Properties of Quarks

Transcription

1 PHY04 Partile Physis 9 Dr C N Booth Properties of Quarks In the earlier part of this ourse, we have disussed three families of leptons but prinipally onentrated on one doublet of quarks, the u and d. We will now introdue other types of quarks, along with the new quantum numbers whih haraterise them. Isospin It was notied that many groupings of partiles of similar mass and properties fitted in to ommon patterns. One way to haraterise these is using isotopi spin or isospin, I. This quantity has nothing to do with the real spin of the partile, but obeys the same addition laws as the quantum mehanial rules for adding angular momentum or spin. When the orientation of an isospin vetor is onsidered, it is in some hypothetial spae, not in terms of the x, y and z axes of normal o-ordinates. Nuleons (p, n), pi mesons (π +, π 0, π ) and the baryons known as ( ++, +, 0, ) are three examples of groups of similar mass partiles differing in harge by one unit. The harge Q in eah ase an be onsidered as due to the orientation of an isospin vetor in some hypothetial spae, suh that Q depends on the third omponent I. Thus the nuleons belong to an isospin doublet: + 0 p I, I =, ; n =,. Similarly the pions form an isospin triplet, π =, ; π =,0 ; π =,. The forms a quadruplet with I =. The rule for eletri harge an then be written Q e( B I ) = +, where B is the baryon number whih is for nuleons and the and 0 for mesons suh as the π. In terms of quarks, the u and d form an isospin doublet, u =, ; d =, (both with B = ). Three quarks with I = an ombine to form I tot = or. I tot = gives the nuleons while I tot = forms the. It is useful to onsider the symmetry of the quarks inside these baryons. The internal wavefuntion an be written as a produt of terms, Ψ = ψspinψspaeψisospinψ olour, and must be antisymmetri overall under interhange (as quarks are fermions). ψ olour is always antisymmetri (as hadrons are olourless); the symmetry of ψ is given by ( ) l and l the orbital angular momentum is spae zero for the long-lived hadrons we onsider, so ψ spae is symmetri. Thus the produt ψspinψ isospin must be symmetri, implying either both must be symmetri or both must be antisymmetri. This explains the orrelation between allowed spin and isospin states for the baryons: the has I = and s = (both symmetri), while the nuleons have I = and s = (both antisymmetri). In strong interations, the total isospin vetor (as well as I ) is onserved. This is not true in eletromagneti or weak interations. The onservation of isospin has observable effets on the relative rates of strong interations, as will be disussed in the letures. 6

2 Strangeness It was observed that some unstable partiles produed in strong interations had a long lifetime. This unusual stability for strongly interating partiles led to the term of strangeness. Suh partiles are always produed in pairs (assoiated prodution), and the quantum number of strangeness, S, was introdued, whih is onserved in strong interations. Thus in the interation π p Λ 0 K 0, the Λ is assigned S = and the K has S = +. These strange partiles an only deay by the weak interation, whih does not onserve strangeness (as we will disuss later). Isospin multiplet B S I <Q>/e Y = B+S π + π 0 π p n 0 ½ ½ ½ Λ Σ + Σ 0 Σ 0 0 K + K 0 0 ½ ½ 0 K K 0 ½ ½ Ξ 0 Ξ ½ ½ Ω 0 Table A seletion of strange and non-strange baryon and meson multiplets. The formula for eletri harge must know be modified to read Q = e I + B + S = e I + Y ( ) ( ) where Y = B + S is known as the hyperharge. (This formula is known as the Gell-Mann Nishijima relation.) Families of partiles with similar properties (e.g. same spin and parity) an be plotted in terms of Y versus I, and form regular geometrial patterns (see figures 9 to ): mesons with spin-parity (J P ) = 0 form an otet; mesons with J P = form a nonet; baryons with J P = ½ + form an otet; baryons with J P = + form a deuplet. The differene in shape (i.e. the allowed ombinations of quarks) between the baryon otet and deuplet is entirely a onsequene of symmetry onstraints, as will be disussed in the letures (and below). In terms of quarks we an introdue a new flavour of quark, the strange quark s. This has harge and baryon number (like a d quark) but I = 0 and S =. It is also somewhat heavier than the u and d quarks. Sine baryons onsist of qqq, it is lear why no positive baryons exist with S >, while negative baryons are found with S = or. 7

3 K 0 ( ds ) Y = S + K + ( us ) π ( du ) η π + ( ud ) π 0 + I K ( su ) K 0 ( sd ) Fig. 9 The lowest-lying pseudosalar-meson states (J P = 0 ), with quark assignments indiated. (The states at the origin are displaed slightly for larity.) K *0 Y = S + K *+ ρ φ ω ρ 0 + ρ + I K * K *0 Fig. 0 The vetor-meson nonet (J P = ). (Quark assignments are the same as above.) 8

4 n Y + p N ( 99) Σ Σ 0 Λ Σ + Σ ( 9) + I Λ ( 6) Ξ Ξ 0 Ξ ( 8) Fig. The baryon otet of spin-parity J P + = (with masses in MeV/ ). I = Y () + I = Σ Σ 0 Σ + + I (84) I = Ξ Ξ 0 (5) I = 0 Ω (67) Fig. The baryon deuplet with spin-parity J P + = (with masses in MeV/ ). 9

5 Further quarks Other, still heavier quarks also exist. The harm quark,, has a harge of, like the u, and an be onsidered as a partner to the s. In dimensions (see figure ) partiles ontaining quarks an be plotted, and again show regular patterns. Fig. Multiplets of hadrons ontaining up, down, strange and harm quarks. The slies through these figures where harm = 0 orrespond to the plane figures already shown in the previous diagrams, though ontaining new partiles in the ase of the mesons omposed of. We thus have doublets or generations of quarks (d, u) and (s, ). Sine there are doublets of leptons, there are theoretial reasons for expeting a third doublet of quarks too. Partiles ontaining b quarks (bottom or beauty) were disovered in 977. The b is an even heavier version of the d. Its partner, the t (top or truth) was first seen in 994, and it has the greatest mass of any known fundamental partile at 74 GeV/. 0

6 Flavour Charge/e B I I S b t Mass d (GeV/ ) ½ ½ u + ½ +½ s b t Table Quark quantum numbers and masses. Note the onvention that quarks with a negative eletri harge arry a negative flavour quantum number. The masses quoted are bare masses when bound in a hadron the effetive masses differ, espeially for the lightest quarks. (Binding and kineti energies mean that the u and d quarks an be treated as effetively equal in mass.) Fig. 4 The partile ontent of the Standard Model of Partile Physis (inluding the Higgs boson whih is not overed in this ourse).

7 Baryons and Quark Symmetry We onsidered above how quarks, with spin (s) ½ and isospin (I) ½, an ombine to make a total wavefuntion (spae, spin, isospin and olour parts) whih is antisymmetri under the interhange of two partiles, as required for fermions. For the lowest lying (l = 0) baryons, we saw that the produt of the spin and isospin wavefuntions must then be symmetri, and the two solutions are the, with I = and s =, and the nuleons, with I = ½ and s = ½. There were a few simplifiations in this aount, so here is a more rigorous explanation. (This more detailed approah for idential quarks is not required for the examination, but understanding it will help your appreiation of the role of symmetry in quark systems. You should be able to disuss symmetri and antisymmetri ombinations of quarks.) First we will look at the ombination of two objets with spin ½ and isospin ½. Sine two quarks do not form a bound state, it is helpful to onsider ombinations of two nuleons as these have similar quantum numbers. In this ase, sine there is no olour part to the total wavefuntion, the produt of spin and isospin states must be antisymmetri. We will denote s z = ½ by and s z = ½ by, and I = ½ by p and I = ½ by n. First onsider just the isospin. There are 4 ombinations of two nuleons: pp, pn, np, nn. The first and last are obviously symmetri under interhange; the other do not have a defined symmetry. We an produe the following ombinations: pp, ( pn + np ), nn, ( pn np ). The first are symmetri, orresponding respetively to (I, I ) = (, ), (, 0), (, ). The last is antisymmetri, (I, I ) = (0, 0). Similarly the spin states are, ( ) +,, ( ), orresponding to (s, s z) = (, ), (, 0), (, ), (0, 0) respetively. (The fators are simply the normalisations.) The allowed states are those where spin and isospin have opposite symmetry; the antisymmetri isospin state and symmetri spin states (s = ) give the bound deuteron, while the omplementary states are only allowed for free pairs of protons and neutrons. (This is disussed further in the Nulear Physis ourse next semester.) We now return to the quarks, denoting I = ½ by u and I = ½ by d. Sine the olour part of the wavefuntion is always antisymmetri, the ombined isospin spin must be symmetri. First we will onsider symmetri isospin ombinations of the three quarks. uuu is (I, I ) = (, ) so must be the ++. uud is I = ½; forming ombinations whih are symmetri under -quark permutations we get uud + udu + duu. This must be (I, I ) = (, ) so the +. ( ) Similarly ( udd + dud + ddu ) is (I, I ) = (, ) so the 0. ddd is (I, I ) = (, ) so the. Note: The first and last state must be I =. There must therefore be symmetri states with I = and intermediate I values, and the above ombinations are the only possibilities!

8 way: The spin states for the must also be symmetri, and we an write these down in an analogous (s, s z ) = (, ) is. (s, s z ) = (, ) is ( + + ). (s, s z ) = (, ) is ( + + ) and (s, s z) = (, ) is. It was implied earlier that the nuleon states were antisymmetri in isospin and antisymmetri in spin. In fat, it is impossible to write down -quark states that are antisymmetri under the interhange of all pairs! (Try it!) However, it is possible to use partially antisymmetri isospin and spin states as a basis to find the required states whih are symmetri overall. First onsider one pair of quarks. Ignoring the normalisation fators, ( u d d u ) is antisymmetri under exhange of isospin labels; ( u d u d ) is antisymmetri under exhange of spin labels; so ( u d u d d u + d u ) is antisymmetri under exhange of isospin or spin labels but symmetri overall (i.e. when both the spin and isospin properties of a pair of partiles are swapped). This ombination has I = 0 and s = 0. We an now ombine this state with a third quark u and symmetrise to find the ombined state representing a proton (I = ½) with spin s z = ½. Simply adding a u on the left gives: u u d u u d u d u + u d u. This must be made symmetri under interhange of pairs of quarks. Permuting first and seond adds on + u u d u u d d u u + d u u, while permuting first and third adds + d u u d u u u d u + u d u. (Note that interhange of nd and rd quarks was already onsidered above). Gathering terms together, and inluding the normalisation fator (equal as usual to the reiproal of the sum of the squares of the oeffiients) gives p = ( u u d u u d u u d 8 + u d u u d u u d u + d u u d u u d u u ). The spin-down state of the proton and the spin states of the neutron an be found in the same way.

9 PHY04 Partile Physis 0 Dr C N Booth Quark Flavour and the Weak Interation As we have already seen, the strong and eletromagneti interations onserve quark flavour, whereas the weak interation may hange it. In many weak deays, the hanges are within a generation, e.g. in beta deay the W ouples a u to a d quark; in the deay D + K 0 π + it ouples a to an s. However, this is not always the ase, e.g. in the deay K π 0 e νe the W ouples an s to a u quark, and it was observed that suh strangeness-hanging deays were slightly weaker than strangeness-onserving weak deays. Cabibbo explained this by proposing that the eigenstates of the weak interation are different from those of the strong interation. The strong interation eigenstates are the u, d, s,, b and t quarks, with well-defined isospin, strangeness et. The eigenstates of the weak interation, whih does not onserve I, S et., are said to be those of weak isospin T. For simpliity, let us first onsider the first generations alone. The weak eigenstates are the leptons and orthogonal linear ombinations of the familiar quarks T T = + ν νe µ = e µ u d s with d = α d + β s s = β d + α s (normalisation α + β = ) α is usually known as os θ, where θ is the Cabibbo angle. A value of sin θ = 0.5 is onsistent with the observed apparent variation of weak oupling onstant with reation type. (Note that by onvention it is only the T = ½ member of the doublet whih is mixed.) The relationship between weak and strong eigenstates in generations an also be expressed as d os θ sin θ d = s sin θ os θ s weak e-states mixing matrix strong e-states The same formalism an be used for generations, and the mixing matrix, known as the Cabibbo- Kabayashi-Maskawa or CKM matrix, an be parameterised in a number of ways. d s b = M d s b The magnitudes of the matrix elements have been determined experimentally, and are given by M 0.974± ± ± = 0.5 ± ± ± ± ± ± Note that the values along the leading diagonal are quite lose to one, those adjaent to it are signifiantly smaller, and the elements in the top-right and bottom-left orners are muh smaller. This 4

10 means that the mixing results in states whih ontain a small admixture of the quark from the next generation, while mixing between st and rd generation quarks is extremely small. Flavour-hanging weak interations always our via the harged urrent. That is, they always involve transitions between the two members of the same weak isospin doublet, e.g. between and s (in either diretion), or between u and d. The mixing of the negative quarks plays a role for both initial and final state quarks. For example, the deay of a quark is always to an s weak eigenstate, whih will be bound in a hadron as one of the strong eigenstates of whih it an be onsidered a mixture. On the other hand, when a hadron ontaining an s quark deays, the s must be onsidered a mixture of d, s and b weak eigenstates, and these deay to u, and t respetively. Physially, the relative probability of produing hadrons ontaining the respetive quarks in a weak deay is determined by the elements of the CKM matrix. For example, when a top quark deays it produes a b quark. This is bound in a hadron by the strong interation, so must be revealed as a strong eigenstate. The b is most likely to result in a partile ontaining a b quark, with a smaller probability of an s quark and almost negligible likelihood of produing a d quark. Therefore, the neardiagonal struture of the CKM matrix means that weak deays are most likely to be within a generation if allowed by onservation of energy (a partile annot deay into one that is heavier) or to the next generation below if this is not allowed. The most likely overall deay hain of a b quark is therefore b s u. When the harged-urrent weak deays are onsidered along with binding into strong eigenstates in the hadrons, the elements of M an be interpreted as giving the effetive transition strengths between quarks as follows: u d u s u b M = d s b. t d t s t b (Again, physial deays must always be to the lighter quark.) For two generations, one parameter was required to desribe the mixing. This was the Cabibbo angle. With three generations, 4 independent parameters are needed to define a general unitary matrix, and the individual matrix elements may have imaginary parts. One possible parameterisation of the CKM matrix is given below. Note that the following material is provided for ompleteness only, and is not examinable! (Further details are provided in the text books.) M = s s iδ sse iδ s se 5 iδ ssse iδ s sse where ij = os θ ij and s ij = sin θ ij, with i and j being generation labels {i,j =,,}. In the limit θ = θ = 0, the third generation deouples, and the situation redues to the usual Cabibbo mixing of the first two generations, with θ identified with the Cabibbo angle. The real angles θ, θ, θ an all be made to lie in the first quadrant by suitable definition of the quark field phases. is known to differ from unity only in the sixth deimal plae. If the parameter δ is non-zero, then the matrix is omplex, and the small degree of CP violation present in the weak interation an be explained naturally. This has not yet been onlusively proven! [The above parameterisation and values are taken from the Partile Physis Data Booklet, from "Review of Partile Physis", Chinese Physis C8, July 04, by the Partile Data Group.] s s s e iδ

11 The K 0 System and Strangeness Regeneration We have seen that the strong and weak eigenstates, when expressed in terms of quarks, are different. We now look at evidene for different eigenstates in the observed, free partiles themselves. This is seen in the neutral kaon system. 0 The K ( ) ds has strangeness, and an be produed in π p ollisions by the strong interation 0 in assoiation with a Λ hyperon (i.e. strange baryon). In ontrast, the K ( ) sd has strangeness, and as there are no baryons of positive strangeness, this an only be formed in higher energy ollisions. A pure sample of K 0 an therefore be produed. The weak interation does not onserve strangeness. It therefore does not distinguish between K 0 and K 0. The eigenstates of the weak interations are not those of strangeness but (approximately) those of CP. It will be shown in the leture that these an be written as ( 0 0 ) ( 0 0 ) K = K K with CP e-value + K = K + K with CP e-value The K deays to two pions, while the K must deay to a -pion final state. As there is muh more phase spae available for the K deay, its deay rate is about 600 times greater than that for the K. Consider a beam of pions striking a thin solid target, mounted in a vauum. Strong interations will our, resulting in the prodution of K 0 (with no K 0 ). These then travel on and deay through the weak interation. The K 0 is equivalent to an equal mixture of K and K, and the K omponent rapidly deays away. If a further solid target (known as a regenerator) is introdued some distane downstream, the remaining kaons will interat with it via the strong interation. Here the surviving K omponent must be seen as an equal mixture of K 0 and K 0, so S = + and states will be produed, even though only K 0 (with S = +) was present initially! In fat, the K 0 will be preferentially absorbed, with only some of the K 0 emerging from the regenerator. Here, weak deays will again our, and an equal mixture of K and K will again be observed, even though previously the entire K omponent had deayed away! Quantum mehanially, the situation is exatly equivalent to a series of Stern-Gerlah experiments, with rossed magneti field gradients. If an unpolarised beam of neutral, spin-½ atoms travelling along the z-axis enounters a region with a field gradient in the x-diretion, it will split into two equal divergent beams with s x = ±½. If one of these is bloked and the other allowed into a seond region where the field gradient is in the y-diretion, it will split again into equal divergent beams with s y = ±½. If now one of these is seleted and enounters a region with a field gradient again in the x-diretion, it will split into two equal beams with s x = ±½ even though one of these omponents had previously been eliminated. 6