About calculation of traces of Dirac γ -matrices contracted with massless vectors in Minkowski space
aa r X i v : . [ h e p - ph ] O c t About calculation of traces of Dirac γ -matrices contractedwith massless vectors in Minkowski space Alexander L. Bondarev and Alexander R. Roslik National Scientific and Educational Center of Particle and High Energy Physicsof the Belarusian State UniversityM.Bogdanovich str.,153, Minsk 220050, Belarus e-mail: [email protected] e-mail: [email protected] Abstract
A new method for calculation of traces of Dirac γ -matrices contracted with masslessvectors in Minkowski space is discussed Calculation of Dirac γ -matrices traces is standard procedure in high energy physics computa-tion. However the expressions for the traces of more than ten γ -matrices are too long.Application of Chisholm [1] identities and Kahane [2] algorithm leads to short expressionsfor traces in 4-spacetime. More compact formulae can be obtained through the new methodproposed in [3] and based on the properties of orthonormal bases in the Minkowski space andisotropic tetrads (see e.g. [4]-[5]) constructed from the vectors of these bases.The purpose of the article is further simplification of the formulae presented in [3] for thecase of Dirac γ -matrices contracted with massless vectors. This task is important today becausemost of analytical calculations in high energy physics are performed at massless approximation(see, for example, [6]).The article is arranged in the following way. The new method of trace computation forarbitrary vectors is shortly described in Section 2. In Section 3 one explores application of thetraces calculation method for Dirac γ -matrices contracted with massless vectors. The formulaeobtained there are good enough to realize an algorithm for any numerical calculation.Note, that the method of calculation of the traces from the [3] is included in the system ofanalytical computation ALHEP [8], and this article contains the proposals for further improve-ment of the similar programs. 1n this paper we use the Feynman metrics: µ = 0 , , , , a µ = ( a , ~a ) , a µ = ( a , − ~a ) , ( ab ) = a µ b µ = a b − ~a~b , and a sign of the Levi-Civita tensor is fixed as ε = +1 . Orientation of the orthonormal basis vectors in Minkowski space( l A l B ) = g AB , ( A, B = 0 , , , ε µνρλ l µ l ν l ρ l λ = +1 . Isotropic tetrads q ± , e ± are constructing in the following way: q + = l + l √ , q − = l − l √ , ( q + q − ) = 1 ; e + = l + il √ , e − = l − il √ , ( e + e − ) = − , e ∗∓ = e ± ;( q + e + ) = ( q + e − ) = ( q − e + ) = ( q − e − ) = 0 . (1) γ -matricescontracted with arbitrary vectors The new method to calculate traces of Dirac γ -matrices contracted with arbitrary vectors a i ( i = 1 , · · · , n ) in Minkowski space is presented in [3]. So the following expressions wereobtained: 12 Tr[(1 − γ )ˆ a ˆ a ] = F ( a , a ) + F ( a , a ) ; (2)12 Tr[(1 − γ )ˆ a ˆ a ˆ a ˆ a ] == F ( a , a ) F ( a , a ) + F ( a , a ) F ( a , a ) + F ( a , a ) F ( a , a ) + F ( a , a ) F ( a , a ) ; (3)where F , F , F , F are some functions (see below) of vectors a i .Hawing calculated Tr[(1 − γ )ˆ a · · · ˆ a n − ˆ a n ] , we may obtain an expression forTr[(1 − γ )ˆ a · · · ˆ a n − ˆ a n ˆ a n +1 ˆ a n +2 ]2rom the previous one by the following replacement: F ( a n − , a n ) → F ( a n − , a n ) F ( a n +1 , a n +2 ) + F ( a n − , a n ) F ( a n +1 , a n +2 ) ,F ( a n − , a n ) → F ( a n − , a n ) F ( a n +1 , a n +2 ) + F ( a n − , a n ) F ( a n +1 , a n +2 ) ,F ( a n − , a n ) → F ( a n − , a n ) F ( a n +1 , a n +2 ) + F ( a n − , a n ) F ( a n +1 , a n +2 ) ,F ( a n − , a n ) → F ( a n − , a n ) F ( a n +1 , a n +2 ) + F ( a n − , a n ) F ( a n +1 , a n +2 ) . (4)The expression for Tr[(1 − γ )ˆ a · · · ˆ a n − ˆ a n ]obtained through the method will contain 2 n terms.Tr[(1 + γ )ˆ a ˆ a · · · ˆ a n ] = (cid:16) Tr[(1 − γ )ˆ a ˆ a · · · ˆ a n ] (cid:17) ∗ . (5)The functions mentioned above have the following forms: F ( a i , a j ) = 2[( a i q − )( a j q + ) − ( a i e + )( a j e − )] == ( a i a j ) + G (cid:18) a i a j l l (cid:19) + i G (cid:18) a i a j l l (cid:19) == 14 Tr[(1 − γ )ˆ q + ˆ q − ˆ a i ˆ a j ] = −
14 Tr[(1 − γ )ˆ e − ˆ e + ˆ a i ˆ a j ] , (6) F ( a i , a j ) = 2[( a i q + )( a j q − ) − ( a i e − )( a j e + )] == ( a i a j ) − G (cid:18) a i a j l l (cid:19) − i G (cid:18) a i a j l l (cid:19) == 14 Tr[(1 − γ )ˆ q − ˆ q + ˆ a i ˆ a j ] = −
14 Tr[(1 − γ )ˆ e + ˆ e − ˆ a i ˆ a j ] , (7) F ( a i , a j ) = 2[( a i e + )( a j q − ) − ( a i q − )( a j e + )] = 2 G (cid:18) a i a j e + q − (cid:19) == − G (cid:18) a i a j l l (cid:19) + i G (cid:18) a i a j l l (cid:19) + G (cid:18) a i a j l l (cid:19) − i G (cid:18) a i a j l l (cid:19) == 14 Tr[(1 − γ )ˆ q − ˆ e + ˆ a i ˆ a j ] , (8) F ( a i , a j ) = 2[( a i e − )( a j q + ) − ( a i q + )( a j e − )] = 2 G (cid:18) a i a j e − q + (cid:19) == − G (cid:18) a i a j l l (cid:19) + i G (cid:18) a i a j l l (cid:19) − G (cid:18) a i a j l l (cid:19) + i G (cid:18) a i a j l l (cid:19) == 14 Tr[(1 − γ )ˆ q + ˆ e − ˆ a i ˆ a j ] , (9)3here G are Gram determinants.There is an example of F i functions for some orthonormal basis. Let’s to fix the last asfollow: l µ = (1 , , , , l µ = (0 , , , , l µ = (0 , , , , l µ = (0 , , , . (10)Then the isotropic tetrads can be expressed as q µ ± = 1 √ , ± , , , ( aq ± ) = 1 √ a ∓ a x ) ,e µ ± = 1 √ , , , ± i ) , ( ae ± ) = − √ a y ± ia z ) . (11)At last F ( a i , a j ) = ( a i a j ) − [ ( a i ) ( a j ) x − ( a i ) x ( a j ) ] + i [ ( a i ) y ( a j ) z − ( a i ) z ( a j ) y ] ,F ( a i , a j ) = ( a i a j ) + [ ( a i ) ( a j ) x − ( a i ) x ( a j ) ] − i [ ( a i ) y ( a j ) z − ( a i ) z ( a j ) y ] ,F ( a i , a j ) = [ ( a i ) ( a j ) y − ( a i ) y ( a j ) ] + i [ ( a i ) x ( a j ) z − ( a i ) z ( a j ) x ] ++ [ ( a i ) x ( a j ) y − ( a i ) y ( a j ) x ] + i [ ( a i ) ( a j ) z − ( a i ) z ( a j ) ] ,F ( a i , a j ) = [ ( a i ) ( a j ) y − ( a i ) y ( a j ) ] + i [ ( a i ) x ( a j ) z − ( a i ) z ( a j ) x ] −− [ ( a i ) x ( a j ) y − ( a i ) y ( a j ) x ] − i [ ( a i ) ( a j ) z − ( a i ) z ( a j ) ] . (12)For clarity one can writes expressions of traces in the following form:12 Tr h (1 − γ )ˆ a ˆ a ˆ a ˆ a · · · ˆ a n − ˆ a n i == Tr " F ( a , a ) F ( a , a ) F ( a , a ) F ( a , a ) · " F ( a , a ) F ( a , a ) F ( a , a ) F ( a , a ) · · · · · " F ( a n − , a n ) F ( a n − , a n ) F ( a n − , a n ) F ( a n − , a n ) . (13)It’s obvious that the trace of 2 n Dirac γ -matrices is reduced to trace of n matrices with 2 × γ ρ ˆ a ˆ a · · · ˆ a n +1 ) Tr( γ ρ ˆ b ˆ b · · · ˆ b m +1 ) , where summing over index ρ is supposed, one can use the Fiertz transform:[(1 ± γ ) γ ρ ] ij [(1 ∓ γ ) γ ρ ] kl = 2(1 ± γ ) il (1 ∓ γ ) kj . (14)4o one achieves the following expressions:Tr[(1 ± γ ) γ ρ ˆ a ˆ a · · · ˆ a n +1 ] · Tr[(1 ∓ γ ) γ ρ ˆ b ˆ b · · · ˆ b m +1 ] == 4 Tr[(1 ∓ γ )ˆ a ˆ a · · · ˆ a n +1 ˆ b ˆ b · · · ˆ b m +1 ] , (15)Tr[(1 ± γ ) γ ρ ˆ a ˆ a · · · ˆ a n +1 ] · Tr[(1 ± γ ) γ ρ ˆ b ˆ b · · · ˆ b m +1 ] == Tr[(1 ± γ ) γ ρ ˆ a ˆ a · · · ˆ a n +1 ] · Tr[(1 ∓ γ ) γ ρ ˆ b m +1 · · · ˆ b ˆ b ] == 4 Tr[(1 ∓ γ )ˆ a ˆ a · · · ˆ a n +1 ˆ b m +1 · · · ˆ b ˆ b ] , (16)Tr( γ ρ ˆ a ˆ a · · · ˆ a n +1 ) · Tr( γ ρ ˆ b ˆ b · · · ˆ b m +1 ) = 2 Tr h ˆ a ˆ a · · · ˆ a n +1 (ˆ b ˆ b · · · ˆ b m +1 + ˆ b m +1 · · · ˆ b ˆ b ) i . (17) γ -matrices contracted withmassless vectors In the method of trace calculation, what was briefly explained in Section 2, vectors a i contractedwith Dirac γ -matrices are arbitrary. But there is a significant simplification of expressions fortraces when vectors a i are massless (see e.g. [6], [7]).For massless vectors the particular equation is true: a i = ( a i l ) − ( a i l ) − ( a i l ) − ( a i l ) = 0 , (18)i.e. ( a i l ) − ( a i l ) = ( a i l ) + ( a i l ) , (19)or ( a i q + )( a i q − ) = ( a i e + )( a i e − ) . (20)Thus the functions F k ( a i , a j ) become to be like the function F ( a i , a j ) multiplied by somefactor: F ( a i , a j ) = 2[( a i e + )( a j q − ) − ( a i q − )( a j e + )] == 2[( a i e + ) ( a j e + )( a j e − )( a j q + ) − ( a i q − )( a j e + ) ( a j q + )( a j q + ) ] = − ( a j e + )( a j q + ) 2[( a i q − )( a j q + ) − ( a i e + )( a j e − )] , (21)5hat is F ( a i , a j ) = − ( a j e + )( a j q + ) F ( a i , a j ) . (22)In the similar way F ( a i , a j ) = − ( a i e − )( a i q − ) · ( a j e + )( a j q + ) F ( a i , a j ) = ( a i e − )( a i q − ) F ( a i , a j ) = − ( a j e + )( a j q + ) F ( a i , a j ) . (23) F ( a i , a j ) = ( a i e − )( a i q − ) F ( a i , a j ) . (24)Formulae of the traces become to be simpler by far.Through the identity(1 ± γ )ˆ qQ (1 ± γ )ˆ q = Tr[(1 ± γ )ˆ qQ ] (1 ± γ )ˆ q , (25)which is valid for any massless vector q and any operator Q , one can obtain12 Tr[(1 − γ )ˆ a ˆ a ˆ a · · · ˆ a n ] = 14( a q − ) Tr[(1 + γ )ˆ q − ˆ a ˆ a ˆ a · · · ˆ a n ˆ a ] == 14( a q − ) · a q − ) · Tr[(1 + γ )ˆ q − ˆ a ˆ a ˆ a (1 + γ )ˆ q − ˆ a · · · ˆ a n ˆ a ] == Tr[(1 + γ )ˆ q − ˆ a ˆ a ˆ a ]4( a q − ) · a q − ) Tr[(1 + γ )ˆ q − ˆ a · · · ˆ a n ˆ a ] = · · · == Tr[(1 + γ )ˆ q − ˆ a ˆ a ˆ a ]4( a q − ) · Tr[(1 + γ )ˆ q − ˆ a ˆ a ˆ a ]4( a q − ) · · · · · Tr[(1 + γ )ˆ q − ˆ a n − ˆ a n ˆ a ]4( a n − q − ) . (26)Further14( a i q − ) Tr[(1 + γ )ˆ q − ˆ a i ˆ a j ˆ a k ] = 116( a i q − )( a j q + ) Tr[(1 + γ )ˆ q − ˆ a i ˆ a j (1 − γ )ˆ q + ˆ a j ˆ a k ] == 132( a i q − )( a j q + ) · Tr[(1 + γ )ˆ q − ˆ a i ˆ a j ˆ q + (1 + γ )ˆ q − ˆ q + ˆ a j ˆ a k ] == 132( a i q − )( a j q + ) · Tr[(1 − γ )ˆ q + ˆ q − ˆ a i ˆ a j ] Tr[(1 + γ )ˆ q − ˆ q + ˆ a j ˆ a k ] = F ( a i , a j ) F ∗ ( a j , a k )2( a i q − )( a j q + ) , (27)and finally from (26), (27) it follows that12 Tr[(1 − γ )ˆ a ˆ a ˆ a · · · ˆ a n ] = F ( a , a ) F ∗ ( a , a ) · · · F ( a n − , a n ) F ∗ ( a n , a )2 n ( a q − )( a q + )( a q − ) · · · ( a n q + ) . (28)From (22), (23) we have F ( a i , a j ) F ∗ ( a j , a k )( a j q + ) = − F ( a i , a j ) F ∗ ( a j , a k )( a j q − ) , (29)6nd (28) takes the form12 Tr[(1 − γ )ˆ a ˆ a ˆ a · · · ˆ a n ] = ( − n · F ( a , a ) F ∗ ( a , a ) · · · F ( a n − , a n ) F ∗ ( a n , a )2 n ( a q − )( a q − )( a q − ) · · · ( a n q − ) . (30)Note thatTr[(1 − γ )ˆ a ˆ a ˆ a · · · ˆ a n ] = Tr[(1 + γ )ˆ a ˆ a · · · ˆ a n ˆ a ] = (cid:16) Tr[(1 − γ )ˆ a ˆ a · · · ˆ a n ˆ a ] (cid:17) ∗ , (31)then (28) leads to the one more expression of traces12 Tr[(1 − γ )ˆ a ˆ a ˆ a · · · ˆ a n ] = F ( a , a ) F ∗ ( a , a ) · · · F ( a n − , a n ) F ∗ ( a n , a )2 n ( a q + )( a q − )( a q + ) · · · ( a n q − ) . (32)At last an identity [see (23), (24)] F ( a i , a j ) F ∗ ( a j , a k )( a j q − ) = − F ( a i , a j ) F ∗ ( a j , a k )( a j q + ) , (33)provide for12 Tr[(1 − γ )ˆ a ˆ a ˆ a · · · ˆ a n ] = ( − n · F ( a , a ) F ∗ ( a , a ) · · · F ( a n − , a n ) F ∗ ( a n , a )2 n ( a q + )( a q + )( a q + ) · · · ( a n q + ) . (34)The expressions for traces (28), (30), (32), (34) are equivalent. Since in the case of masslessvectors | F ( a i , a j ) | = 2 q ( a i a j )( a i q − )( a j q + ) , | F ( a i , a j ) | = 2 q ( a i a j )( a i q + )( a j q − ) , | F ( a i , a j ) | = 2 q ( a i a j )( a i q − )( a j q − ) , | F ( a i , a j ) | = 2 q ( a i a j )( a i q + )( a j q + ) , (35)then (cid:12)(cid:12)(cid:12)(cid:12)
12 Tr[(1 − γ )ˆ a ˆ a ˆ a · · · ˆ a n ] (cid:12)(cid:12)(cid:12)(cid:12) = 2 n q ( a a )( a a ) · · · ( a n a ) , (36)(see also [6]). uncer-tainty The appearance of uncertainty type 00 is possible during numerical calculation by reason ofdenominators presence in the formulae (28), (30), (32), (34).7or example, when the formula (34) is being used and orthonormal basis is chosen in ac-cording to (10), the uncertainty will appear, if a i = const · q + i.e. if ( a i ) = ( a i ) x . The simplest way to avoid the appearing of uncertainties in such points of phase space isto use for calculation here another basis vectors, because obtained formulas are correct for anyorthonormal basis and numerical results received are independent from it’s choice. But suchapproach results in unnecessary complicating of the computer program.There are three solutions of this problem: At the points of phase space, where denominators are 0, one should use common formulaefrom Section 2 (see e.g. [8]). One can choose an arbitrary 4-vector t normalized by condition t = 1 to perform theidentical transformation of the initial expression:12 Tr[(1 − γ )ˆ a ˆ a ˆ a · · · ˆ a n ] = 12 Tr[(1 + γ )ˆ t ˆ a ˆ t ˆ t ˆ a ˆ t ˆ t ˆ a ˆ t · · · ˆ t ˆ a n ˆ t ] == 12 Tr[(1 + γ )ˆ a ′ ˆ a ′ ˆ a ′ · · · ˆ a ′ n ] = 12 Tr[(1 − γ )ˆ a ′ ˆ a ′ · · · ˆ a ′ n ˆ a ′ ] , (37)where a ′ i = − a i + 2( a i t ) t . (38)Thus ( a ′ i ) = ( a i ) − a i t ) + 4( a i t ) t = ( a i ) = 0 , (39)so one can use the formulae from Section 3.1 to calculate the transformed expression. At thesame time the denominators take forms( a ′ i q ± ) = − ( a i q ± ) + 2( a i t )( tq ± ) , (40)where the second term in (40) is positive. It is possible to transform the assumption formula. Let us suppose that (34) is using forcalculation and for i = 2 s ( a s q + ) = 0 . In this situation one can replace [see (33)] F ( a s − , a s ) F ∗ ( a s , a s +1 )( a s q + ) → − F ( a s − , a s ) F ∗ ( a s , a s +1 )( a s q − ) . (41)Notice that in this case ( a s q − ) = 0 . Conclusion
The simple and compact formulae are proposed to calculate traces of Dirac γ -matrices con-tracted with massless vectors. These formulae may be easily implemented as a simple yetefficient computer algorithm. References [1] J.S.R. Chisholm, Nuovo Cim. 30 (1963) 426.[2] J. Kahane, J. Math. Phys. 9 (1968) 1732.[3] A.L. Bondarev, Nucl. Phys. B 733 (2006) 48; E-print arXiv: hep-ph/0504223.[4] V.I. Borodulin, R.N. Rogalyov, S.R. Slabospitsky, Preprint IHEP-95-90 (Protvino, 1995);E-print arXiv: hep-ph/9507456.[5] V.V. Andreev, Phys. Rev. D 62 (2000) 014029; E-print arXiv: hep-ph/0101140.[6] R. Gastmans and T.T. Wu,