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- ​ - Next topic: [[documentation:​guide:​mappings_of_indices]] - <​html>​ - ​ - ​ - ​ - - ---- - - - Besides general features for manipulation with tensors and a [[list of transformations|wide range of general-purpose transformations]],​ Redberry provides a set of transformations and functions required for high energy physics (HEP). - - ====Calculation of one-loop counterterms==== - Redberry provides a set of tools for calculation of one-loop counterterms in curved space-time. These tools allow to calculate one-loop counterterms of an arbitrary theory (with second or fourth order operator in Lagrangian quadratic form) and background in four dimensions in curved space-time in the dimensional regularization. ​ - - For details see [[documentation:​guide:​calculating_one-loop_counterterms]] and the following tutorials: - * [[documentation:​tutorials:​vector_field]] - * [[documentation:​tutorials:​minimal_second_order_operator]] - * [[documentation:​tutorials:​minimal_fourth_order_operator]] - * [[documentation:​tutorials:​squared_vector_field]] - * [[documentation:​tutorials:​Gravity]] - ====Solving equations==== - Tensorial equations are often occur for example when one need to calculate propagator or find a projector operator etc. Redberry provides a function [[documentation:​ref:​Reduce]] which reduces a system of tensorial equations to a system of symbolic equations and allows to solve the last one with external "​scalar"​ CAS. - - For examples, see tutorials: - * [[documentation:​tutorials:​spin-3_propagator]] - * [[documentation:​tutorials:​Projectors for tensor particle]] - ====Dirac & SU(N) algebra==== - Redberry provides a set of common transformations needed for simplification expressions with Dirac and SU(N) matrices. ​ - - ===DiracTrace=== - evaluates trace of gamma matrices: - - defineMatrices '​G_m',​ '​G5',​ Matrix1.matrix - println DiracTrace[[Gamma:​ '​G_m'​]] >> '​Tr[G_m*G_n]'​.t - ​ - - > 4*g_{mn} - ​ - See [[documentation:​ref:​DiracTrace]]. - - ---- - ===SpinorsSimplify=== - simplifies Dirac spinors: - - defineMatrices '​G_a',​ '​G5',​ Matrix1.matrix,​ '​cu',​ Matrix1.covector - def sSimplify = SpinorsSimplify[[uBar:​ '​cu',​ Momentum: '​p_a',​ Mass: '​m'​]] - println sSimplify >> '​cu*G^a*p_a'​.t - ​ - - > -m*cu*G_{b}+2*cu*p_{b} - ​ - See [[documentation:​ref:​SpinorsSimplify]]. - - ---- - ===DiracSimplify=== - simplifies products of gamma matrices: - - defineMatrices '​G_a',​ '​G5',​ Matrix1.matrix - println DiracSimplify >> '​G5*G_a*G5*G_b*G^a*G^b'​.t - ​ - - > -8 - ​ - See [[documentation:​ref:​DiracSimpllify]]. - - ---- - ===DiracOrder=== - order products of gamma matrices: - - defineMatrices '​G_a',​ '​G5',​ Matrix1.matrix - println DiracOrder >> '​G5*G_c*G_b*G_a'​.t - ​ - - > G_{a}*G_{b}*G_{c}*G5+2*G_{b}*G5*g_{ca}-2*G_{c}*G5*g_{ba}-2*G_{a}*G5*g_{cb} - ​ - See [[documentation:​ref:​DiracOrder]]. - - ---- - ===LeviCivitaSimplify=== - simplifies combinations with Levi-Civita tensors: - - println LeviCivitaSimplify.minkowski['​e_abcd'​.t] >> '​e_abcm*e^abcn'​.t - ​ - - > -6*d_{m}^{n} - ​ - See [[documentation:​ref:​ LeviCivitaSimplify]]. - - ---- - ===setMandelstam=== - generates a list of mass shell and Mandelstam substitutions:​ - - def mandelstam = - ​setMandelstam([k1_a:​ '​m1',​ k2_a: '​m2',​ k3_a: '​m3',​ k4_a: '​m4'​]) - println mandelstam >> '​k1_a*k2^a + k3_b*k1^b'​.t - ​ - - > (1/​2)*(-m2**2-m1**2+s)-(1/​2)*(t-m3**2-m1**2) - ​ - See [[documentation:​ref:​setMandelstam]]. - - ---- - ===setMandelstam5=== - generates a list of mass shell and general Mandelstam substitutions for 2->3 processes: - - def mandelstam = - ​setMandelstam5([k1_a:​ '​m1',​ k2_a: '​m2',​ k3_a: '​m3',​ k4_a: '​m4',​ k5_a: '​m5'​]) - println mandelstam >> '​k1_a*k2^a + k3_b*k1^b + k5_a*k1^a'​.t - ​ - - > (1/​2)*(s-m2**2-m1**2)+(1/​2)*(-t1+m3**2+m1**2)+(1/​2)*(s+t1-m4**2-m2**2-m3**2+t2-m1**2) - ​ - See [[documentation:​ref:​setMandelstam5]]. - - ---- - ===UnitarySimplify=== - simplifies combinations of unitary matrices and SU(N) structural and $d$-constants - - defineMatrices '​T_A',​ Matrix2.matrix - println UnitarySimplify[[Matrix:​ '​T_A'​]] >> '​T_A*T^A'​.t - ​ - - > (1/​2)*N**(-1)*(N**2-1) - ​ - See [[documentation:​ref:​UnitarySimplify]]. - - ---- - ===UnitaryTrace=== - evaluates trace of unitary matrices: - - defineMatrices '​T_A',​ Matrix2.matrix - println UnitaryTrace[[Matrix:​ '​T_A'​]] >> '​Tr[T_A*T_B]'​.t - ​ - - > (1/​2)*g_{BA} - ​ - See [[documentation:​ref:​ UnitaryTrace]]. - - For examples, see tutorials: - * [[documentation:​tutorials:​compton_scattering_in_qed]] - * [[documentation:​tutorials:​compton_scattering_in_qcd]] - * [[documentation:​tutorials:​b_c_to_u_bar_d_gamma|$B_c \to u \bar d \gamma$]] - ====See also==== - * Related guides: [[documentation:​guide:​list_of_transformations]],​ [[documentation:​guide:​calculating_one-loop_counterterms]],​ [[documentation:​guide:​programming_with_redberry]]