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58-410: CKM may refer to: Cabibbo–Kobayashi–Maskawa matrix in particle physics CKM (magazine) , a Polish men's magazine C. K. McClatchy High School Cotton Keays & Morris , an Australian pop music band Creatine kinase , muscle, an enzyme CKM (gene) , a gene that in humans encodes the enzyme creatine kinase, muscle Topics referred to by

116-547: A {\textstyle a} , b {\textstyle b} are real numbers), that are used for generalization of this notion to other domains: Non-negativity, positive definiteness, and multiplicativity are readily apparent from the definition. To see that subadditivity holds, first note that | a + b | = s ( a + b ) {\displaystyle |a+b|=s(a+b)} where s = ± 1 {\displaystyle s=\pm 1} , with its sign chosen to make

174-407: A + b | = s ⋅ ( a + b ) = s ⋅ a + s ⋅ b ≤ | a | + | b | {\displaystyle |a+b|=s\cdot (a+b)=s\cdot a+s\cdot b\leq |a|+|b|} , as desired. Some additional useful properties are given below. These are either immediate consequences of the definition or implied by

232-517: A CP-violating phase angle (   δ   ). θ 1 is the Cabibbo angle. For brevity, the cosines and sines of the angles θ k are denoted c k and s k , for k = 1,   2,   3 respectively. A "standard" parameterization of the CKM matrix uses three Euler angles (   θ 12 , θ 23 , θ 13   ) and one CP-violating phase (   δ 13   ). θ 12

290-451: A function, and d d x | f ( x ) | = f ( x ) | f ( x ) | f ′ ( x ) {\displaystyle {d \over dx}|f(x)|={f(x) \over |f(x)|}f'(x)} if another function is inside the absolute value. In the first case, the derivative is always discontinuous at x = 0 {\textstyle x=0} in

348-604: A modern series of experiments under way at the Japanese BELLE and the American BaBar experiments, as well as at LHCb in CERN, Switzerland. Four independent parameters are required to fully define the CKM matrix. Many parameterizations have been proposed, and three of the most common ones are shown below. The original parameterization of Kobayashi and Maskawa used three angles (   θ 1 , θ 2 , θ 3   ) and

406-721: A more common and less ambiguous notation. For any real number x {\displaystyle x} , the absolute value or modulus of x {\displaystyle x} is denoted by | x | {\displaystyle |x|} , with a vertical bar on each side of the quantity, and is defined as | x | = { x , if  x ≥ 0 − x , if  x < 0. {\displaystyle |x|={\begin{cases}x,&{\text{if }}x\geq 0\\-x,&{\text{if }}x<0.\end{cases}}} The absolute value of x {\displaystyle x}

464-519: A need to explain two observed phenomena: Cabibbo's solution consisted of postulating weak universality (see below) to resolve the first issue, along with a mixing angle θ c , now called the Cabibbo angle , between the d and s quarks to resolve the second. For two generations of quarks, there can be no CP violating phases, as shown by the counting of the previous section. Since CP violations had already been seen in 1964, in neutral kaon decays,

522-465: A prediction of the standard model: They are free parameters . At present, there is no generally-accepted theory that explains why the angles should have the values that are measured in experiments. The constraints of unitarity of the CKM-matrix on the diagonal terms can be written as separately for each generation j . This implies that the sum of all couplings of any one of the up-type quarks to all

580-429: A set X  ×  X is called a metric (or a distance function ) on  X , if it satisfies the following four axioms: The definition of absolute value given for real numbers above can be extended to any ordered ring . That is, if  a is an element of an ordered ring  R , then the absolute value of  a , denoted by | a | , is defined to be: where − a is the additive inverse of 

638-517: Is | z | = r . {\displaystyle |z|=r.} Since the product of any complex number z {\displaystyle z} and its complex conjugate z ¯ = x − i y {\displaystyle {\bar {z}}=x-iy} , with the same absolute value, is always the non-negative real number ( x 2 + y 2 ) {\displaystyle \left(x^{2}+y^{2}\right)} ,

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696-430: Is negative (in which case negating x {\displaystyle x} makes − x {\displaystyle -x} positive), and | 0 | = 0 {\displaystyle |0|=0} . For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero. Generalisations of

754-413: Is a special case of multiplicativity that is often useful by itself. The real absolute value function is continuous everywhere. It is differentiable everywhere except for x = 0 . It is monotonically decreasing on the interval (−∞, 0] and monotonically increasing on the interval [0, +∞) . Since a real number and its opposite have the same absolute value, it is an even function , and

812-506: Is an example of a continuous function that achieves a global minimum where the derivative does not exist. The subdifferential of  | x | at  x = 0 is the interval  [−1, 1] . The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann equations . The second derivative of  | x | with respect to  x

870-750: Is defined by | z | = Re ⁡ ( z ) 2 + Im ⁡ ( z ) 2 = x 2 + y 2 , {\displaystyle |z|={\sqrt {\operatorname {Re} (z)^{2}+\operatorname {Im} (z)^{2}}}={\sqrt {x^{2}+y^{2}}},} the Pythagorean addition of x {\displaystyle x} and y {\displaystyle y} , where Re ⁡ ( z ) = x {\displaystyle \operatorname {Re} (z)=x} and Im ⁡ ( z ) = y {\displaystyle \operatorname {Im} (z)=y} denote

928-536: Is different from Wikidata All article disambiguation pages All disambiguation pages Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa matrix In 1963, Nicola Cabibbo introduced the Cabibbo angle ( θ c ) to preserve the universality of the weak interaction . Cabibbo was inspired by previous work by Murray Gell-Mann and Maurice Lévy, on the effectively rotated nonstrange and strange vector and axial weak currents, which he references. In light of current concepts (quarks had not yet been proposed),

986-466: Is doubly antisymmetric, Up to antisymmetry, it only has 9 = 3 × 3 non-vanishing components, which, remarkably, from the unitarity of V , can be shown to be all identical in magnitude , that is, so that Since the three sides of the triangles are open to direct experiment, as are the three angles, a class of tests of the Standard Model is to check that the triangle closes. This is the purpose of

1044-461: Is hence not invertible . The real absolute value function is a piecewise linear , convex function . For both real and complex numbers the absolute value function is idempotent (meaning that the absolute value of any absolute value is itself). The absolute value function of a real number returns its value irrespective of its sign, whereas the sign (or signum) function returns a number's sign irrespective of its value. The following equations show

1102-455: Is only one parameter, which is a mixing angle between two generations of quarks. Historically, this was the first version of CKM matrix when only two generations were known. It is called the Cabibbo angle after its inventor Nicola Cabibbo . For the Standard Model case ( N  = 3), there are three mixing angles and one CP-violating complex phase. Cabibbo's idea originated from

1160-404: Is the non-negative value of x {\displaystyle x} without regard to its sign . Namely, | x | = x {\displaystyle |x|=x} if x {\displaystyle x} is a positive number , and | x | = − x {\displaystyle |x|=-x} if x {\displaystyle x}

1218-483: Is the Cabibbo angle. Couplings between quark generations j and k vanish if θ jk = 0 . Cosines and sines of the angles are denoted c jk and s jk , respectively. The 2008 values for the standard parameters were: and A third parameterization of the CKM matrix was introduced by Lincoln Wolfenstein with the four real parameters λ , A , ρ , and η , which would all 'vanish' (would be zero) if there were no coupling. The four Wolfenstein parameters have

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1276-421: Is thus always either a positive number or zero , but never negative . When x {\displaystyle x} itself is negative ( x < 0 {\displaystyle x<0} ), then its absolute value is necessarily positive ( | x | = − x > 0 {\displaystyle |x|=-x>0} ). From an analytic geometry point of view,

1334-495: Is zero everywhere except zero, where it does not exist. As a generalised function , the second derivative may be taken as two times the Dirac delta function . The antiderivative (indefinite integral ) of the real absolute value function is where C is an arbitrary constant of integration . This is not a complex antiderivative because complex antiderivatives can only exist for complex-differentiable ( holomorphic ) functions, which

1392-418: The Pythagorean theorem : for any complex number z = x + i y , {\displaystyle z=x+iy,} where x {\displaystyle x} and y {\displaystyle y} are real numbers, the absolute value or modulus of z {\displaystyle z} is denoted | z | {\displaystyle |z|} and

1450-419: The Standard Model that emerged soon after clearly indicated the existence of a third generation of quarks, as Kobayashi and Maskawa pointed out in 1973. The discovery of the bottom quark at Fermilab (by Leon Lederman 's group) in 1976 therefore immediately started off the search for the top quark , the missing third-generation quark. Note, however, that the specific values that the angles take on are not

1508-442: The square root symbol represents the unique positive square root , when applied to a positive number, it follows that | x | = x 2 . {\displaystyle |x|={\sqrt {x^{2}}}.} This is equivalent to the definition above, and may be used as an alternative definition of the absolute value of real numbers. The absolute value has the following four fundamental properties (

1566-415: The 4-tensor ( α , β ; i , j ) ≡ Im ⁡ ( V α i V β j V α j ∗ V β i ∗ ) {\displaystyle \;(\alpha ,\beta ;i,j)\equiv \operatorname {Im} (V_{\alpha i}V_{\beta j}V_{\alpha j}^{*}V_{\beta i}^{*})\;}

1624-408: The CKM matrix is unitary, its inverse is the same as its conjugate transpose , which the alternate choices use; it appears as the same matrix, in a slightly altered form. To generalize the matrix, count the number of physically important parameters in this matrix V which appear in experiments. If there are N generations of quarks (2 N flavours ) then For the case N  = 2, there

1682-420: The CKM-matrix can be written in the form For any fixed and different i and j , this is a constraint on three complex numbers, one for each k , which says that these numbers form the sides of a triangle in the complex plane . There are six choices of i and j (three independent), and hence six such triangles, each of which is called a unitary triangle . Their shapes can be very different, but they all have

1740-402: The Cabibbo angle where the various | V ij | represent the probability that the quark of flavor j decays into a quark of flavor i . This 2×2  rotation matrix is called the "Cabibbo matrix", and was subsequently expanded to the 3×3 CKM matrix. In 1973, observing that CP-violation could not be explained in a four-quark model, Kobayashi and Maskawa generalized the Cabibbo matrix into

1798-421: The Cabibbo angle is related to the relative probability that down and strange quarks decay into up quarks ( | V ud |   and   | V us |  , respectively). In particle physics terminology, the object that couples to the up quark via charged-current weak interaction is a superposition of down-type quarks, here denoted by d′ . Mathematically this is: or using the Cabibbo angle: Using

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1856-542: The Cabibbo–Kobayashi–Maskawa matrix (or CKM matrix) to keep track of the weak decays of three generations of quarks: On the left are the weak interaction doublet partners of down-type quarks, and on the right is the CKM matrix, along with a vector of mass eigenstates of down-type quarks. The CKM matrix describes the probability of a transition from one flavour j quark to another flavour i quark. These transitions are proportional to | V ij | . As of 2023,

1914-407: The absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers , the quaternions , ordered rings , fields and vector spaces . The absolute value is closely related to the notions of magnitude , distance , and norm in various mathematical and physical contexts. In 1806, Jean-Robert Argand introduced

1972-418: The absolute value of x {\textstyle x} is generally represented by abs( x ) , or a similar expression. The vertical bar notation also appears in a number of other mathematical contexts: for example, when applied to a set, it denotes its cardinality ; when applied to a matrix , it denotes its determinant . Vertical bars denote the absolute value only for algebraic objects for which

2030-520: The absolute value of a complex number z {\displaystyle z} is the square root of z ⋅ z ¯ , {\displaystyle z\cdot {\overline {z}},} which is therefore called the absolute square or squared modulus of z {\displaystyle z} : | z | = z ⋅ z ¯ . {\displaystyle |z|={\sqrt {z\cdot {\overline {z}}}}.} This generalizes

2088-410: The absolute value of a real number is that number's distance from zero along the real number line , and more generally the absolute value of the difference of two real numbers (their absolute difference ) is the distance between them. The notion of an abstract distance function in mathematics can be seen to be a generalisation of the absolute value of the difference (see "Distance" below). Since

2146-469: The absolute value of the difference of two real or complex numbers is the distance between them. The standard Euclidean distance between two points and in Euclidean n -space is defined as: This can be seen as a generalisation, since for a 1 {\displaystyle a_{1}} and b 1 {\displaystyle b_{1}} real, i.e. in a 1-space, according to

2204-403: The alternative definition for reals: | x | = x ⋅ x {\textstyle |x|={\sqrt {x\cdot x}}} . The complex absolute value shares the four fundamental properties given above for the real absolute value. The identity | z | 2 = | z 2 | {\displaystyle |z|^{2}=|z^{2}|}

2262-415: The alternative definition of the absolute value, and for a = a 1 + i a 2 {\displaystyle a=a_{1}+ia_{2}} and b = b 1 + i b 2 {\displaystyle b=b_{1}+ib_{2}} complex numbers, i.e. in a 2-space, The above shows that the "absolute value"-distance, for real and complex numbers, agrees with

2320-596: The best determination of the individual magnitudes of the CKM matrix elements was: Using those values, one can check the unitarity of the CKM matrix. In particular, we find that the first-row matrix elements give: | V u d | 2 + | V u s | 2 + | V u b | 2 = .999997 ± .0007   ; {\displaystyle |V_{\mathrm {ud} }|^{2}+|V_{\mathrm {us} }|^{2}+|V_{\mathrm {ub} }|^{2}=.999997\pm .0007~;} Making

2378-402: The complex absolute value function is not. The following two formulae are special cases of the chain rule : d d x f ( | x | ) = x | x | ( f ′ ( | x | ) ) {\displaystyle {d \over dx}f(|x|)={x \over |x|}(f'(|x|))} if the absolute value is inside

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2436-419: The currently accepted values for   | V ud |   and   | V us |   (see below), the Cabibbo angle can be calculated using When the charm quark was discovered in 1974, it was noticed that the down and strange quark could transition into either the up or charm quark, leading to two sets of equations: or using the Cabibbo angle: This can also be written in matrix notation as: or using

2494-451: The definition given at the top for the real absolute value cannot be directly applied to complex numbers. However, the geometric interpretation of the absolute value of a real number as its distance from 0 can be generalised. The absolute value of a complex number is defined by the Euclidean distance of its corresponding point in the complex plane from the origin . This can be computed using

2552-399: The down-type quarks is the same for all generations. This relation is called weak universality and was first pointed out by Nicola Cabibbo in 1967. Theoretically it is a consequence of the fact that all SU(2) doublets couple with the same strength to the vector bosons of weak interactions. It has been subjected to continuing experimental tests. The remaining constraints of unitarity of

2610-443: The experimental results in line with the theoretical value of 1. The choice of usage of down-type quarks in the definition is a convention, and does not represent a physically preferred asymmetry between up-type and down-type quarks. Other conventions are equally valid: The mass eigenstates u , c , and t of the up-type quarks can equivalently define the matrix in terms of their weak interaction partners u′ , c′ , and t′ . Since

2668-506: The fact that the Nobel Prize committee failed to reward the work of Cabibbo , whose prior work was closely related to that of Kobayashi and Maskawa. Asked for a reaction on the prize, Cabibbo preferred to give no comment. Absolute value In mathematics , the absolute value or modulus of a real number x {\displaystyle x} , denoted | x | {\displaystyle |x|} ,

2726-407: The first case and where f ( x ) = 0 {\textstyle f(x)=0} in the second case. The absolute value is closely related to the idea of distance . As noted above, the absolute value of a real or complex number is the distance from that number to the origin, along the real number line, for real numbers, or in the complex plane, for complex numbers, and more generally,

2784-405: The four fundamental properties above. Two other useful properties concerning inequalities are: These relations may be used to solve inequalities involving absolute values. For example: The absolute value, as "distance from zero", is used to define the absolute difference between arbitrary real numbers, the standard metric on the real numbers. Since the complex numbers are not ordered ,

2842-671: The notion of an absolute value is defined, notably an element of a normed division algebra , for example a real number, a complex number, or a quaternion. A closely related but distinct notation is the use of vertical bars for either the Euclidean norm or sup norm of a vector in R n {\displaystyle \mathbb {R} ^{n}} , although double vertical bars with subscripts ( ‖ ⋅ ‖ 2 {\displaystyle \|\cdot \|_{2}} and ‖ ⋅ ‖ ∞ {\displaystyle \|\cdot \|_{\infty }} , respectively) are

2900-600: The parameters ρ and η . Using the values of the previous section for the CKM matrix, as of 2008 the best determination of the Wolfenstein parameter values is: In 2008, Kobayashi and Maskawa shared one half of the Nobel Prize in Physics "for the discovery of the origin of the broken symmetry which predicts the existence of at least three families of quarks in nature". Some physicists were reported to harbor bitter feelings about

2958-410: The property that all are of order 1 and are related to the 'standard' parameterization: Although the Wolfenstein parameterization of the CKM matrix can be as exact as desired when carried to high order, it is mainly used for generating convenient approximations to the standard parameterization. The approximation to order λ , good to better than 0.3% accuracy, is: Rates of CP violation correspond to

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3016-535: The real and imaginary parts of z {\displaystyle z} , respectively. When the imaginary part y {\displaystyle y} is zero, this coincides with the definition of the absolute value of the real number x {\displaystyle x} . When a complex number z {\displaystyle z} is expressed in its polar form as z = r e i θ , {\displaystyle z=re^{i\theta },} its absolute value

3074-401: The relationship between these two functions: or and for x ≠ 0 , Let s , t ∈ R {\displaystyle s,t\in \mathbb {R} } , then and The real absolute value function has a derivative for every x ≠ 0 , but is not differentiable at x = 0 . Its derivative for x ≠ 0 is given by the step function : The real absolute value function

3132-660: The result positive. Now, since − 1 ⋅ x ≤ | x | {\displaystyle -1\cdot x\leq |x|} and + 1 ⋅ x ≤ | x | {\displaystyle +1\cdot x\leq |x|} , it follows that, whichever of ± 1 {\displaystyle \pm 1} is the value of s {\displaystyle s} , one has s ⋅ x ≤ | x | {\displaystyle s\cdot x\leq |x|} for all real x {\displaystyle x} . Consequently, |

3190-560: The same area, which can be related to the CP violating phase. The area vanishes for the specific parameters in the Standard Model for which there would be no CP violation . The orientation of the triangles depend on the phases of the quark fields. A popular quantity amounting to twice the area of the unitarity triangle is the Jarlskog invariant (introduced by Cecilia Jarlskog in 1985), For Greek indices denoting up quarks and Latin ones down quarks,

3248-402: The same term [REDACTED] This disambiguation page lists articles associated with the title CKM . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=CKM&oldid=929562939 " Category : Disambiguation pages Hidden categories: Short description

3306-442: The standard Euclidean distance, which they inherit as a result of considering them as one and two-dimensional Euclidean spaces, respectively. The properties of the absolute value of the difference of two real or complex numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion of a distance function as follows: A real valued function d on

3364-690: The term module , meaning unit of measure in French, specifically for the complex absolute value, and it was borrowed into English in 1866 as the Latin equivalent modulus . The term absolute value has been used in this sense from at least 1806 in French and 1857 in English. The notation | x | , with a vertical bar on each side, was introduced by Karl Weierstrass in 1841. Other names for absolute value include numerical value and magnitude . In programming languages and computational software packages,

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