Violation of CP symmetry. Charles Dunkl (University of Virginia)

Transcription

1 Violation of CP symmetry form a point of view of a mathematician Karol Życzkowski in collaboration with Charles Dunkl (University of Virginia) J. Math. Phys. 50, (2009). ZFCz IF UJ, Kraków January 24, 2012 KŻ (UJ) CP violation & unistochastic matrices / 18

2 Discussion with a mathematician: A communication issue: How to explain a mathematician a) What is the CKM Matrix? b) What the term violation of CP symmetry actually means? c) Is it related with the fact that the CKM Matrix is of size 3 (and not 2)? KŻ (UJ) CP violation & unistochastic matrices / 18

3 A mathematical perspective for CP violation I 1) In high energy physics two kinds of interactions are important. They are called strong and weak interactions. 2) Both interactions can be described by Hamiltonians represented by Hermitian matrices H s and H w. 3) Elementary particles consists of quarks, which can be divided into classes, called generations. Initially one used N = 2 generations of quarks, but new particles discovered in sixties forced us to include the third generation. Several later attempts to discover the fourth generation failed, so N = 3 (right now). 4) Both Hamiltonians are represented by generic Hermitian matrices of order N, so they do not commute, [H s, H w ] 0. KŻ (UJ) CP violation & unistochastic matrices / 18

4 A mathematical perspective for CP violation II 5) Representing Hermitian matrices by their eigendecompositions H w = N i=1 w i ψ i ψ i and H s = N j=1 s j φ j φ j one introduces a unitary matrix V U(N) which transforms one eigenbasis into another one: V ij := (ψ i,φ j ) = ψ i φ j. V CKM U(3) CKM matrix (Cabbibo-Kobayashi-Maskawa 1973) 6) Physicists are interested in this very matrix, but they can only measure transition probabilities, B ij = V ij 2, for i, j = 1,...,N. 7) The matrix B is bistochastic: has positive elements B ij 0, sum in each column (each row) equal to unity, i B ij = j B ij = 1. 8) A natural mathematical question arises: Given a bistochastic matrix B find out if there exists a corresponding unitary matrix V such that V ij 2 = B ij and check, whether such a unitary V is orthogonal. KŻ (UJ) CP violation & unistochastic matrices / 18

5 Set B N of bistochastic matrices of size N The Birkhoff polytope A square matrix B is called bistochastic (doubly stochastic) if it has positive elements B ij 0, the sum in each column and each row is equal to unity, i B ij = j B ij = 1. Birkhoff theorem. Every bistochastic matrix can be written as a convex combination of permutation matrices P k, B = k q kp k. Thus the set B N is called the Birkhoff polytope In general a matrix B B N is described by (N 1) 2 parameters, so the Birkhoff polytope B N Ê (N 1)2. Bistochastic matrices for N = 2 [ ] a 1 a B 2 = (a) = = a½ + (1 a)p 1 a a 12, for a [0, 1]. Thus N = 2 Birkhoff polytope is equivalent to unit interval, B 2 = [0, 1]. KŻ (UJ) CP violation & unistochastic matrices / 18

6 Set B N of bistochastic matrices of size N Bistochastic matrices for N = 3, for which B 3 Ê 4 b 1 b 2 1 b 1 b 2 B 3 (b 1, b 2, b 3, b 4 ) := b 3 b 4 1 b 3 b 4 B 3 1 b 1 b 3 1 b 2 b 4 4 i=1 b i 1 The set B 3 is the convex hull of 3! = 6 permutation matrices, {½, P = P 123, P 2 = P 132, P 12, P 13, P 23 }. W denotes the flat bistochastic matrix, W = 1 3 [1, 1, 1; 1, 1, 1; 1, 1, 1], located at the center of the Birkhoff polytope B 3. KŻ (UJ) CP violation & unistochastic matrices / 18

7 Unistochastic and orthostochastic matrices Definitions A bistochastic matrix B B N is called unistochastic if there exists a unitary U U(N) such that B ij = U ij 2, written B = f (U). A bistochastic matrix B B N is called orthostochastic if there exists an orthogonal O O(N) such that B ij = O ij 2, written B = f (O). Let U N and O N denote the sets of unistochastic and orthostochastic matrices of size N, respectively. By definition the following inclusion relation hold O N U N B N. KŻ (UJ) CP violation & unistochastic matrices / 18

8 Uni- and ortho-stochastic matrices for N = 2 Proposition For N = 2 all three sets coincide, O 2 = U 2 = B 2 = [0, 1]. [ Proof. Take any a [0, 1] and set B = a 1 a 1 a a The corresponding[ orthogonal matrix ] reads cos ϑ sinϑ O =, where a = cos sinϑ cos ϑ 2 ϑ. ]. In other words every N = 2 bistochastic matrix is orthostochastic (and thus also unistochastic). KŻ (UJ) CP violation & unistochastic matrices / 18

9 Uni- and ortho-stochastic matrices for N = 3 Proposition. For N = 3 both inclusion relations O 3 U 3 B 3 are proper. a) b) Proof by demonstration a) Fourier matrix F 3 = ω ω 2 with ω = e 2πi/3 is unitary and 1 ω 2 ω corresponds to the flat bistochastic matrix W = f (F 3 ), as W ij = 1/3. Thus W is unistochastic but not orthostochastic, since for N = 3 there are no Hadamard matrices. b) Example of Schur: the bistochastic matrix B S, B S = P+P2 2 = is bistochastic but not unistochastic KŻ (UJ) CP violation & unistochastic matrices / 18

10 Unistochasticity and chain link condition for N = 3 Unitarity condition UU = ½ can be written as u µ u ν = δ µ,ν Diagonal elements u µ u µ = 1, impose bistochasticity, i B iµ = 1 while orthogonality relation u µ u ν = 0 imposes further constraints for elements of B = f (U)! What constraints for unistochasticity? As the sum u 1 u 2 = 3 j=1 U j1uj2 = L 1e iχ 1 + L 2 e iχ 2 + L 3 e iχ 3 of three complex numbers should vanish, their (ordered) moduli L 1 L 2 L 3 satisfy the following chain link condition (triangle inequality) L 1 L 2 + L 3 with L k := B 1k B 2k. a) unistochastic matrix with a positive area of the unitarity triangle, A 2 > 0, b) limiting case: an orthostochastic matrix with A 2 = 0, c) bistochastic matrix B not included into U 3 for which A 2 < 0. KŻ (UJ) CP violation & unistochastic matrices / 18

11 Unitarity triangle formed by links L 1, L 2, L 3 The length of the links of the unitarity triangle read L 1 = b 1 b 2, L 2 = b 3 b 4, L 3 = (1 b 1 b 2 )(1 b 3 b 4 ), (1) Let p = (L 1 + L 2 + L 3 )/2 denotes its semiperimeter. Making use of the Heron s formula for the area of the triangle A = p(p L 1 )(p L 2 )(p L 3 ), (2) we arrive with a compact expression for the squared area A 2, A 2 = [4b 1 b 2 b 3 b 4 (b 1 + b 2 + b 3 + b 4 1 b 1 b 4 b 2 b 3 ) 2 ]/16. (3) The chain links conditions are equivalent to a single condition for unistochasticity: A 2 (B) 0 (if a triangle exists its area is real and positive!) KŻ (UJ) CP violation & unistochastic matrices / 18

12 The set U 3 of unistochastic matrices of size N = 3 cross-sections of U 3 (implied by A 2 (B) 0 ) Nonconvex 3-Hypocycloid obtained by the cross-section of U 3 along the plane spanned by the equilateral triangle (P, P 2,½), b) a similar cross-section along totally orthogonal plane, c) a view from above. The set O 3 of orthostochastic matrices Proposition. For N = 3 the set O 3 of orthostochastic matrices forms the boundary of the 4D set U 3 of unistochastic matrices. KŻ (UJ) CP violation & unistochastic matrices / 18

13 Unistochastic matrices are useful for quantizing classical dynamical systems (which lead to bistochastic transition matrices). Prot Pakoński, Ph.D. Thesis 2002, Pakoński, Życzkowski, Kuś, 2001, The set U 3 of N = 3 unistochastic matrices was investigated in Bentgsson, Ericsson, Kuś, Tadej, Życzkowski, Commun. Math. Phys. (2005). The set U 3 of unistochastic matrices of size N = 3 occupies (with respect to the Lebesgue measure) more than 3/4 of the corresponding Birkhoff polytope B 3, Dunkl, Życzkowski, 2009 vol(u 3 ) vol(b 3 ) = 8π2 = (4) 105 KŻ (UJ) CP violation & unistochastic matrices / 18

14 Unitarity triangle and Jarlskog invariant (for N = 3) Jarlskog invariant For any unitary U U(3) define the number J(U) := Im(U 11 U 22 U 12U 21) called Jarlskog invariant. Equivalent unitary matrices Two unitary matrices U and U are called equivalent if there exist two diagonal unitary matrices, D A and D B, and two permutations P A and P B such that U U = D A P A UP B D B (5) The following relation holds: if U U then J(U) = J(U ), Jarlskog 1985 Simple calculation shows that the Jarlskog invariant is related to the area of the unitarity triangle, J 2 (U) = 4A 2 (B), where B = f (U). KŻ (UJ) CP violation & unistochastic matrices / 18

15 Jarlskog invariant: extreme values Squared Jarlskog invariant, J 2 = 4A 2, computed for bistochastic matrices takes values between J 2 min = J2 (B s ) = 1/64 for the Schur bistochastic matrix (not unistochastic) to J 2 max = J 2 (W) = 1/108 for the flat bistochastic matrix W = f (F 3 ). If J 2 (B) = 0 the matrix B is orthostochastic. Jarlskog invariant: probability distribution a) J=2A Ι b) 2 2 J =4A Ι * P P 2 P P 2 Distributions a) P( J ) and b) P(J 2 ) at the cross-section of U 3 along the plane formed by two permutation matrices and the identity. KŻ (UJ) CP violation & unistochastic matrices / 18

16 Jarlskog invariant: probability distribution analytical formulae for moments and probability distribution P( J ) given explicitly as an integral of a hypergeometric function 2 F 1, Distributions P( J ) for random unistochastic matrices with respect to a) flat measure (dashed line); b) Haar measure (solid line). KŻ (UJ) CP violation & unistochastic matrices / 18

17 Jarlskog invariant for CKM matrix Experimental data for the CKM matrix V CKM give J obs (V CKM ) = 3.08 [+0.16, 0.18] 10 5 This implies the following probability that a random unitary matrix U U(3) distributed according to Haar measure yields a smaller value of the Jarlskog invariant P { J J obs } = 7.74 [+0.41, 0.45] Such a statement improves the estimation made in the 2009 paper of Gibbons, Gielen, Pope and Turok. Hence CKM matrix V CKM should not be considered as a generic unitary matrix drawn at random with respect to the Haar measure on U(3). KŻ (UJ) CP violation & unistochastic matrices / 18

18 Concluding Remarks A bistochastic matrix B corresponds to a unitary matrix if it is unistochastic, B = f (U) so that B ij = U ij 2. The set U N of unistochastic matrices is characterized for N = 3 as B U 3 A 2 (B) 0. For N 4 the unistochasticity problem for remains open. for N = 2 every bistochastic matrix is orthostochastic. For N = 3 a generic unistochastic matrix is not orthostochastic, so the underlying unitary is not real. Thus the symmetry with respect to complex conjugation is violated. For N = 3 we computed the volume of the set U 3 and the average value of the Jarlskog invariant J for a random Haar unitary matrix U U(3). Cumulative probability P { J J obs } is small, what (according to Gibbons et al.) may provide an argument that the CKM matrix V CKM is not a generic random unitary of order N = 3. KŻ (UJ) CP violation & unistochastic matrices / 18