) α (cid:21) , (A10)which is centered at p . The wavepacket function in coordinate space is φ ( x ) = (cid:104) x | φ (cid:105) = ˆ d k (2 π ) φ ( k ) e i k · x , (A11)where we have used Eq. (A5).If we displace the particle state by b in coordinate space, the new wavepacket functionis given by φ (cid:48) ( x ) = φ ( x − b ) = ˆ d k (2 π ) φ ( k ) e i k · ( x − b ) = (cid:104) x | φ (cid:48) (cid:105) , (A12)where the new wavepacket state is | φ (cid:48) (cid:105) = ˆ d k (2 π ) √ E k φ ( k ) e − i k · b | k (cid:105) . (A13)For spin-1/2 particles, the single particle state | k , λ (cid:105) has a spin index λ which is the spinalong a quantization direction. The orthogonality and completeness conditions in (A3) now4become (cid:104) k (cid:48) , λ (cid:48) | k , λ (cid:105) = 2 E k (2 π ) δ (3) ( k − k (cid:48) ) δ λ,λ (cid:48) , ˆ d p (2 π ) E p (cid:88) λ | p , λ (cid:105) (cid:104) p , λ | . (A14)The wavepacket has the form | φ, λ (cid:105) = ˆ d k (2 π ) √ E k φ ( k ) | k , λ (cid:105) , (A15)and satisfies the normalization condition (cid:104) φ, λ | φ, λ (cid:105) = 1 similar to Eq. (A9). Appendix B: Expansion of f A and f B in impact parameter We can make an expansion of f A ( X c + y c,T / , p c,A ) f B ( X c − y c,T / , p c,B ) in y c,T = (0 , b c ) if | b c | is small compared with the range in which f A and f B change slowly. The variableswith the subscript ’c’ are defined in the CMS of the scattering, while those without ’c’are defined in the lab frame. We assume that the system has reached local equilibrium inmomentum and the phase space distributions depend on the space-time through the fluidvelocity u µ ( x ) and temperature T ( x ) in the form f ( x, p ) = f [ β ( x ) p · u ( x )] .To the linear order in y c,T , we have f A (cid:16) X c + y c,T , p c,A (cid:17) f B (cid:16) X c − y c,T , p c,B (cid:17) ≈ f A ( X c , p c,A ) f B ( X c , p c,B )+ 12 y µc,T (cid:20) ∂f A ( X c , p c,A ) ∂X µc f B ( X c , p c,B ) − f A ( X c , p c,A ) ∂f B ( X c , p c,B ) ∂X µc (cid:21) = f A ( X c , p c,A ) f B ( X c , p c,B ) + 12 y µc,T ∂ ( βu c,ρ ) ∂X νc × (cid:20) p ρc,A f B ( X c , p c,B ) df A ( X c , p c,A ) d ( βu c · p c,A ) − p ρc,B f A ( X c , p c,A ) df B ( X c , p c,B ) d ( βu c · p c,B ) (cid:21) = f A ( X, p A ) f B ( X, p B ) + 12 y µc,T ∂X ν ∂X µc ∂ ( βu ρ ) ∂X ν × (cid:20) p ρA f B ( X, p B ) df A ( X, p A ) d ( βu · p A ) − p ρB f A ( X, p A ) df B ( X, p B ) d ( βu · p B ) (cid:21) , (B1)where in the second equality we have boosted to the lab frame using f A ( X, p A ) = f A ( X c , p c,A ) and f B ( X, p B ) = f B ( X c , p c,B ) . We look closely at the term5 y µc,T [ ∂ ( βu c,ρ ) /∂X µc ] p ρc,A , y µc,T p ρc,A ∂ ( βu ρ ) ∂X µc = 14 y [ µc,T p ρ ] c,A (cid:20) ∂ ( βu c,ρ ) ∂X µc − ∂ ( βu c,µ ) ∂X ρc (cid:21) + 14 y { µc,T p ρ } c,A (cid:20) ∂ ( βu c,ρ ) ∂X µc + ∂ ( βu c,µ ) ∂X ρc (cid:21) = − y [ µc,T p ρ ] c,A (cid:36) ( c ) µρ + 14 y { µc,T p ρ } c,A (cid:20) ∂ ( βu c,ρ ) ∂X µc + ∂ ( βu c,µ ) ∂X ρc (cid:21) = − L µρ ( c ) (cid:36) ( c ) µρ + 14 y { µc,T p ρ } c,A (cid:20) ∂ ( βu c,ρ ) ∂X µc + ∂ ( βu c,µ ) ∂X ρc (cid:21) , (B2)where [ µρ ] and { µρ } denote the anti-symmetrization and symmetrization of two indicesrespectively, L µρ ( c ) ≡ y [ µc,T p ρ ] c,A is the OAM tensor, and ω ( c ) µρ ≡ − (1 / ∂ X c µ ( βu c,ρ ) − ∂ X c ρ ( βu c,µ )] is the thermal vorticity. We see that the coupling term of the OAM and vorticity appearin Eq. (B1). The second term in last line of Eq. (B2) is related to the Killing conditionrequired by the thermal equilibrium of the spin.Using X µc = Λ µν X ν and X µ = [Λ − ] µν X νc , so we have ∂X ν ∂X µc = [Λ − ] νµ = Λ νµ and then Eq.(B1) becomes f A (cid:16) X c + y c,T , p c,A (cid:17) f B (cid:16) X c − y c,T , p c,B (cid:17) = f A ( X, p A ) f B ( X, p B ) + 12 y µc,T [Λ − ] νµ ∂ ( βu ρ ) ∂X ν × (cid:20) p ρA f B ( X, p B ) df A ( X, p A ) d ( βu · p A ) − p ρB f A ( X, p A ) df B ( X, p B ) d ( βu · p B ) (cid:21) . (B3)In Appendix C we give the exact form of Λ µν and [Λ − ] µν . Appendix C: Lorentz transformation
In the lab frame two colliding particles have on-shell momenta p A = ( E A , p A ) and p B =( E B , p B ) . The Lorentz transformation for the energy-momentum from the lab frame to theCMS of two colliding particles is p c,i = p i + ( γ bst − v bst (ˆ v bst · p i ) − γ bst v bst E i ,E c,i = γ bst ( E i − v bst · p i ) . (C1)where i = A, B , v bst is the boost velocity or the velocity of CMS in the lab frame and isgiven by v bst = p A + p B E A + E B , (C2)6and γ bst = (1 − | v bst | ) − / , (C3)is the Lorentz contraction facror corresponding to v bst . Equation (C1) defines the Lorentztransformation matrix Λ µν . The reverse transformation to (C1) from the CMS to the labframe can be obtained by flipping the sign of ˆ v bst , p i = p c,i + ( γ bst − v bst (ˆ v bst · p c,i ) + γ bst v bst E c,i ,E i = γ bst ( E c,i + v bst · p c,i ) . (C4)The above defines the Lorentz transformation matrix [Λ − ] µν .The Lorentz transformation for x A = ( t A , x A ) and x B = ( t B , x B ) is x c,i = x i + ( γ bst − v bst (ˆ v bst · x i ) − γ bst v bst t i ,t c,i = γ bst ( t i − v bst · x i ) . (C5)The difference of two space-time points in the CMS are expressed in lab frame variables, ∆ t c = t c,A − t c,B = γ bst (∆ t − v bst · ∆ x ) , ∆ x c = ∆ x + ( γ bst − v bst (ˆ v bst · ∆ x ) − γ bst v bst ∆ t, (C6)where ∆ t = t A − t B and ∆ x = x A − x B . We then express the impact parameter as b c = ∆ x c · (1 − ˆp c,A ˆp c,A ) . (C7)Let us look at the CMS constraint δ (∆ t c ) δ (∆ x c,L ) in Eq. (10) (we have recovered thesubscript ’c’). The condition ∆ t c = 0 leads to ∆ t = v bst · ∆ x , (C8)while the condition ˆp c,A · ∆ x c = 0 leads to ( v A − v B ) · ∆ x = 0 , (C9)where we have used ∆ x c = ∆ x + ( γ − − v bst (ˆ v bst · ∆ x ) , (C10)which is the result of Eqs. (C6,C8). The condition in Eq. (C9) means that ( x A − x B ) ⊥ ( v A − v B ) . Equation (C8) and (C9) are the lab frame version of the constraint δ (∆ t c ) δ (∆ x c,L ) .7 Appendix D: Integration over impact parameter and Delta Functions in Eq. (21)
We carry out the integration over the impact parameter and show how to remove thedelta functions by integration in Eq. (21).Substitute Eq. 23 into Eq. 21, we have the integration of b c in the following form I ( b c ) = i ˆ d b c exp ( i a · b c ) 1 b c b c,j b c,k b c,l = − ∂∂ a l ∂∂ a j ∂∂ a k ˆ d b c exp ( i a · b c ) 1 b c = − π ∂∂ a l ∂∂ a j ∂∂ a k ˆ b db c b c J ( ab c ) , (D1)where b c ≡ | b c | , b is the cutoff of b c , a = k (cid:48) c,A − k c,A , and J ( ab c ) = 12 π ˆ π dφ exp ( iab c cos φ ) . (D2)Then we carry out the derivatives on a j , a k and a l , I ( b c ) = − π a Q Ljkl ˆ w dww J (cid:48)(cid:48)(cid:48) ( w ) − π a Q Tjkl ˆ w dw [ wJ (cid:48)(cid:48) ( w ) + J ( w )] , (D3)where we have used w = ab with b being the upper limit or cutoff of b c , J i ( i = 0 , , )are Bessel functions, and Q Ljkl = a l a j a k a ,Q Tjkl = 1 a (cid:0) a a k δ lj + a a l δ jk + a a j δ lk − a l a j a k (cid:1) . (D4)Note that the overall minus sign of Eq. (D3) cancels the one in Eq. (21).We carry out the integration to remove the delta functions. First we integrate over k c,B and k (cid:48) c,B to remove six delta functions in three momenta, the result is to make followingreplacement in the integrand k c,B = p c, + p c, − k c,A , k (cid:48) c,B = p c, + p c, − k (cid:48) c,A . (D5)We are left with two delta functions for energy conservation which can be removed by theintegration over k Lc,A and k (cid:48) Lc,A , where ’L’ means the longitudinal direction along p c,A . To this8purpose, we express the energies in terms of longitudinal and transverse momenta E c,A = (cid:113) ( k Lc,A ) + ( k Tc,A ) + m ,E c,B = (cid:113) ( p Tc, + p Tc, − k Tc,A ) + ( p Lc, + p Lc, − k Lc,A ) + m ,E (cid:48) c,A = (cid:113) ( k (cid:48) Lc,A ) + ( k (cid:48) Tc,A ) + m ,E (cid:48) c,B = (cid:113) ( p Tc, + p Tc, − k (cid:48) Tc,A ) + ( p Lc, + p Lc, − k (cid:48) Lc,A ) + m . (D6)So two delta functions for energy conservation become I ( δE ) = δ ( E c,A + E c,B − E c, − E c, )= 1 | Ja( k Lc,A (1)) | δ [ k Lc,A − k Lc,A (1)] + 1 | Ja( k Lc,A (2)) | δ [ k Lc,A − k Lc,A (2)] I ( δE (cid:48) ) = δ ( E (cid:48) c,A + E (cid:48) c,B − E c, − E c, )= 1 | Ja( k (cid:48) Lc,A (1)) | δ [ k (cid:48) Lc,A − k (cid:48) Lc,A (1)] + 1 | Ja( k (cid:48) Lc,A (2)) | δ [ k (cid:48) Lc,A − k (cid:48) Lc,A (2)] (D7)where the Jacobians of two delta functions are given by
Ja( k Lc,A ) = ∂∂k
Lc,A ( E c,A + E c,B − E c, − E c, )= k Lc,A (cid:18) E c,A + 1 E c,B (cid:19) − E c,B ( p Lc, + p Lc, ) , Ja( k (cid:48) Lc,A ) = ∂∂k (cid:48)
Lc,A ( E (cid:48) c,A + E (cid:48) c,B − E c, − E c, )= k (cid:48) Lc,A (cid:32) E (cid:48) c,A + 1 E (cid:48) c,B (cid:33) − E (cid:48) c,B ( p Lc, + p Lc, ) , (D8)and k Lc,A ( i = 1 , and k (cid:48) Lc,A ( i = 1 , are two roots of the energy conservation equation E c,A + E c,B − E c, − E c, = 0 and E (cid:48) c,A + E (cid:48) c,B − E c, − E c, = 0 , respectively. The explicitforms of k Lc,A ( i = 1 , and k (cid:48) Lc,A ( i = 1 , are k Lc,A (1 ,
2) = C ± C ,k (cid:48) Lc,A (1 ,
2) = k Lc,A (1 , k Tc,A → k (cid:48) Tc,A ] , (D9)where C and C are given by C = 12 · p Lc, + p Lc, ( E c, + E c, ) − ( p Lc, + p Lc, ) × (cid:2) ( E c, + E c, ) − ( p Lc, + p Lc, ) +2( p Tc, + p Tc, ) · k Tc,A − ( p Tc, + p Tc, ) (cid:3) ,C = − · E c, + E c, ( E c, + E c, ) − ( p Lc, + p Lc, ) √ H, (D10)9with H being defined by H = ( E c, + E c, ) + 4 m ( p Lc, + p Lc, ) + ( p c, + p c, ) +4( k Tc,A ) ( p c, + p c, ) − p c, + p c, ) [ k Tc,A · ( p Tc, + p Tc, )] − E c, + E c, ) × [2 m + 2( k Tc,A ) − k Tc,A · ( p Tc, + p Tc, ) + ( p c, + p c, ) ] . (D11) Appendix E: Some formula for Dirac spinors
The Hamiltonian for a Dirac fermion with the mass m is given by H = α · p + γ m = m σ · p σ · p − m , (E1)where γ µ = ( γ , γ ) are Dirac gamma-matrices, α ≡ γ γ , and σ = ( σ , σ , σ ) are Paulimatrices. The energy eigenstate can be found from the equation H χφ = ± E p χφ , (E2)where E p = (cid:112) p + m , the sign ± in the right-hand side corresponds to positive/negativeenergy state, χ and φ are Pauli spinors which form a Dirac spinor ( χ, φ ) . We can express χ in terms of φ and vice versa, χ = σ · p ηE p − m φ,φ = σ · p ηE p + m χ, (E3)where η = ± correspond to the positive and negative energy state respectively. So thepositive energy solution becomes u ( s, p ) = (cid:112) E p + m χ s σ · p E p + m χ s , (E4)0where s = ± is the spin orientation of the Pauli spinor and n = (sin θ cos φ, sin θ sin φ, cos θ ) is the spin quantization direction. The spin eigenstates along n are given by χ + = e − iφ cos θ sin θ ,χ − = − e − iφ sin θ cos θ , (E5)which satisfy σ · n = cos θ e − iφ sin θe iφ sin θ − cos θ , ( σ · n ) χ s = sχ s . (E6)The negative energy solution can be put into the form ˜ v ( s, p ) = (cid:112) E p + m − σ · p E p + m χ s χ s , (E7)The Dirac spinor for anti-particles can be defined by v ( s, p ) = ˜ v ( − s, − p ) = (cid:112) E p + m σ · p E p + m χ − s χ − s , (E8)or defined in terms of the positive energy solution, v ( s, p ) = iγ u ∗ ( s, p ) = − i (cid:112) E p + m σ · p E p + m σ χ ∗ s σ χ ∗ s . (E9)The two Dirac spinors in (E8) and (E9) are actually the same up to a sign.Now we rewrite the Dirac spinor of a moving particle in the way of a Lorentz transfor-mation of the one in the particle’s rest frame. The Lorentz transformation matrix for theDirac spinor is given by Λ / ( p ) = exp (cid:18) − η p α · ˆ p (cid:19) = cosh (cid:18) η p (cid:19) − ( α · ˆ p ) sinh (cid:18) η p (cid:19) , Λ − / ( p ) = Λ / ( − p ) = exp (cid:18) η p α · ˆ p (cid:19) , (E10)1where ˆ p ≡ p / | p | is the momentum direction, η p is the rapidity satisfying E p = m cosh( η p ) , | p | = m sinh( η p ) , v p = tanh( η p ) , E p + m = 2 m cosh (cid:0) η p (cid:1) , E p − m = 2 m sinh (cid:0) η p (cid:1) . So u ( s, p ) can be expressed by a Lorentz boost of u ( s, ) for the particle at rest, u ( s, p ) = (cid:112) E p + m χ s σ · p E p + m χ s = Λ / ( − p ) u ( s, )= √ m cosh (cid:0) η p (cid:1) χ s ( σ · ˆ p ) sinh (cid:0) η p (cid:1) χ s . (E11)In the same way we can rewrite v ( s, p ) as v ( s, p ) = (cid:112) E p + m σ · p E p + m χ − s χ − s = Λ / ( − p ) v ( s, )= √ m ( σ · ˆ p ) sinh (cid:0) η p (cid:1) χ − s cosh (cid:0) η p (cid:1) χ − s . (E12)With Eqs. (E11,E12) we have following formula (cid:88) s u ( s, p )¯ u ( s, q ) = Λ / ( − p ) (cid:34)(cid:88) s u ( s, )¯ u ( s, ) (cid:35) Λ − / ( − q )= m Λ / ( − p )(1 + γ )Λ − / ( − q ) , (cid:88) s v ( s, p )¯ v ( s, q ) = Λ / ( − p ) (cid:34)(cid:88) s v ( s, )¯ v ( s, ) (cid:35) Λ − / ( − q )= m Λ / ( − p )( γ − − / ( − q ) , (E13)where we have used ¯ u ( s, q ) = ¯ u ( s, )Λ − / ( − q ) , ¯ v ( s, q ) = ¯ v ( s, )Λ − / ( − q ) , (cid:80) s u ( s, )¯ u ( s, ) = m (1 + γ ) and (cid:80) s v ( s, )¯ v ( s, ) = m ( − γ ) .The spin projector is defined by Π( s, n ) = 12 (1 + sγ n σ γ σ ) (E14)where n σ is the Lorentz boost of the polarization vector (0 , n ) in the particle’s rest framesatisfying n · p = 0 and n = − . In the particle’s rest frame, we have Π rest ( s, n ) = 12 (1 + s n · Σ ) ≡ s n · σ
00 1 − s n · σ . (E15)2We have following properties for the spin projector Π( s, n ) u ( s, p ) = u ( s, p ) , Π( s, n ) v ( s, p ) = v ( s, p ) , Π( s, n ) u ( − s, p ) = 0 , Π( s, n ) v ( − s, p ) = 0 . (E16)As an example, we can explicitly verify the first one as Π( s, n ) u ( s, p ) = 12 Λ / ( − p ) u ( s, )+ 12 sn σ γ Λ / ( − p )Λ − / ( − p ) γ σ Λ / ( − p ) u ( s, )= 12 Λ / ( − p ) u ( s, )+ 12 s Λ / ( − p ) γ n σ Λ νσ ( − p ) γ ν u ( s, )= 12 Λ / ( − p ) u ( s, )+ 12 s Λ / ( − p ) γ ( n · Σ ) u ( s, )= Λ / ( − p )Π rest ( s, n ) u ( s, )= u ( s, p ) , (E17)where we have used Λ − / ( − p ) γ σ Λ / ( − p ) = Λ νσ ( − p ) γ ν and Λ νσ ( − p ) = Λ νσ ( p ) . Using thespin projector, we have the following relation Π( s , n ) (cid:88) s u ( s, p )¯ u ( s, p ) = Π( s , n ) ( p · γ + m ) | p µ =( E p , p ) = u ( s , p )¯ u ( s , p ) , (E18)where p · γ ≡ p µ γ µ . Appendix F: Polarized amplitudes for quarks in 2-to-2 parton scatterings
In this appendix, we give polarized amplitudes for quarks in all 2-to-2 parton scatteringslisted in Table I. We assume the same quark mass m for all flavors and that the externalgluon is massless. We introduce a mass into internal gluons or gluon propagators in the tand u channel to regulate the possible divergence.3All kinematic variables are defined in the CMS in this appendix, for notational simplicitywe will suppress the subscript ’c’ for all variables, for example, p A actually means p cA . Thevalues of color factors, denoted as C AB → CD for the process A + B → C + D , are given inTable II. q a q b → q a q b with a (cid:54) = b Following the Feynman diagram in Table I, we obtain the difference in the squared am-plitude between the spin state s = 1 / and s = − / for q b in the final state, ∆ I q a q b → q a q b M = I q a q b → q a q b M ( s = 1 / − I q a q b → q a q b M ( s = − / C q a q b → q a q b g s m q q (cid:48) × Tr (cid:104) ( p · γ + m ) γ µ Λ / ( − k A )( γ + 1)Λ − / ( − k (cid:48) A ) γ ν (cid:105) × Tr (cid:104) γ ( n · γ )( p · γ + m ) γ µ Λ / ( − k B )( γ + 1)Λ − / ( − k (cid:48) B ) γ ν (cid:105) , (F1)where q = k A − p and q (cid:48) = k (cid:48) A − p are momenta in the propagators. ¯ q a q b → ¯ q a q b with a (cid:54) = b For the polarization of q b , we obtain ∆ I ¯ q a q b → ¯ q a q b M = C ¯ q a q b → ¯ q a q b g s m q q (cid:48) × Tr (cid:104) γ µ ( p · γ − m ) γ ν Λ / ( − k (cid:48) A )( γ − − / ( − k A ) (cid:105) × Tr (cid:104) γ ( n · γ )( p · γ + m ) γ µ Λ / ( − k B )( γ + 1)Λ − / ( − k (cid:48) B ) γ ν (cid:105) , (F2)where q = k A − p and q (cid:48) = k (cid:48) A − p are momenta in the propagators.4 ¯ q a q a → ¯ q a q a For the polarization of q a in the final state, we obtain ∆ I ¯ q a q a → ¯ q a q a M = C (1)¯ q a q a → ¯ q a q a g s m q q (cid:48) × Tr (cid:104) γ ( n · γ )( p · γ + m ) γ µ Λ / ( − k B )( γ + 1)Λ − / ( − k (cid:48) B ) γ ν (cid:105) × Tr (cid:104) ( p · γ − m ) γ ν Λ / ( − k (cid:48) A )( γ − − / ( − k A ) γ µ (cid:105) − C (2)¯ q a q a → ¯ q a q a g s m q q (cid:48) × Tr (cid:104) γ ( n · γ )( p · γ + m ) γ µ Λ / ( − k B )( γ + 1)Λ − / ( − k (cid:48) B ) γ ν × Λ / ( − k (cid:48) A )( γ − − / ( − k A ) γ µ ( p · γ − m ) γ ν (cid:105) − C (2)¯ q a q a → ¯ q a q a g s m q q (cid:48) × Tr [ γ ( n · γ )( p · γ + m ) γ µ ( p · γ − m ) γ ν × Λ / ( − k (cid:48) A )( γ − − / ( − k A ) γ µ Λ / ( − k B )( γ + 1)Λ − / ( − k (cid:48) B ) γ ν (cid:105) + C (1)¯ q a q a → ¯ q a q a g s m q q (cid:48) × Tr (cid:104) Λ / ( − k (cid:48) A )( γ − − / ( − k A ) γ µ Λ / ( − k B )( γ + 1)Λ − / ( − k (cid:48) B ) γ ν (cid:105) × Tr [ γ ( n · γ )( p · γ + m ) γ µ ( p · γ − m ) γ ν ] , (F3)where q = k A − p , q = k A + k B , q (cid:48) = k (cid:48) A − p and q (cid:48) = k (cid:48) A + k (cid:48) B are momenta in thepropagators. q a q a → q a q a For the polarization of q a in the final state, we obtain5 ∆ I q a q a → q a q a M = C (1) q a q a → q a q a g s m q q (cid:48) × Tr (cid:104) ( p · γ + m ) γ µ Λ / ( − k A )( γ + 1)Λ − / ( − k (cid:48) A ) γ ν (cid:105) × Tr (cid:104) γ ( n · γ )( p · γ + m ) γ µ Λ / ( − k B )( γ + 1)Λ − / ( − k (cid:48) B ) γ ν (cid:105) − C (2) q a q a → q a q a g s m q q (cid:48) × Tr (cid:104) ( p · γ + m ) γ µ Λ / ( − k A )( γ + 1)Λ − / ( − k (cid:48) A ) γ ν × γ ( n · γ )( p · γ + m ) γ µ Λ / ( − k B )( γ + 1)Λ − / ( − k (cid:48) B ) γ ν (cid:105) − C (2) q a q a → q a q a g s m q (cid:48) q × Tr (cid:104) γ µ Λ / ( − k A )( γ + 1)Λ − / ( − k (cid:48) A ) γ ν ( p · γ + m ) × γ µ Λ / ( − k B )( γ + 1)Λ − / ( − k (cid:48) B ) γ ν γ ( n · γ )( p · γ + m ) (cid:105) + C (1) q a q a → q a q a g s m q (cid:48) q × Tr (cid:104) γ ( n · γ )( p · γ + m ) γ µ Λ / ( − k A )( γ + 1)Λ − / ( − k (cid:48) A ) γ ν (cid:105) × Tr (cid:104) Λ / ( − k B )( γ + 1)Λ − / ( − k (cid:48) B ) γ ν ( p · γ + m ) γ µ (cid:105) , (F4)where q = k A − p , q = k A − p , q (cid:48) = k (cid:48) A − p and q (cid:48) = k (cid:48) A − p are momenta in propagators. gg → ¯ q a q a In principle, the ghost diagrams should also contribute. However, its contribution iscanceled when we calculate ∆ I gg → ¯ q a q a M . For the polarization of q a in the final state, we6obtain ∆ I gg → ¯ q a q a M = C (1) gg → ¯ q a q a g s q − m )( q (cid:48) − m ) I + C (2) gg → ¯ q a q a g s q − m )( q (cid:48) − m ) I − C (3) gg → ¯ q a q a g s q − m ) q (cid:48) I + C (2) gg → ¯ q a q a g s q (cid:48) − m )( q − m ) I + C (1) gg → ¯ q a q a g s q − m )( q (cid:48) − m ) I + C (3) gg → ¯ q a q a g s q − m ) q (cid:48) I − C (3) gg → ¯ q a q a g s q (cid:48) − m ) q I + C (3) gg → ¯ q a q a g s q (cid:48) − m ) q I + C (4) gg → ¯ q a q a g s q q (cid:48) I , (F5)where q = k A − p , q = p − k A , q = k A + k B , q (cid:48) = k (cid:48) A − p , q (cid:48) = p − k (cid:48) A and q (cid:48) = k (cid:48) A + k (cid:48) B are momenta in propagators, and the terms I ρi for i = 1 , , · · · , are given by I = Tr[ γ ( n · γ )( p · γ + m ) γ ν ( q · γ + m ) γ µ × ( p · γ − m ) γ µ (cid:48) ( q (cid:48) · γ + m ) γ ν (cid:48) ] g µµ (cid:48) g νν (cid:48) (F6) I = Tr[ γ ( n · γ )( p · γ + m ) γ ν ( q · γ + m ) γ µ × ( p · γ − m ) γ ν (cid:48) ( q (cid:48) · γ + m ) γ µ (cid:48) ] g µµ (cid:48) g νν (cid:48) (F7) I = Tr[ γ ( n · γ )( p · γ + m ) γ ν ( q · γ + m ) γ µ ( p · γ − m ) γ σ (cid:48) ] g µµ (cid:48) g νν (cid:48) × [ g σ (cid:48) µ (cid:48) ( − q (cid:48) − k (cid:48) A ) ν (cid:48) + g µ (cid:48) ν (cid:48) ( k (cid:48) A − k (cid:48) B ) σ (cid:48) + g ν (cid:48) σ (cid:48) ( k (cid:48) B + q (cid:48) ) µ (cid:48) ] (F8) I = Tr[ γ ( n · γ )( p · γ + m ) γ µ ( q · γ + m ) γ ν × ( p · γ − m ) γ µ (cid:48) ( q (cid:48) · γ + m ) γ ν (cid:48) ] g µµ (cid:48) g νν (cid:48) (F9)7 I = Tr[ γ ( n · γ )( p · γ + m ) γ µ ( q · γ + m ) γ ν × ( p · γ − m ) γ ν (cid:48) ( q (cid:48) · γ + m ) γ µ (cid:48) ] g µµ (cid:48) g νν (cid:48) (F10) I = Tr[ γ ( n · γ )( p · γ + m ) γ µ ( q · γ + m ) γ ν ( p · γ − m ) γ σ (cid:48) ] g µµ (cid:48) g νν (cid:48) × [ g σ (cid:48) µ (cid:48) ( − q (cid:48) − k (cid:48) A ) ν (cid:48) + g µ (cid:48) ν (cid:48) ( k (cid:48) A − k (cid:48) B ) σ (cid:48) + g ν (cid:48) σ (cid:48) ( k (cid:48) B + q (cid:48) ) µ (cid:48) ] (F11) I = Tr[ γ ( n · γ )( p · γ + m ) γ σ ( p · γ − m ) γ µ (cid:48) ( q (cid:48) · γ + m ) γ ν (cid:48) ] g µµ (cid:48) g νν (cid:48) × [ g σµ ( − q − k A ) ν + g µν ( k A − k B ) σ + g νσ ( k B + q ) µ ] (F12) I = Tr[ γ ( n · γ )( p · γ + m ) γ σ ( p · γ − m ) γ ν (cid:48) ( q (cid:48) · γ + m ) γ µ (cid:48) ] g µµ (cid:48) g νν (cid:48) × [ g σµ ( − q − k A ) ν + g µν ( k A − k B ) σ + g νσ ( k B + q ) µ ] (F13) I = Tr[ γ ( n · γ )( p · γ + m ) γ σ ( p · γ − m ) γ σ (cid:48) ] × [ g σµ ( − q − k A ) ν + g µν ( k A − k B ) σ + g νσ ( k B + q ) µ ] × [ g σ (cid:48) µ (cid:48) ( − q (cid:48) − k (cid:48) A ) ν (cid:48) + g µ (cid:48) ν (cid:48) ( k (cid:48) A − k (cid:48) B ) σ (cid:48) + g ν (cid:48) σ (cid:48) ( k (cid:48) B + q (cid:48) ) µ (cid:48) ] × g µµ (cid:48) g νν (cid:48) (F14) gq a → gq a In principle, the ghost diagram should also contribute. However, its contribution iscanceled when we calculate ∆ I gq a → gq a M . For the polarization of q a in the final state, we8obtain ∆ I gq a → gq a M = C (1) gq a → gq a g s m q q (cid:48) I + C (2) gq a → gq a g s m q ( q (cid:48) − m ) I − C (2) gq a → gq a g s m q ( q (cid:48) − m ) I + C (2) gq a → gq a g s m q (cid:48) ( q − m ) I + C (3) gq a → gq a g s m q − m )( q (cid:48) − m ) I + C (4) gq a → gq a g s m q − m )( q (cid:48) − m ) I − C (2) gq a → gq a g s m q (cid:48) ( q − m ) I + C (4) gq a → gq a g s m q (cid:48) − m )( q − m ) I + C (3) gq a → gq a g s m q − m )( q (cid:48) − m ) I (F15)where q = k A − p , q = p − k A , q = k A + k B , q (cid:48) = k (cid:48) A − p , q (cid:48) = p − k (cid:48) A and q (cid:48) = k (cid:48) A + k (cid:48) B are momenta in propagators, and the terms I ρi for i = 1 , , · · · , are given by I = Tr[ γ ( n · γ )( p · γ + m ) γ σ Λ / ( − k B )( γ + 1)Λ − / ( − k (cid:48) B ) γ σ (cid:48) ] × g µµ (cid:48) g νν (cid:48) [ g µν ( k A + p ) σ + g νσ ( q − p ) µ + g σµ ( − q − k A ) ν ] × [ g µ (cid:48) ν (cid:48) ( k (cid:48) A + p ) σ (cid:48) + g ν (cid:48) σ (cid:48) ( q (cid:48) − p ) µ (cid:48) + g σ (cid:48) µ (cid:48) ( − q (cid:48) − k (cid:48) A ) ν (cid:48) ] (F16) I = Tr[ γ ( n · γ )( p · γ + m ) γ σ Λ / ( − k B )( γ + 1)Λ − / ( − k (cid:48) B ) × γ ν (cid:48) ( q (cid:48) · γ + m ) γ µ (cid:48) ] g µµ (cid:48) g νν (cid:48) × [ g µν ( k A + p ) σ + g νσ ( q − p ) µ + g σµ ( − q − k A ) ν ] (F17) I = Tr[ γ ( n · γ )( p · γ + m ) γ σ Λ / ( − k B )( γ + 1)Λ − / ( − k (cid:48) B ) × γ µ (cid:48) ( q (cid:48) · γ + m ) γ ν (cid:48) ] g µµ (cid:48) g νν (cid:48) × [ g µν ( k A + p ) σ + g νσ ( q − p ) µ + g σµ ( − q − k A ) ν ] (F18)9 I = Tr[ γ ( n · γ )( p · γ + m ) γ µ ( q · γ + m ) γ ν × Λ / ( − k B )( γ + 1)Λ − / ( − k (cid:48) B ) γ σ (cid:48) ] g µµ (cid:48) g νν (cid:48) × [ g µ (cid:48) ν (cid:48) ( k (cid:48) A + p ) σ (cid:48) + g ν (cid:48) σ (cid:48) ( q (cid:48) − p ) µ (cid:48) + g σ (cid:48) µ (cid:48) ( − q (cid:48) − k (cid:48) A ) ν (cid:48) ] (F19) I = Tr[ γ ( n · γ )( p · γ + m ) γ µ ( q · γ + m ) γ ν × Λ / ( − k B )( γ + 1)Λ − / ( − k (cid:48) B ) × γ ν (cid:48) ( q (cid:48) + m ) γ µ (cid:48) ] g µµ (cid:48) g νν (cid:48) (F20) I = Tr[ γ ( n · γ )( p · γ + m ) γ µ ( q · γ + m ) γ ν × Λ / ( − k B )( γ + 1)Λ − / ( − k (cid:48) B ) × γ µ (cid:48) ( q (cid:48) · γ + m ) γ ν (cid:48) ] g µµ (cid:48) g νν (cid:48) (F21) I = Tr[ γ ( n · γ )( p · γ + m ) γ ν ( q · γ + m ) γ µ × Λ / ( − k B )( γ + 1)Λ − / ( − k (cid:48) B ) γ σ (cid:48) ] g µµ (cid:48) g νν (cid:48) × [ g µ (cid:48) ν (cid:48) ( k (cid:48) A + p ) σ (cid:48) + g ν (cid:48) σ (cid:48) ( q (cid:48) − p ) µ (cid:48) + g σ (cid:48) µ (cid:48) ( − q (cid:48) − k (cid:48) A ) ν (cid:48) ] (F22) I = Tr[ γ ( n · γ )( p · γ + m ) γ ν ( q · γ + m ) γ µ × Λ / ( − k B )( γ + 1)Λ − / ( − k (cid:48) B ) × γ ν (cid:48) ( q (cid:48) · γ + m ) γ µ (cid:48) ] g µµ (cid:48) g νν (cid:48) (F23) I = Tr[ γ ( n · γ )( p · γ + m ) γ ν ( q · γ + m ) γ µ × Λ / ( − k B )( γ + 1)Λ − / ( − k (cid:48) B ) × γ µ (cid:48) ( q (cid:48) · γ + m ) γ ν (cid:48) ] g µµ (cid:48) g νν (cid:48) (F24) ¯ q a q a → ¯ q b q b with a (cid:54) = b For the polarization of q b in the final state, we obtain ∆ I ¯ q a q a → ¯ q b q b M = C ¯ q a q a → ¯ q b q b g s m q q (cid:48) × Tr (cid:104) Λ / ( − k (cid:48) A )( γ − − / ( − k A ) γ µ × Λ / ( − k B )( γ + 1)Λ − / ( − k (cid:48) B ) γ ν (cid:105) × Tr [ γ ( n · γ )( p · γ − m ) γ µ ( p · γ − m ) γ ν ] , (F25)0where q = k A + k B and q (cid:48) = k (cid:48) A + k (cid:48) B are momenta in propagators. 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