D V Bugg 1 Queen

A helpful analogy between the covalent bond and Particle Spectroscopy

Keywords: numbers: 12.39.Ki, 12.40.Yx, 13.25.Ft, 14.20.Gk

It is proposed that meson resonances are linear combinations of q q and meson-meson (MM) baryon resonances are combinations of qqq and meson-baryon (MB). Mixing between these combinations arises via decays of con ned states to meson-meson or meson-baryon. There is a precise analogy with the covalent bond in molecular physics it helps to visualise what is happening physically. One eigenstate is lowered by the mixing the other moves up and normally increases in width. Cusps arise at thresholds. At sharp thresholds due to S-wave 2-particle decays, these cusps play a conspicuous role in many sets of data. The overall pattern of light mesons is consistent with nearly linear Regge trajectories, hence q q component s . T h e r e i s n o o b vious reason why this pattern should arise from dynamically generated states without q q content.

Introduction

In the early days of Particle Physics, Chew, Goldberger and others tried to account for resonances in terms of particle exchanges 1]. This met with partial success. For N elastic scattering, Donnachie and Hamilton 2] showed that exchanges of N, (1232), and provide long range attraction in P 33 , D 13 , D 15 and F 15 partial waves where prominent resonances are observed. Furthermore, these exchanges account for repulsive partial waves P 13 , P 31 , P 11 , D 33 and D 35 . However, meson exchanges failed to account for the meson. This approach w as therefore quickly overtaken by the quark model, which has provided a semi-quantitative picture of most of the known resonances.

However, there are cracks in this framework. Firstly, it does not account for the and poles, a 0 (980) and f 0 (980). Secondly, there are many examples where resonances appear at or close to sharp S-wave thresholds: e.g. f 0 (980) and a 0 (980) at the K K threshold, f 2 (1565) at the !! threshold, X(3872) at the D 0 D 0 threshold, S 11 (1650) and D 13 (1700) close to the !N threshold.

There is a straightforward explanation of how S-wave thresholds attract these resonances 3].

Consider f 0 (980) as an example. Its amplitude for ! KKis given to rst approximation by the Flatt e f o r m ula 4]:

A 12 = T 12 p 1 2 = G 1 G 2 p 1 2 M 2 ; s ; i G 2 1 1 (s) + G 2 2 2 (s)] = N(s) D(s) [START_REF] Chew | The analytic S-matrix: a basis for nuclear democracy[END_REF] where phase space is factored out of T. Here G i = g i F i (s) a n d g i are coupling constants, F i are form factors. Writing D(s) = M 2 ; s ; i (s), a more exact form for D(s) i s M 2 ; s ; Re (s) ; i (s), where Re KK (s) = 1 P Z 1 4m 2 K ds 0 G 2 KK (s 0 ) KK (s 0 ) (s 0 ; s) :

(2)

1 email: d.bugg@rl.ac.uk Fig. 1(a) illustrates the dispersive term Re (s) f o r f 0 (980) using F KK = e x p (;3k 2 KK ), where k KK is KKmomentum in GeV/c. Re acts as an e ective attraction. Parameters of f 0 (980) are known. Ref. 3] gives tables of pole positions when M, G 1 and G 2 are varied. If M is as low as 500 MeV, there is still a pole at 806 ; i78 MeV for M in the range 850{1100 MeV, there is a pole within 23 MeV of the KKthreshold. The moral is that a strong threshold can move a resonance a surprisingly long way. F or f 0 (980), G 2 It is important to realise that Re is not an `optional extra'. It is a rigorous consequence of analyticity f o r a l l s-channel decay processes. In principle these terms are required for all resonance decays. It is then logical to include also ta n d u-channel exchanges.

P-wave thresholds lead to broader e ects because of the k 3 momentum dependence of phase space. There are in principle contributions to (770) and a 1 (1260), but in practice these e ects may be accomodated by tted form factors describing decay w i d t h s .

The virtue of Eq. (2) is twofold. Firstly, it is easily evaluated, secondly it illustrates graphically the e ect of the form factor F. Re (s) goes negative close to the peak of G 2 and subsequently has a minimum at 1:7 GeV thereafter it slowly rises to 0 as M ! 1 .

There are alternatives to evaluating Eq. (2). The same result may be obtained by e v aluating the loop diagram of Fig. 1(b). Secondly, solving the Bethe-Salpeter equation is equivalent to evaluating all loop diagrams. It is straightforward in principle to include sand t-channel exchanges in solving this equation.

Several authors have adopted a similar approach. Ja e 5] gives the equations and discusses many of the implications in Section 2C of his paper. The Hamiltonian for a q q state decaying to meson-meson obeys

H = H 11 V V H 22 ! (3)
H 11 describes short-range con gurations H 22 refers to ingoing and outgoing states and should include tand u-channel exchanges V accounts for the coupling between them due to s-channel decays.

Weinstein and Isgur pursued the connection between q q, qq q q and meson components in their work on K K molecules 6]. Van Beveren and Rupp 7] construct a model where the short-range attraction in H 11 is approximated by a harmonic oscillator potential, which couples at radius R to ingoing and outgoing waves corresponding to decay c hannels. Despite the approximations, this gives valuable insight. Their algebraic solution satis es the Schr odinger equation and is therefore fully analytic. It includes e ects of thresholds fully, although not yet the sand t-channel exchanges.

Barnes and Swanson 8] consider meson loops due to pairs of D, D , D s and D s mesons, using the 3 P 0 model for decays. For 1S, 1P and 2P charmonium states, they nd that large mass shifts due to these loops may be `hidden' in the valence quark model by a c hange of parameters. The important conclusion from their work is that two-meson continuum components of charmonium states may be quite large, with the result that the constituent quark model predicts masses which are too high, particularly near the thresholds of opening channels.

Oset and collaborators demonstrate in a series of papers that some resonances may be understood as `dynamically driven states' 9] 10] 11] 12] 13] 14] 15] 16] due to s, t and u-channel exchanges. They take the view that qq and qqqcomponents are not needed at all in these cases. This takes us full circle back to the approach tried by Chew.

How is it that Oset et al. are able to reproduce known resonances (approximately) with meson exchanges alone? They use S-wave form factors which are adjusted to get one predicted resonance of each paper at the right mass. Resulting amplitudes are strong. The form factors may be mocking up short range q q components. The importance of their work is the demonstration that components derived from meson loops are large, and should be taken into account.

On the other hand, the well known J= , 3 P 0 (3415), 3 P 1 (3510), 3 P 2 (3556), (2S) and (3770) are interpreted naturally as c c states (with tiny admixtures of the mesonic states to which t h e y decay). Therefore it is logical to include the c c component i n to all other resonances unless there is a good reason why not. One should not be deterred from invoking q q and qqq components to get all resonances with their correct masses and widths.

The central premise of the present paper is that both H 11 and H 22 play essential roles in all cases. This is di erent from approaches where only one of the two components in the Hamiltonian contributes, for example the approach based on four-quark mesonic states.

An approach which has recently been popular is to suppose that `molecules' are formed from q qqq con gurations 17] 18] 19] 20] 21]. The well known question over this approach i s w h y so few tetraquarks are observed. Vijande et al. 22] throw l i g h t on this issue. They study the stability of pure c cn n and cc n n states in the absence of diquark interactions. They nd that all 12 c cn n states with J = 0, 1 or 2 are unstable. Their calculation points to the conclusion that such molecules are rare unless either (i) there are attractive diquark interactions, or (ii) coupling to meson-meson nal states contributes, as proposed here.

The layout of remaining sections is as follows. Section 2 considers a useful analogy with the covalent b o n d i n c hemistry. This analogy helps visualise the main features of mixing. Section 3 reviews approximations to be used in tting data. Section 4 considers , , a 0 (980) and f 0 (980). Section 5 discusses rstly X(3872) and concludes, as do several authors, that data require a linear combination of c c and DD . Next it reviews the status of light mesons with J P = 1 ; , 2 + and 0 + . The objective is to demonstrate that observed states lie close to straight line trajectories when masses squared are plotted against radial excitation number. There are some signi cant deviations attributable to thresholds. However, the overall picture is consistent with rather regularly spaced q q states consistent with Regge trajectories. Such trajectories are usually attributed to ux tubes joining q q pairs and expanding as spin J increases. There is no obvious reason why dynamically generated states should follow s u c h a regular sequence. Section 6 makes remarks on desirable experiments and Section 7 draws conclusions.

A helpful analogy with the covalent b o n d i n c hemistry

The wave function of Eq. ( 3) is a linear combination of q q (or qqq) and uncon ned MM (or MB). The key point is that two attractive c o m p o n e n ts H 11 and H 22 lower the eigenstate via the mixing. This is a purely quantum mechanical e ect. There is a direct analogy with the covalent bond in chemistry. The solution of Eq. ( 3) is given by the Breit-Rabi equation:

E = ( E 1 + E 2 )=2 q (E 1 ; E 2 ) 2 + jV j 2 (4)
where E 1 and E 2 are eigenvalues of separate H 11 and H 22 . One linear combination is pulled down in energy.

In chemistry, H is in principle known exactly. The discussion of the hydrogen molecule (and more complex ones) is given in many textbooks on Physical Chemistry, for example the one of Atkins 23]. Consider two h ydrogen atoms labelled A and B, combining to make a h ydrogen molecule H 11 and H 22 describe A and B. The equation describing this pair is precisely the same as Eq. (3). except that H 11 and H 22 have a di erent radial dependence to the mesonic case. Fig. 2(a) sketches contours of electron density for the lower state of Eq. ( 4). The e ect of the mixing is that the electron density adjusts so that the two electrons are somewhat concentrated between the two ions. In more detail, the wave-function for the atomic molecule is expanded in terms of a complete set of atomic H orbitals. For hydrogen, these are just the energy levels of a single hydrogen atom. For a carbon atom, they are replaced by w ave functions allowing for shielding of the Coulomb eld of the nucleus by electrons in the lowest atomic levels. It is also necessary to anti-symmetrise wave functions. Atkin gives algebraic forms which approximate the electronic wave functions. With modern computing techniques, the Variational method adjusts the coe cients of the expansion in terms of excited states to describe wave functions with great accuracy. Two k ey points are that (a) the extension of the wave function into the overlap region lowers momenta of electrons, hence their kinetic energy, (b) the whole system shrinks slightly and the binding of electrons to both nuclei increases.

In the Particle Physics case, the procedure is conceptually identical. The mesonic wave function for the ground state is sucked into the region of overlap, producing an attractive interaction between q q and meson-meson. This implies that q q will decay to nal states where the interaction is indeed attractive. Fig. 2(b) shows the electron density for the upper state of Eq. ( 4). In this case, electrons are repelled from the overlap region, increasing their momenta and kinetic energies, hence the energy eigenvalue. The second e ect is that the increased binding for the lower state draws the quarks slightly down the Coulomb i c p a r t o f t h e Q C D p o t e n tial, shrinking the radius of the state. The converse happens for the upper state. Calculation of the mixing requires the radial wave functions of q q states, which c a n b e e v aluated from the funnel potential. A di culty, h o wever, is that the radial form of the mixing element V between q q and MMparts of the wave function is unknown, and has to be guessed. The calculations of Oset et al. concern purely the mesonic part of the wave function.

The conceptual analogy is simple: mixing between q q and meson-meson will lower the eigen- value of the lower of the two states given by ( 4). The Variational Principle governs this eigenvalue and the wave function. The numerator of Eq. ( 2) is positive de nite. It is therefore unavoidable that Re (s) goes negative at large s. T h e V ariational Principle will cut o the high mass tail of the resonance, so as to minimise the repulsive p a r t o f R e ( s), hence reducing the resonance width. A feature of light q q and qqq resonances is that their widths are roughly equal to the spacing between radial excitations. It is likely that widths are restricted to this value by a feedback mechanism which creates an orderly sequence of resonances. A general feature of decays of high mass resonances is that S-wave decays to low mass nal states are weak. Decays tend to be to high mass con gurations with small momenta. High spin states generally decay with large angular momenta, where the centrifugal barrier delays the opening of the threshold.

There is little evidence for decays to I = 2 pairs or I = 3 =2 K , w h e r e i n teractions are repulsive 24]. The commonly observed SU(3) octets and decuplets are those whose decays do not lead to such repulsive nal states. Higher representions such a s 27, 10 and 10 do lead to such repulsive nal states. The natural interpretation is that repulsive nal states actively inhibit formation of representations higher than octets and decuplets. The Variational Principle arranges that the con gurations produced are those where repulsive nal states are suppressed.

Approximations

A di culty at present is that the form factor used in Fig. 1 is not known precisely. The usual Flatt e formula, Eq. ( 1), serves as an approximate tting function where M and g 2 are tted empirically. H o wever, the cusp changes slope abruptly at the KKthreshold. Analysis of data then requires a precise knowledge of experimental resolution if the cusp is included in the t. This is illustrated for a 0 (980) in Ref. [START_REF] Bugg | [END_REF], where Crystal Barrel data on pp ! 0 0 are tted including the cusp. The mass resolution, 9.5 MeV, is known accurately in this case, but is large enough to smear out the cusp seriously. The cusp plays a strong role if the energy resolution is as good as 1 MeV. A further detail is that separate thresholds for K + K ; and K 0 K 0 have not been used in Fig. 1 for simplicity these two thresholds may be taken into account straightforwardly and the equations are given by A c hasov and Shestakov 2 6 ] .

Broad Thresholds

Although broad thresholds may p l a y a role in forming a resonance, the dispersive t e r m R e eventually has only a small e ect on the experimental line-shape in most cases. There is, however, a crucial detail which has often been neglected in ts to data.

Let us consider as an example ! . The Breit-Wigner amplitude in this case is f = 1 k M q ; (s) q ; 4 (s) M 2 ; s ; Re (s) ; iM; total (s) :

(5)

Here, k is the centre of mass momentum (allowing for the incident ux). The phase space is approximately constant. However, it is essential to allow for the rapid s-dependence of ; 4 . Many experimental analyses ignore this point and t jfj 2 to a Breit-Wigner resonance of constant width. This is a critical point for many resonances in the mass range 1{2 GeV, where thresholds are opening. Fig. 3(a) shows line-shapes of f 0 (1370) ! and 4 as an example. There is a large di erence between them. This is the source of the large spread in masses reported by the Particle Data Group (PDG) for f 0 (1370) 27]. Anisovich et al. 28] determine the K-matrix pole position 1306 20 MeV from a combined analysis of data on ; p ! 0 0 n and K Kn, + ; ! + ; , and Crystal Barrel data for pp at rest ! 3 0 , 0 , + ; 0 , K + K ; 0 , K 0 S K 0 S 0 , K + K 0 S ; and pn ! ; ; + , K 0 S K ; 0 and K 0 S K 0 The conclusion is that experimentalists can safely omit Re as a rst approximation. However, phase shifts depart signi cantly from a Breit-Wigner amplitude of constant width. At l o w mass, where 4 phase space is small, ; total is small and the phase shift varies rapidly at high mass it varies more slowly. F or high quality data, a second pass including Re (s) is desirable Likewise (1405) and (1475) may be tted as two decay modes of a single (1440) 30]. The (1475) is seen only in KK (890), where phase space rises from threshold near 1385 MeV as momentum cubed in the nal state the (1405) is seen in K S-waves and where phase space changes slowly. 4 , , a 0 (980) and f 0 (980)

The and poles are well predicted in both mass and width by the Roy equations 31] 32], which are based on ta n d u-channel exchanges. Exchange of (770) and K (890) make strong contributions. The Julich group of Janssen et al . 33] showed that meson exchanges account for f 0 (980) and a 0 (980). It seems unavoidable that all four states , , a 0 (980) and f 0 (980) are strongly driven by meson exchanges. They are conspicuous because q q states lie in the mass range 1300{1700 MeV. There is one feature, however, which i s h i d d e n i n t h e m e s o n e x c hanges going into the Roy equations. They impose the Adler zero coming from chiral symmetry breaking. This is a short-range e ect.

Ja e has proposed 34] that , , a 0 (980) and f 0 (980) are colourless 4-quark states made from a coloured SU(3) 3 combination of qq and a 3 combination of q q. This naturally leads to a light , a n i n termediate and the highest (degenerate) masses for a 0 (980) and f 0 (980), in agreement with experiment. Note, however that meson-meson con gurations lead to a similar spectrum except that the a 0 (980) might lie at the threshold. This does not happen because of the nearby Adler zero at s = m 2 ;m 2 =2 the a 0 (980) migrates to the KKthreshold because the Adler zero in this case is distant, at s = m 2 K =2 35]. Ja e's model does not agree well with the observed decay branching ratio ( ! KK)=( ! ) near 1 GeV. 36] A serious problem is that, from the width of the pole, the width is predicted to be (236 39) MeV, much l e s s than the latest value: 758 10(stat) 44(syst) MeV 24].

A further point is that Maiani et al. 17] extend Ja e's scheme to cq] qq] con gurations 6 3.

They give a rm prediction for the observation of analogues of a 0 (980) in c sn n with I = 1 and charges 0, +1 and +2. There is no evidence for such s t a t e s a s y et. Although experimental line-shapes are not a ected strongly for broad resonances, one should not jump to the conclusion that broad resonances are pure q q states. The calculations of Oset et al. show that mesonic components are potentially large. A related point is what happens to the upper energy combinations appearing in Eq. ( 4). If mesonic components are large, these upper combinations are moved upwards substantially. As Ja e remarks, they become broad and are likely to fall apart, creating a broad high mass background. The high mass tail of the does behave in this sort of way a b o ve 1 GeV, due to coupling to 4 29] however, the precise form of this high mass behaviour is poorly known because of lack o f d a t a o n ! 4 .

Applications

X(3872)

It is evident from the width of the cusp in Re of Fig. 1 that a cusp alone fails to t the 3 MeV width of X(3872) 27] the coupling to DD is weaker than that of f 0 (980) to KK, but the shape of the dispersion curve is similar.

Several authors conclude that a linear combination of c c and DD is likely in X(3872). Eichten et al . 37] remark that this explains the low m a s s o f X(3872) compared to early calculations based purely on c c. Suzuki 38] points out that the large production rate of X(3872) in CDF data from the Tevatron requires that it has a large wave function at the origin, hence a substantial c c component for a pure molecular state the observed production rate is nearly 2 orders of magnitude smaller than CDF observe. Conversely, the molecular re-arrangements DD ! J= and !J= account naturally for the weak decays which are observed here, the strength of DD binding is unimportant. Bignamini et al. con rm that the CDF production rate is about 2 orders of magnitude too large to be explained by a molecular component alone 40]. Swanson 41] also favoures mixing between cc and a molecular component.

Recently Lee et al. 42] have made a detailed t to existing data, solving the Bethe-Salpeter equation -equivalent t o e v aluating the dispersion integral of Eq. (2). The binding energy of X(3872) arises essentially from DD loop diagrams. The magnitude of exchange is known from the width of the decay D ! D other exchanges are modelled. However, meson exchanges are not the essential source of binding. They simply need to be attractive, so that D and D approach one another. Both D 0 D 0 and D + D ; channels contribute, though the X(3872) appears at the lower threshold. The binding energy is controled sensitively by the form factor.

Kalashnikova and Nefediev conclude that X(3872) has substantial components of both cc and DD 43]. They point out that Babar data for the ratio of branching ratios BR X(3872) ! 0 (2S)]=BR X(3872) ! J= ] = 3:4 1:4 44] agrees within a factor 2 with estimates for a cc state 45]. The prediction of Swanson for a pure molecule is ' 4 10 ;3 46]. Also Gutsche et al. conclude that some component o f c c is essential to explain the large rate for X(3872) ! 0 47]. So there seems to be widespread agreement t h a t X(3872) is a linear combination of c c and D D . The precise combination is not yet agreed.

The narrow width of X(3872) arises because its decay m o d e s t o + ; J= , !J= and possibly (3510) are OZI suppressed, therefore weak. Its coupling to D 0 D 0 over the width of the resonance is also weak. However, the coupling to DD rises rapidly above threshold and produces the binding via virtual loop diagrams. Kalashnikova and Nefediev conclude that the cc state is attracted to the DD threshold. Ortega et al. 48] reach a similar conclusion that X(3872) must have a large DD component.

Not all cusps are resonances

There is a cusp at the d threshold 49], but no resonance. The exotic Z + (4430) of Belle 50] is at the threshold for D (2007)D 1 (2420) and has a width close to that of D 1 (2420). The data can be tted as a resonance, but can also be tted successfully by a non-resonant cusp, see Fig. 6 of 3]. Additionally, Babar do not con rm the existence of the Z(4430).

Light V ector Mesons

Crystal Barrel data in ight, taken together with other data at lower energies, provide evidence that resonance masses squared for each spin-parity lie close to straight line trajectories 51]. Updated examples are shown here in Fig. 3. They resemble Regge trajectories, except they are drawn for one set of quantum numbers at a time. There is a striking agreement for all J P , with a common slope of 1:143 0:013 GeV 2 for each unit of excitation. A similar regularity is observed for baryon resonances with similar slope 52]. Such regularity agrees with states having a large q q component, but there is no direct connection with molecules or dynamically generated states. An application of the idea that some resonances mix strongly with channels to which t h e y decay concerns (1900). This state lies close to the N N threshold and it is well known that the pp 3 S 1 interaction is strongly attractive. Babar and E687 observe it in decays to 3 + 3 ; and 2( + ; 0 ) 27]. These are strong decay m o d e s i n N N annihilation. It is natural to interpret (1900) as the n = 3 3 S 1 n n state mixed with pp. Then other states fall into place as follows: (ii) (2000) = 3 D 1 n = 2 . It is observed in three sets of data: + ; , ! and a 0 !. There are extensive di erential cross section and polarisation data on pp ! + ; from the PS 172 experiment d o wn to a mass of 1910 MeV (a beam momentum of 360 MeV/c) 53]. There are further similar data above a beam momentum of 1 GeV/c from an experiment at the Cern PS of Eisenhandler et al 54]. The polarisation data determine the ratio of decay amplitudes to 3 D 1 and 3 S 1 pp con gurations: r D=S = g pp ( 3 D 1 )=g pp ( 3 S 1 ) = 0 :70 0:32 for the low a vailable momentum in pp, this is a rather large 3 D 1 component. (iii) (2150) = 3 S 1 n = 4 . I t i s s e e n i n + ; data of 53] and 54] and in Crystal Barrel data for a 0 (980) and in GAMS and Babar data 27] (the PDG incorrectly lists the 1988 MeV state of Hasan 55] under (2150), but it is the (2000)). For (2150), r D=S = ;0:05 0:42.

(iv) (2265) = 3 D 1 , n = 3. It is observed only in two sets of data, + ; and in Crystal Barrel data for a 2 ! and therefore needs con rmation it has a large error for r D=S . T h e (1700), ( 2000) and ( 2265) are consistent within errors with a straight trajectory with the same slope as other states, see Fig. 2(b). (v) The Y (2175) 27] observed by BES 2 and Babar in f 0 (980) and K + K ; f 0 (980) makes a natural s s partner for (2000). Note that there is su cient momentum in the nal state to allow a 3 D 1 state.

J P = 2 + light mesons

The f 2 (1565) lies at the !! threshold. The PDG quotes an average mass of 1562 13 MeV, although Baker et al. 56] nd a mass of 1598 11(stat) 9(syst) MeV when Re (s) is included in the analysis. This is distinctly lower than the mass of a 2 (1700), in the range 1660{1732 MeV. For almost massless quarks, one expects f 2 and a 2 masses to be close. So f 2 (1565) has clearly been attracted to the !!threshold. This requires a molecular component. The f 2 (1565) appears clearly in , as observed by s e v eral groups 27]. It should appear in with g 2 = 3 g 2 !! by SU(2) symmetry, w h i c h predicts g( 0 0 ) = ;g(!!) because of the similar masses of light quarks and the close masses of (770) and !(782).

The f 2 (1640) ! !! observed by GAMS and VES 27] may be tted by folding the line-shape of f 2 (1565) with !! phase space and a reasonable form factor 56], together with the dispersive term Re (s). There is no need for separate f 2 (1565) and f 2 (1640) this has confused a number of theoretical predictions of the sequence of 2 + states. Fig. 2(b) shows trajectories for 2 ++ states, including those above the pp threshold from Crystal Barrel data in ight, using trajectories with a slope of 1.14 GeV 2 . T h e P D G m a k es a number of serious errors in reporting the Crystal Barrel publications. Those mistakes distort conclusions to be drawn from the data. It lists f 2 (2240) under f 2 (2300), which is observed in and KKby all other groups. The f 2 (2300) is naturally interpreted as an s s state. Both f 2 (2240) and f 2 (2293) are observed in a combined analysis of ten sets of data: four sets of PS172 and Eisenhandler et al., together with Crystal Barrel data for 0 0 , 0 0 0 , , 0 0 , and 0 .

The data from the last 3 channels are tted to a linear combination cos jn n > + sin js s > and the mixing angle is determined to be = 7 :5 for f 0 (2240) and = ;14:8 for f 2 (2293) 57]. So the f 2 (2240) is certainly not an s s state. From polarisation data, the f 2 (2240) is dominantly 3 P 2 with r F= P = 0 :46 0:09 (de ned like r D=S ) and the f 2 (2300) is largely 3 F 2 with r F= P = ;2:2 0:6. The PDG fails to list the f 2 (2293) at all, despite many prompts over a 6 year period. It is observed in 5 channels: , , 0 , f 2 and a 2 (1320) .

A further comment is that the f 2 (2150) is conspicuous by its absence from Crystal Barrel data in ight. All s s states such a s f 2 (1525) are produced very weakly in pp interactions. The f 2 (2150) is observed mostly in K K and channels. It is therefore naturally interpreted as an s s state, the partner of f 2 (1905).

An important systematic observation is that pp states tend to decay with the same L as the initial pp state. There is a simple explanation, namely good overlap of the initial and nal states in impact parameter. This observation may be useful to those calculating decays, hence mesonic contributions to eqns. (1){(4).

Light 0 + mesons

There is extensive evidence for a radial excitation of f 0 (1370) at 1790 MeV. The facts run as follows. In BES 2 data for J= ! !K + K ; , there is a clear f 0 (1710) ! KK 58]. In high statistics data for J= ! ! + ; , 59] there is no visible f 0 (1710), setting a limit on branching ratios: BR(f 0 (1710) ! )=BR(f 0 (1710) ! KK) < 0:11 with 95% con dence. Thirdly, i n J= ! + ; , there is a peak requiring an additional f 0 (1790) decaying to but weakly to KK 60]. There is ample independent evidence for it in J= ! 4 61] 62] 63] and pp ! 0 in ight 64]. BES 2 also report an ! peak of 95 events at 1812 MeV J P = 0 + is favoured 65]. It is con rmed by VES data at the Hadron09 conference 66]. The BES data may be tted well with the f 0 (1790) line-shape folded with ! phase space and a form factor exp ;3k 2 ! . There is some scatter on Fig. 4(c) of masses about the line of standard slope this may w ell be because masses of 0 + states tend to be the most di cult to deterine due to isotropic angular distributions.

A comment is needed on f 2 (1810) of the Particle Data Tables. It does not t in naturally in Fig. 2(b). The spin analysis of the GAMS group 67] nds a very marginal di erence between spin 0 and spin 2. It rests on a ne distinction in the angular distribution depending strongly on experimental acceptance however, no Monte Carlo of the acceptance is shown. With the bene t of hindsight, it seems possible that this was in fact the rst observation of f 0 (1790).

Summarising subsections 5.3 to 5.5, there is strong evidence in Figs. 4(a)-(c) that resonances lie close to straight trajectories as a function of mass squared. These may be redrawn as Regge trajectories for 1 ; , 2 + , 3 ; , etc., see Fig. 56 of 51]. Regge trajectories are naturally explained by a ux tube joining q and q the energy stored in the ux tube explains the linear relation between J and mass squared. There is no clear reason why molecules or dynamically generated states should follow s u c h trajectories.

Glueballs

Morningstar and Peardon 68] predict glueball masses in the quenched approximation where q q are omitted. When mixing with q q is included, mixing is likely to lower glueball masses.

Remarks on further experiments

Further progress towards a complete spectroscopy of light mesons and baryons is important for an understanding of con nement -one of the key phase transitions in physics. Progress is possible by measuring transverse polarisation in formation processes. Consider pp as an example. The high spin states appear clearly as peaks, e.g. f 4 (2050) and f 4 (2300). These serve as interferometers f o r l o wer states. However, di erential cross sections measure only real parts of interferences. This leaves the door open to two-fold ambiguities in relative phases and large errors if resonances happen to be orthogonal. A measurement of transverse polarisation normal to the plane of scattering measures T r<A y A > , w h e r e A is the amplitude. This measures the imaginary part of interferences. What appears to be less well known is that transverse polarisation in the plane of scattering gives additional information for three and four-body nal states with a decay plane di erent to the plane de ned by the beam and initial state polarisation. This depends on T r<A x A > , and measures the real parts of exactly the same interferences as appear from the y operator. Longitudinal polarisation depends only on di erences of two intensities and is less useful.

An example of a simple experiment w h i c h w ould pay a rich dividend is to measure such polarisations with the Crystal Barrel detector at the forthcoming GSI p source, over the same mass range as used at LEAR. An extracted beam with these momenta will be available at the FLAIR ring. Such measurements could indeed have been made at LEAR if it had not been sacri ced to the funding of the LHC. The present situation is that the amplitudes for I = 0 , C = +1 states are unique for all expected J P . F or I = 1 , C = ;1, they are nearly complete, but there are some weaknesses for low spin states, notably 3 S 1 which leads to a at decay angular distribution. For I = 1 , C = +1 there is a two-fold ambiguity for nal states and crucial J P = 0 + states are missing. For I = 0 , C = ;1 there are many missing states.

The measurements required are to (i) and

(I = 1 , C = +1), (ii) ! (I = 1 , C = ;1), (iii) ! and ! 0 (I = 0 , C = ;1). A measurement o f 0 0 would cross-check the existing solution and provide information on interferences between singlet and triplet pp states. As well as locating missing states, this would build up a clear picture of the many d e c a ys observable, for comparision with meson exchange processes. All of these channels can be measured simultaneously with the existing Crystal Barrel detector.

A Monte Carlo simulation of results extrapolated from existing analyses predicts a unique set of amplitudes for all quantum numbers. Data are required from 2 GeV/c down to the lowest possible momentum 360 MeV/c. Seven of the nine momenta studied at LEAR were run in 3 months of beam time, so it is not a long experiment, nor does it demand beam intensities above 5 10 4 p=s. A Monte Carlo study shows that backgrounds from heavy nuclei in the polarised target (and its cryostat) should be at or below a n a verage level of 10% this is comparable with cross-talk between nal states and is easily measured from a dummy target.

Baryon spectrocopy w ould also bene t from similar p polarisation measurements in inelastic channels. A transverse magnetic eld is required for compatibility with the polarised target. Rates are enormous, so running time is governed by d o wn-time required for polarising the target and changing momenta. Data at 30 MeV steps of mass appear su cient, except close to 2-body thresholds such a s !N.

Conclusions

The objective of this paper has been to make a case for what appears logically necessary, n a m e l y that both quark combinations at short range and decay c hannels at large range contribute to the eigenstates. The X(3872) is a prime example of mixing between c c and meson-meson in the form of DD .

In The data on a 0 (980), f 0 (980), f 2 (1565) and (1900) t naturally into this picture. There must be a large mesonic contribution to the nonet of , , a 0 (980) and f 0 (980), but there could be a modest diquark component a s w ell. It is likely that there will be a small q q component, but this is suppressed by the L = 1 centrifugal barrier for 3 P 0 combinations.

Experimentalists must take c a r e t o t t h e s-dependence of the numerator of Breit-Wigner resonances due to phase space, e.g. f 0 (1370) ! 2 has a very di erent line-shape to f 0 (1370) ! 4 . The denominator may be tted as a rst approximation with a Breit-Wigner resonances of constant width however, for high quality data, the e ect of the dispersive component in the real part of the denominator matters. For sharp thresholds, e.g. f 0 ! KK, the Flatt e f o r m u l a i s a n approximation with high quality data, the correction due to the cusp in Re (s) is important, but requires precise information on experimental resolution. Further experiments on transverse polarisation in inelastic processes are needed and appear to be practicable without large cost.

Figure 1 :

 1 Figure 1: (a) Re KK (s) a n d G 2 KK KK (s) f o r f 0 (980), normalised to 1 at the peak of G 2 KK KK

Figure 2 :

 2 Figure 2: Sketch of the electron density in the hydrogen molecule for (a) lower and (b) upper states of Eq. (4).

S;Figure 3 :

 3 Figure 3: (a) Line-shapes of f 0 (1370) for ! (full), a Breit-Wigner amplitude of constant width (dotted), and ! 4 (dashed) (b) the Argand diagram, with masses marked in GeV.

Figure 4 :

 4 Figure 4: Trajectories of nn resonances: (a) I = 1 , J P C = 1 ;; , ( b ) I = 0 , J P C = 2 ++ , (c) I = 0 , J P C = 4 ++ and 0 ++ .

  view of the calculations of Oset et al. and Barnes and Swanson, it seems likely that many resonances contain large mesonic contributions. The straight trajectories of Figs. 2(a)-(c) are naturally explained by Regge phenomenology molecules and dynamically generated states provide no obvious explanation of these trajectories. The large mass shift between f 2 (1565) and a 2 (1700) indicates meson mixing into f 2 (1565), lowering the eigenvalue in analogy with the covalent b o n d i n c hemistry.


Recent results on strangeness production from NA49 Michael Mitrovski for the NA49 Collaboration

Keywords: numbers: 25.75.-q, 25.75.Ag, 25.75.Dw, 25.75.Gz, 25.75.Nq

We present a summary of measurements of strange particle production performed by the experiment NA49 in inelastic p+p interactions, as well as semicentral C+C and Si+Si, central Pb+Pb, and minimum bias Pb+Pb collisions in the energy range √ s N N = 6.3 -17.3 GeV. New results on π -, K + and K -production in minimum bias Pb+Pb collisions at √ s NN = 8.7 and 17.3 are shown. Furthermore the strangeness enhancement factor at √ s NN = 17.3 GeV is presented and compared to the results from NA57 and STAR. Energy dependence of strange particle yields normalized to pion yields is presented. New data on K * (892) 0 production are shown at √ s NN = 17.3 GeV. Furthermore we present the energy dependence of K/π and K/p fluctuations. The data are compared with model predictions.

Introduction

Relativistic heavy ion collisions offer a possibility to study nuclear matter under extreme collisions, at high temperatures and densities, in the laboratory. It is expected that at high temperature and baryon density, nuclear matter melts into a state of free quarks and gluons, known as the quark gluon plasma (QGP) [1,2,3].

The study of strange hadron production is a good diagnostic tool to investigate the properties of matter created in heavy ion collisions. The measured final distributions of strange hadrons may give insight into strangeness production in the hot and dense stage of the reaction.

Almost 25 years ago the authors in [4,5] suggested that strange particle production should be enhanced in collisions with QGP production in comparison to those without QGP and could thus serve as a unique signature of the quark gluon plasma. This enhancement can be quantified by the ratio of yields in nucleus+nucleus to p+p collisions where the system is very small and the QGP creation is not expected.

The NA49 experiment

The NA49 detector [6] is a large acceptance hadron spectrometer at the CERN SPS, featuring four large volume TPCs as tracking detectors. Two of them are positioned inside two superconducting dipole magnets. The ionization energy loss (dE/dx) measurements are used for particle identification in the TPCs with a typical resolution of 4% at forward rapidities. The dE/dx resolution allows in the laboratory frame of the Main-TPCs hadron identification for p > 4 GeV/c with typical K/π and K/p separation power of ≈ 1.5 σ. Furthermore the two time-of-flight walls (resolution σ T OF = 60 ps) improve the separation power in selected regions around midrapidity. The centrality of a given reaction is determined on the basis of the energy of the projectile spectator nucleons measured by the forward calorimeter.

Results

The NA49 experiment has collected data on central Pb+Pb collisions at 5 beam energies in the range √ s NN = 6.

System-size dependence of particle yields

The NA49 experiment features a large acceptance in the forward hemisphere allowing measurements of rapidity spectra from midrapidity up to almost beam rapidity. Figures 1 and2 show rapidity spectra for π -, K + , K -, Λ and Λ [7] for different centralities in Pb+Pb collisions at √ s NN = 8.7 and 17.3 GeV, where the closed points represent the measured data and the open symbols represent the reflected points at midrapidity. The total multiplicity is extracted by fitting a (double-) gaussian distribution to the rapidity spectra. It is visible that the width of the pion rapidity-spectra changes with centrality. The distribution become broader when moving to peripheral collisions which might indicate that the influence of resonance decays is much more pronounced for peripheral collisions. In comparison the width of the charged kaons does not change with the centrality of the collision.

Figure 3 shows the strangeness enhancement factor E at midrapidity (|y| ≤ 0.5) for mesons (left) and baryons (right) as a function of the number of wounded nucleons N W , calculated within the Glauber model [8], at √ s NN = 17.3 GeV [9,10,[START_REF] Afanasiev | The NA49 Collaboration[END_REF]12,13,14].

The strangeness enhancement E is defined as the yield per wounded nucleon in Pb+Pb The grey boxes represent systematic errors.

collisions normalized to the corresponding yield in p+p interactions:

E = Y ield N W A+A / Y ield 2 p+p ( 1
)
An enhancement is visible for mesons and baryons. For mesons a clear hierarchy is visible from particles without strange quarks (π) to particles with two strange quarks (φ) when going from p+p to Pb+Pb collisions. A similar behavior is visible for baryons, except the Λ which does not show any enhancement. Furthermore the enhancement factor increases fast for N W < 60 and then approximately saturates. Various approaches have been used to explain the centrality/system size dependence of particle production. One approach is the Core-Corona model [15,16,17,18,19]. It is assumed that the core of a heavy ion collisions is fully thermalized and strangeness is produced according to equilibrium hadron gas. The core is surrounded by a corona which can be modeled as a superposition of elementary p+p reactions. Another approach is using a hydrodynamic model with a transport model as afterburner [20]. One finds that strangeness equilibration is only reached in Pb+Pb collisions. The equilibrium is also reached in peripheral Pb+Pb collisions, whereas for C+C and Si+Si interactions a better description is obtained with a binary scattering transport approach like UrQMD [20,21] which does not assume equilibration. Both approaches provide a good description of the data.

Figure 4 shows the enhancement factor versus the strangeness content at midrapidity (|y| ≤ 0.5) for mesons (left) and baryons (right) in central Pb+Pb/Au+Au collisions from the NA49 [9,10,[START_REF] Afanasiev | The NA49 Collaboration[END_REF]12,13,14] (closed circles) and NA57 [22] (open between top SPS and top RHIC energies is observed. For baryons a different behavior is visible. Compared to top RHIC energies the net-baryon density at SPS energies is higher and therefore the Λs are getting absorbed during the late stage of the evolution [20,26].

For other strange baryons a hierarchy with the strangeness content is visible but a higher enhancement is observed at top SPS compared to top RHIC energies. The strangeness enhancement factor increases further when going from the SPS to AGS energies. This trend can be understood due to an increasing suppression of strangeness production in p+p interactions with decreasing energy because of energy conservation [27]. Thus the old idea of strangeness enhancement as QGP signal [4,5] does not hold at low energies.

Energy dependence of particle yields

Figure 5 shows the energy dependence of relative strange particle yields at midrapidity (|y| ≤ 0.5). The K -/π -, Λ/π, Ξ+ /π, Ω -+ Ω+ /π and φ/π ratios rise continuously from AGS to RHIC energies. In contrast the K + /π + , Λ/π and Ξ -/π ratio shows a distinct maximum in the energy dependence. The results are compared to the transport model UrQMD v2.3 [21]. UrQMD provides a good description for the K -/π -and Λ/π ratio. For other ratios large deviations are observed. Statistical hadron gas models provide better fits to the data (see e.g. [28,29].

Resonances

Short lived resonances are an efficient tool to probe the properties of the hot and dense medium which is created in heavy ion collisions. Resonances can be sensitive to two effects in a dense medium. If they decay before the kinetic freeze-out they may not be reconstructed due to the rescattering of the daughter particles and therefore their yield is reduced due to destruction between chemical and kinetic freeze-out. The other possibility is that after chemical freeze-out pseudo-elastic interactions [30] among hadrons could increase the resonance population. This resonance regeneration depends on the cross-section of the interacting hadrons in the medium. Figure 6 (left) shows the K * (892) 0 / K + and the K * (892) 0 / K -ratios as a function of the system size at √ s NN = 17.3 GeV. It is clearly visible that these ratios decrease with the increasing system size. The right panel of Fig. 6 shows the comparison of the total yield of the resonances K * (892) 0 ( K * (892) 0 ), Λ(1520) and φ to predictions from a hadron gas model [31]. The deviation of the hadron gas model from the data becomes larger with decreasing lifetime of the resonance. This leads to the conclusion that a large part of the reduction of the K * (892) 0 yields could be due to rescattering of the decay daughters during the hadronic stage of the fireball and indicates that this stage lasts for a time GeV as a function of N W . Right panel: The total yield of K * (892) 0 ( K * (892) 0 ), Λ(1520) and φ divided by a hadron gas model [31] as a function of the lifetime cτ .

Figure 7.

Energy dependence of σ dyn for the K/π (left panel) and K/p (right panel) ratio [35,36,[START_REF] Schuster | for the NA49 collaboration[END_REF]. The measurements are compared to the transport models UrQMD [START_REF] Schuster | for the NA49 collaboration[END_REF] and HSD [38].

comparable to the lifetime of the resonance. These results are comparable to published results from the STAR experiment [32,33].

Fluctuations

Event-by-event hadron yield ratios characterize the chemical composition of the fireball in each event and fluctuations of net baryon number and strangeness are sensitive to the properties of the early stage. Compared to other conserved quantities, like charge [34], these fluctuations are less strongly affected by hadronic reinteractions in the later stage of the collisions. This means that possible signatures of a phase transition are less washed out. The left panel of Fig. 7 shows the energy dependence of σ dyn for the K/π ratio [35,36,[START_REF] Schuster | for the NA49 collaboration[END_REF].

A positive value for σ dyn for the K/π ratio is observed for the whole energy range. From top SPS to top RHIC energies σ dyn is constant. The rise at low SPS energies could be a signal for the onset of deconfinement [35]. When comparing measurements to the predictions of the transport models UrQMD [START_REF] Schuster | for the NA49 collaboration[END_REF] and HSD [38] one finds that UrQMD does not describe the increase towards lower SPS energies, whereas HSD provides a better description of the increasing trend but does not describe the data at higher SPS energies. Due to large differences between the transport model predictions no conclusions can be drawn. Furthermore it is visible that applying different acceptances (in this example NA49 and STAR acceptance) does not change the predictions for σ dyn . The value of σ dyn for the K/p ratio is shown in Fig. 7

Figure 1 .

 1 Figure 1. Rapidity distributions for π -, K + , K -, Λ and Λ [7] measured for different centralities in Pb+Pb collisions at √ s NN = 8.7 GeV. The closed symbols indicate measured points, whereas the open symbols are reflected with respect to midrapidity.

Figure 2 .

 2 Figure 2. Rapidity distributions for π -, K + , K -, Λ and Λ [7] measured for different centralities in Pb+Pb collisions at √ s NN = 17.3 GeV. The closed symbols indicate measured points, whereas the open symbols are reflected with respect to midrapidity.

Figure 3 .

 3 Figure 3. Strangeness enhancement E at midrapidity (|y| ≤ 0.5) for mesons (left) and baryons (right) as a function of N W at √ s NN = 17.3 GeV [9, 10, 11, 12, 13, 14].

Figure 4 .

 4 Figure 4. Enhancement factor E versus strangeness content at midrapidity (|y| ≤ 0.5) for mesons (left) and baryons (right) in central Pb+Pb/Au+Au collisions from the NA49 [9, 10, 11, 12, 13, 14] (closed circles) and NA57 [22] (open circles) Collaborations at √ s NN = 17.3 GeV and the STAR [23, 24, 25] (stars) Collaboration at √ s NN = 200GeV (Note NA57 uses p+Be instead of p+p as baseline). The grey boxes represent the systematic error.

Figure 5 .

 5 Figure 5. The energy dependence of the midrapidity (|y| ≤ 0.5) yields of strange hadrons, normalized to the pion yields (π = 1.5 (π -+ π + )), in central Pb+Pb/Au+Au collisions. The data is compared to the string hadronic model UrQMD v2.3 [21].

Figure 6 .

 6 Figure 6. Left panel: The total yield of K * (892) 0 ( K * (892) 0 ) divided by the total yields of K + ( K -) in p+p, C+C, Si+Si and Pb+Pb collisions at √ s NN = 17.3

  (right panel). Two sign changes are seen for this observable when looking at the energy dependence. Between √ s NN = 7.6 -17.3 a negative value is observed, whereas at √ s NN = 6.3, 62.4 and 200 GeV it is positive. Comparing the results to UrQMD one finds that the model describes the overall trend except the lowest energy NA49 point.

† Presented at Strangeness in Quark Matter 2009, Buzios, Rio de Janeiro, Brazil Confidential: not for distribution. Submitted to IOP Publishing for peer review 15 March 2010

Acknowledgments

This work was supported by the US Department of Energy Grant DE-FG03-97ER41020/A000, the Bundesministerium fur Bildung und Forschung, Germany, the Virtual Institute VI-146 of Helmholtz Gemeinschaft, Germany, DESY, Germany, the Polish Ministry of Science and Higher Education (1 P03B 006 30, 1 P03B 127 30, 0297/B/H03/2007/33, NN202 078735), the Hungarian Scientific Research Foundation (T032648, T032293, T043514), the Hungarian National Science Foundation, OTKA, (F034707), the Bulgarian National Science Fund (Ph-09/05), the Croatian Ministry of Science, Education and Sport (Project 098-0982887-2878) and Stichting FOM, the Netherlands.


G D Jones
email: jx@ns.ph.liv.ac.uk

High alignment following 219 Rn and 211 Bi decays 1 High alignment following 219 Rn and 211 Bi decays

Keywords: 23, 20, En; 23, 60, + e

An analysis of the 219 Rn (ground state) 215 Po * (402 keV) 215 Po (ground state) angular correlation and the corresponding linear polarization angular distribution shows that the 402 keV excited state is highly aligned in substates. A similar analysis for the 211 Bi (ground state) 207 Tl * (351 keV) 207 Tl (ground state) ! angular correlation and a linear polarization measurement at 90°shows that, in contrast, the 351 keV excited state is highly aligned in ! substates.

The results enable hypotheses to be advanced about the shapes of nuclei in these excited states.

Introduction

The three processes of (i) α-decay followed by γ-decay (ii) α-capture followed by γ-decay and (iii) α-decay of a polarized system have been related by Rowley et al. [1].

For process (i) they considered favoured α-transitions (hindrance factor < 4) between parent and daughter states in odd-even nuclei. In the case of a prolate even-even core (positive quadrupole moment, Q) coupled to a deformation aligned nucleon they found that high alignment was produced in the excited state of the daughter, with preferential population of the higher substates. The analogous result for the oblate deformation aligned system (-Q) was a high alignment, with preferential population of the lower substates.

Two problems, which were not addressed, were firstly the degree of alignment produced, following favoured α-decay, when the even-even core was very weakly deformed (Q 0) and secondly the degree of alignment produced following non-favoured α-decay.

One can go some way to answering both these questions, from an experimental perspective, by considering transitions in the 227 Th decay chain. The 219 Rn (ground state) 215 Po * (402 keV) transition is a favoured α-decay (hindrance factor = 3.4). 215 Po consists of 7 nucleons outside a doubly magic 208 Pb core and more importantly the ground and excited states are well described by the single particle shell model hence the core quadrupole moment should be relatively small.

The 211 Bi (ground state)

Tl * (351 keV) transition is non-favoured and the ground state and excited states of 207 Tl are also well described by the single particle shell model, in terms of holes in the doubly magic 208 Pb core. So, not only does one have a non-favoured α-transition one should also have a relatively small core quadrupole moment.

Experimental α-γ angular correlation measurements have been reported [2,3] for the 219 Rn (ground state) 215 Po * (402 keV)

" 215 Po (ground state) decay. The corresponding α-γ-ray linear polarization angular distribution has also been measured by Krien et al [4]. Unfortunately, an extensive theoretical analysis by Krien et al [4] of the combined angular correlation measurements of Letessier et al [2] and the linear polarization measurements of Krien et al [4] was unable to produce the correct spins for the decay sequence. The spins given above are taken from the work of compilers [3] and consequently the analysis presented in section 4 is the first using the correct spin sequence.

Confidential: not for distribution. Submitted to IOP Publishing for peer review 1 November 2010

In the case of the 211 Bi (ground state) 207 Tl * (351 keV)

" 207
Tl (ground state) ! decay the correct spins were employed by Gorodetzky et al [5] but it is observed in section 6 that their definition of the α-γ-ray linear polarization is opposite in sign to that usually employed (see section 3) and therefore calls into question their theoretical analysis particularly since this is far from transparent [5]. Since historical angular correlation analyses have not been specifically concerned with the degree of alignment of excited states following α-decay, we outline the method required to extract the information from experimental angular correlation measurements in section 2.

In section 3, we outline the standard definitions and formulae used for α-γ-ray linear polarization distributions pertinent to the present analyses.

Angular correlation theory for α-γ decay.

Consider α-decay from a parent nuclear state of spin J p to the excited state of a daughter nucleus of spin J 1 followed by γ-decay to a lower state of the daughter nucleus of spin J 2 . If the α-particles are undetected, the excited daughter nuclei of spin J 1 are randomly orientated in space and the γ-ray angular distribution is isotropic. A non-isotropic γ-ray angular distribution depends on (i) the (non-statistical) population of the state of spin J 1 and on (ii) the intrinsic properties of the states of spin J 1 and J 2 .

The seminal article on γ-ray angular correlations by Rose and Brink [6] shows that factors (i) and (ii) are separable and gives examples which consider different modes of excitation of the electromagnetically decaying state of spin J 1 . However, Rose and Brink [6] did not specifically address the problem of α-γ angular correlations which we now outline using their notation.

The direction of emission of an α-particle from a radioactive source and its subsequent detection provides a quantization axis which is also an axis of cylindrical symmetry. The subsequent γ-ray angular distribution relative to this axis of symmetry may then be written as

# $ % $ & ' ( ) ) * ) +,-% $ (2.
1) where i is the index of data points of solid angle corrected, coincidence intensity W i , i the angle between the directions of α-particle and γ-ray emission and K is even (most frequently K 4). The Legendre polynomial coefficients A K can be determined by the evaluation of the variance-covariance (error) matrix [START_REF] Ferguson | Angular Correlation Methods in Gamma-Ray Spectroscopy[END_REF][START_REF] Bevington | Data Reduction and Error Analysis for the Physical Sciences[END_REF] using weights, 1/ σ i 2 where σ i are the experimental errors. The reaction plane of such a direction-direction angular correlation is defined by the directions of emission of both α-particle and γ-ray.

Normalized Legendre polynomial coefficients, a K , can be determined from the coefficients above. i.e. a o = 1, a 2 = A 2 /A 0 , a 4 = A 4 /A 0 with the appropriate errors [START_REF] Ferguson | Angular Correlation Methods in Gamma-Ray Spectroscopy[END_REF][START_REF] Bevington | Data Reduction and Error Analysis for the Physical Sciences[END_REF]. Values of a 2 and a 4 calculated in this way are the ones quoted in the literature [3] since they are determined without any knowledge of the spins of the states involved.

Theoretical values of a 2 and a 4 , dependent on J 1 , J 2 and the multipole mixing ratio between J 1 and J 2 , must be generated so that theoretically generated coincidence intensities can be compared with experimental data by means of least squares fits.

These theoretical coefficients a K can be factored into two separable parts a K =B K (population) Γ K (gamma decay) [1,6] where

. ) / & ' 0 1 2! 3 4 / ! / / 1 2 1567 (2.
2) depends only on the population parameters w(M) of the magnetic substates M of J 1 with respect to the direction of emission of the α-particle (quantization axis). i.e. on the degree of alignment of the excited state of spin J 1 .

For B K (J 1 ) the sum over M runs from -J 1 to +J 1 and the normalization of the population parameters is such that ' 0 1 & ! and also B 0 (J 1 )=1.

α-particles have an intrinsic spin of zero and hence cannot be polarized. Further for the usual case of an unpolarized radioactive source in a state of spin J P with definite parity then α-decay must align [6] [6].

Equation (2.3) is the most general parameterization of the population condition for the state of spin J 1 requiring no prior knowledge of how the state was formed, the only strict condition being that the state is aligned.

The only limitation imposed by α-decay is that if J P <J 1 , the maximum substate populated should be M = J P since α-particles are spinless and any projection of orbital angular momentum (connecting J P and J 1 ) onto the quantization axis is zero.

Usually only the two lowest order multipoles L 1 and L 2 = L 1 +1 are considered for the γ-decay from the state of spin J 1 to the state of spin J 2 and for the general case one writes

; ) / / & <= ) > > / / ?= ) > > / / ? = ) > > / / @ ! ? (2.5)
where δ is the relevant mixing ratio. For the majority of cases and for the examples considered in the present work L 1 = 1 and L 2 = 2 i.e. dipole and quadrupole radiation respectively. The R K coefficients in (2.5) can be written explicitly as

= ) > > / / & 2! 3 4 3 A B A B 4 ) C / ! > ! > ! D 4 A > > ! 2 !567 # / / > > E 6/ (2.
6) These R K coefficients can be calculated directly but are also given as tabulated coefficients [6].

For the case of 215 Po * decay the γ-transition is pure electric quadrupole with

J 1 = 5/2 + , J 2 =9/2 + . Therefore Γ K (5/2,9/2) = R K (2 2 5/2,9/2) so that F ) & . ) G H; ) G I H & ' C0 1 : ) I 1 D 9 J A 9 4 A = ) G I H (2.7)
Where the w(M) are then the variable parameters to be used for least squares fitting to the experimental data.

α-γ linear polarization theory.

The prescription for obtaining the linear polarization distribution in a direction-polarization measurement was detailed in the definitive article by Fagg and Hanna [START_REF] Fagg | [END_REF].

The γ-ray linear polarization angular distribution relative to the quantization axis (direction of α-particle emission) may be written as

# %I K & # % LMN K ' O ) * ) LMN % )9 IP (3.1)
where the +ve sign is taken for the E2 and E2/M1 radiation considered in the following work, W(θ) is the polarization insensitive intensity measured at a given angle θ, ξ is the angle between the plane containing the electric vector and the reaction plane. Only 0°and 90°are considered for ξ so that cos(2ξ) = ±1. P K (cosθ) = 7.5(-1+8cos 2 θ -7cos 4 θ). [START_REF] Fagg | [END_REF] noted that the coefficients C K in (3.1) can be factored into two separable parts in this case C K = B K (population) G K (polarized γ-decay) where B K (J 1 ) is identical to the form given in (2.3) since we are dealing with the identical population of the state of spin J 1 via α-decay.

Fagg and Hanna

The form of G K can be written explicitly for dipole/quadrupole radiation from Fagg and Hanna

[9] Q / / & ! = ! ! / / 2 ! ?= ! / / ? = / / <! ? @ (3.2)
Q P / / & 2 ? ! = P / / <! ? @ (3.3) where again the R K coefficients can be evaluated from (2.6) or obtained from tables [6].

Fagg and Hanna following earlier authors proposed that the natural quantity for the observation of linear polarization was

* % & <# %I K & 7R 2 # %I K & 7R @<# %I K & 7R # %I K & 7R @ (3.4) If a γ-
ray under investigation is 100% plane polarized with its electric vector in the reaction plane (ξ=0°) then P(θ) = +1. Likewise if a γ-ray is 100% plane polarized with the electric vector normal to the reaction plane (ξ=90°) then P(θ) = -1. All other degrees of polarization lie between ±1.

Substituting ( (3.11) The most general theoretical form of P(θ) for mixed dipole/quadrupole transitions, allowing all physical values of a 2 and a 4 and using trigonometric equalities to expand P K (2) (cosθ) and P K (cosθ) has been given by Jones [10].

In order to develop experimental measurements of γ-ray linear polarization which can be linked to the primary definition given in (3.4) one has to have a phenomenon which is sensitive to the degree of polarization of an incident γ-ray.

The sensitivity of Compton scattering to the linear polarization of an incident γ-ray has long been known and is well described by the Klein-Nishina formula [9,11 and references therein]. This formula enables the relevant differential cross section to be evaluated for scattering of γ-rays through different angles. This shows that the scattering is a maximum in the plane normal to the plane containing the electric vector of the incident γ-ray. An experimental measurement of the type

( % & <[ %I \ & 7R 2 [ %I \ & 7R @<[ %I \ & 7R [ %I \ & 7R @ (3.
12) is expected to be proportional to the theoretical value of (3.4). [ %I \ & 7R is the number of Compton scattered events in the reaction plane and [ %I \ & 7R the number scattered normal to the reaction plane.

We can experimentally determine the proportionality between (3.4) and (3.12) [ the polarization sensitivity Q(E γ )] by utilizing γ-rays of known linear polarization. Several measurements are required at various E γ since Q(E γ ) is energy dependent and the appropriate value can be interpolated from the calibration measurements.

We then may deduce an experimentally measured polarization * % & ( % ] ^

The decay of 219 Rn -angular correlation measurements

We use the spins / _ `& I / `& I / `& as given in the abstract with a pure E2, γ-decay. We have taken published Legendre polynomial coefficients [3] to generate data points at 10°intervals from 0°to 90°using the function # $ % $ & !777<! F * LMN % $ F P * P LMN % $ @ with pseudo-statistical errors of a $ & ! # % $ . These data have been least squares fitted by generating a K from (2.7) with w(M) as the variables.

Firstly we generated data points using the experimentally obtained Legendre polynomial coefficients namely a 2 = -0.198 ± 0.024 and a 4 = -0.005 ± 0.030 determined by Hesselink [3]. The best fit obtained was w(1/2) = 0.030, w(3/2) = 0.048, w(5/2) = 0.422 giving a sum of squares of residuals of ∆ 2 = 3.4x10 -4 . These population parameters produced theoretical coefficients of a 2 = -0.198, a 4 = -0.005 i.e. the same as the experimental coefficients.

Secondly we generated data points from the experimentally obtained Legendre polynomial coefficients namely a 2 = -0.162, a 4 = -0.069 obtained by Letessier et al [2]. Unfortunately, the errors associated with a 2 and a 4 are not reported in that work. The best fit was obtained from w(1/2) = 0.069, w(3/2) = 0, w(5/2) = 0.431 with a sum of the squares of residuals of ∆ 2 = 6.4. These population parameters produced theoretical coefficients of a 2 = -0.192, a 4 = -0.008.

Figure 1 maps the full range of a 2 and a 4 for mathematically allowed population parameters and for the decay sequence studied here. We plot also the best fit results to the Hesselink coefficients, [labelled (a)] and to the Letessier coefficients, [labelled (b)]. Figure 1. Mathematically allowed w(1/2), w(3/2), w(5/2) for an aligned J 1 = 5/2 state mapped onto a 2 and a 4 assuming a subsequent pure E2 γ-decay to a J 2 = 9/2 state. The upper boundary lines are defined by the condition w(5/2) = 0, the lower boundary lines by the condition w(3/2) = 0 for any given w(1/2). The labelled points are determined by least squares fits to the data of, [point a] Hesselink [3] and [point b] Letessier et al [2] assuming aligned systems with population parameters as variables.

It is unlikely that the results of fit (b) are physical, since the experimentally measured Legendre polynomial coefficient a 4 = -0.069 is outside the theoretical range of a 4 allowed for the given spin sequence. The mathematical constraints that the w(M) are positive and that w(1/2) + w(3/2) + w(5/2) = 0.5 are sufficient to force fit (b) close to fit (a). Unfortunately, ∆ 2 is much larger for fit (b) compared to fit (a) and was probably a contributory factor to the original difficulties in assigning spins to the states.

The decay of 219 Rn -linear polarization measurements

Krien et al [4] analysed the angular correlation data of Letessier et al [2] simultaneously with their own linear polarization distribution measurements. Unfortunately, the angular correlation data of Hesselink et al [3] post dated the work of Krien et al [4] and so could not be used. Interestingly, Krien et al [4] also analysed the linear polarization angular distribution of the 402keV γ-ray in the form <[ %I \ & 7R @<[ %I \ & 7R @ not in the standard form given by (3.12). We are unable to convert the published data into a standard P E (θ) since Q(E γ ) is not determined. However we can calculate A(θ) and obtain Q for each measurement in order to determine whether the Q are consistent with the w(M) obtained in section (4). The angles at which the linear polarization measurements were taken can be obtained from Weigt et al [11].

We are able to extract A(90°) = -0.13 ± 0.01, A(115.7°) = -0.12 ±0.015, A(141.4°) = -0.04 ± 0.02 from the published data [4]. The published 180 °point is merely a check against differences in detection and electronic For the population parameters obtained from the Hesselink data we obtain from (2.7 and 3.8) P(90°) = -0.27, P(115.7°) = -0.23, P(141.4°) = -0.12 and from the Letessier data we obtain P(90°) = -0.27, P(115.7°) = -0.22, P(141.4°) = -0.11 i.e. very similar. The polarization sensitivity, Q, is determined as 0.48 ± 0.04, 0.52 ± 0.07, 0.33 ± 0.17 respectively, from the Hesselink data, giving a weighted average of 0.48 ± 0.03. Therefore, the measurements of Q are statistically consistent. This means, of course, that the theoretical linear polarization distribution calculated using the high alignment parameters is consistent with the shape of the experimental linear polarization distribution.

The errors on P(90˚), for example, can readily be deduced, from the errors on the experimental asymmetry, as ~10%. Similar errors may be deduced from the experimental Legendre polynomial coefficients.

The decay of 211 Bi

We use the spin sequence J P = 9/2 -, J 1 = 3/2 + , J 2 = 1/2 + given in the abstract [5,12]. Legendre polynomial coefficients of a 2 = 0.004 ± 0.005 (a 4 = 0) were obtained from angular correlation measurements [5] and a γ-ray linear polarization measurement was also carried out at 90°. The negative of the linear polarization sign convention used in the present work (3.4 and 3.12) was employed [5] and in order to comply with the theory used here the published experimental polarization should be taken as P E (90°) = -0.623 ± 0.023.

We could carry out a similar analysis to that performed in section 4 which would only require a step search since here w(1/2) + w(3/2) = 0.5 but since only a 2 was determined from experiment we can employ a more transparent method.

If we use the coefficients [6],

ρ 2 (3/2 1/2) = -2, ρ 2 (3/2 3/2) = 2 in (2.3) and the coefficient R 2 (1 1 3/2 1/2 ) = 1/2, R 2 (1 2 3/2 1/2) = 3/2 and R 2 (2 2 3/2 1/2) = -1/2 in (2.5
) we obtain from the product of (2.3) and (2.5) that

F & 0 -0 ! ! c ? 2 ? ! ? (6.1) Since a 2 is zero within experimental error, either w(1/2) w(3/2), or δ 2 -2 3δ -1 0. If w(1/2) w(3/2)
then the linear polarization would be very small which is patently incorrect. Therefore, if we solve the quadratic equation in δ to give an approximate solution we find δ = -0.268 or +3.732. The higher solution for δ is ruled out by internal coefficient measurements [5].

More accurately, we can obtain a simple expression for P(90°) using (3.11) with θ = 90°and (2.3) for B 2 (3/2) we find

* 7R & < F 2 X d V0 1 : 1 W= ! ! ? ! ? @ 9 e 9 2 F
Using the coefficients given above we find

* 7R & V F 2 X 0 2 0 ! c ?f ! ? W 2 F (6.2)
Inserting the values P(90°) = -0.623 ± 0.023 and a 2 = 0.004 ± 0.005 we solve the simultaneous equations (6.1) and (6.2) to obtain δ = -0.271 ± 0.004 and [w(1/2) -w(3/2)] = 0.36 ± 0.01. Since w(1/2) + w(3/2) = 0.5 we find w(1/2) =0.43 and w(3/2) = 0.07 i.e. high alignment in the ± 1/2 substates. We note that Gorodetzky et al. [5] obtained the same numerical value for δ but the opposite sign. Rose and Brink [6] have noted the differences in the sign of δ by different authors relative to their own, phase consistent approach. Unfortunately, the theoretical analysis by Gorodetzky et al. [5] is not clear. The parent and daughter states of spin J P and J 1 , respectively connect by the vector equation J P = J 1 + L α . The L α define the allowed α-particle partial waves. Hence, the minimum partial wave > & 5/ _ 2 / 5 Mg 5/ _ 2 / 5 ! which is even when no parity change occurs between J P and J 1 and odd if a parity change occurs.

Since the ground state of 211 Bi has J P π = 9/2 -and the 351 keV state of 207 Tl has J 1 π = 3/2 + then only two interfering partial waves L = 3 and L = 5 are allowed and the population parameters are given by 0 1 & <h hi = LMN j = i @ ! = where X, Y are Clebsch-Gordan coefficients h & / _ / 1 2 15 7 and i & / _ / 1 2 15 7 , = & 5( f( 5 is the ratio of the amplitudes of the partial waves and Φ σ 5 -σ 3 +nπ is the relevant phase between waves of angular momentum 3 and 5 where σ 5 and σ 3 are Coulomb phases [1,6].

In figure 2 we plot w(1/2) and P(90°) as a function of θ = tan -1 R from 0°to 90°, with the value of the mixing ratio δ = -0.271. Figure 2. Plots of the variation of w(1/2) with θ and P(90˚) with θ, where tan θ = R (the ratio of partial wave amplitudes) for the 351 keV decay in 207 Tl calculated with a γ-ray multipole mixing ratio of δ(E2/M1) = -0.271.

The solid lines denote a fixed Coulomb phase difference of (σ 5 -σ 3 ) the dashed lines (σ 5 -σ 3 +π). The points indicate the experimentally measured linear polarization at 90˚and the best fit w(1/2), as calculated in the text.

We find that the phase difference between the partial waves needed to satisfy both w(1/2) = 0.43 and P(90°) = -0.623 requires the choice Φ = (σ 5 -σ 3 ) Also, we find R = 0.27 (in agreement with [5]) satisfies both conditions. This value for R leads to amplitudes of A 3 = 0.97 and A 5 = 0.26, assuming the normalization condition that the sum of the squares of the amplitudes is unity. A second solution can be found to satisfy both conditions on population and polarization with R > 1. This solution is considered much less likely due to decreasing penetrability through the angular momentum barrier for increasing L.

Discussion

The rudimentary picture for the onset of prolate deformation is sometimes visualized as the orbit of a single hole in an otherwise closed shell spherical nucleus giving rise to a 'pinched' waist. In this scenario the total angular momentum vector, J, is perpendicular to the plane of the hole's orbit i.e. along the symmetry axis. This semi-classical picture implies that the nucleus would be fully aligned in M = ± J substates. Allowing for the possibility of Coulomb interference between the 3 α-partial waves involved in the 219 Rn ground state decay, the values w(1/2) = 0.03, w(3/2) = 0.048 and w(5/2) = 0.422 for the population of the 402 keV state in 215 Po must be extremely close to the maximum alignment possible. The low hindrance factor, (HF = 3.4), for the α-decay also implies that the ground state of 219 Rn and the 402 keV excited state of 215 Po have similar structure. The close analogy to the semi-classical picture above would perhaps explain the spin assignments of the states in terms of the single particle shell model rather than the Nilsson model. Rowley et al. [1] showed that for favoured α-decays (as is the case for the 219 Rn decay here) the restriction to the two lowest allowed partial waves i.e. L=0 and L=2 would favour a phase difference of (σ 2 -σ 0 + π) for prolate daughter nuclei. This was found to be the case. The best fit parameters for a two partial wave fit to the data of Hesselink [3] were w(1/2) = 0.010, w(3/2) = 0.076 and w(5/2) = 0.414 for R = 0.682 giving a sum of the squares of residuals ∆ 2 = 0.006 and P(90 ˚) = -0.27. These population parameters and derived polarization are clearly quite close to those obtained by the more exact approach. It should perhaps be pointed out again that the weak deformation above is not in the regime of strong core quadrupole deformation examined by Rowley et al. [1].

In the case of the 351 keV state in 207 Tl, the proton structure is believed to consist of (core)(2d 3/2 ) 3 (3s 1/2 ) 2 which decays to the ground state of proton structure (core)(2d 3/2 ) 4 (3s 1/2 ) 1 . Therefore, logically one might expect to observe the above semi-classical picture again since 208 Pb is the neighbouring doubly closed shell nucleus (although the hole is deeper in the core, for the excited state, rather than in the outer shell). However, the large alignment in ±1/2 substates [w(1/2) = 0.43] implies that the orbital angular momentum vector must be largely perpendicular to the quantization axis. This, perhaps surprisingly, further implies that the shape of the nucleus in the 351 keV state is possibly oblate. (The quantization axis is defined by α-emission which takes place radially outwards around the equatorial plane of an oblate nucleus).

Unlike the case of 219 Rn, the decay of 211 Bi is not a favoured α-decay. The phase shift (σ 5 -σ 3 ) is required to obtain w(1/2) = 0.43 and P(90 ˚) = -0.623 as shown in Figure 2. It is an open question whether any physics is obtained from a determination of this phase shift for such non-favoured transitions. However, it should perhaps be noted that the ground state of 211 Bi is probably also slightly oblate ( 2 neutrons and 1 proton outside a 208 Pb core) and the α-decay is between j = l -1/2 states.

It should be emphasized that perhaps the most remarkable aspect of the results obtained here is the strong tendency to high alignment in these weakly deformed nuclei following the disruptive effect of α-decay.

Conclusions

Alpha-gamma angular correlations and linear polarization measurements have been analyzed following the decays of 219 Rn and 211 Bi with special emphasis on the population parameters of the excited states in the daughter nuclei. The analyses are consistent with the spins assigned by compilers. The multipole mixing ratio and the alpha partial wave amplitude ratio are the same as obtained in previous work for 211 Bi decay [5]. The 402 keV excited state in 215 Po and the 351keV excited state in 207 Tl are found to be highly aligned following α-decay. It is shown that the alignment conditions may be useful when speculating about the shapes of nuclei in excited states.

It is hoped that the advent of radioactive ion beam facilities will enable transitions to be studied which are accessible to both α-decay and Coulomb excitation in order to check the above hypothesis. Since α-decay gives access to states which are sometimes not readily accessible by Coulomb excitation and since the analysis of α-γ angular correlation is relatively simple this method would be a useful complement to Coulomb excitation for the study of nuclear shapes. Figure 2. Plots of the variation of w(1/2) with θ and P(90˚) with θ, where tan θ = R (the ratio of partial wave amplitudes) for the 351 keV decay in 207 Tl calculated with a γ-ray multipole mixing ratio of δ(E2/M1) = -0.271. The solid lines denote a fixed Coulomb phase difference of (σ 5 -σ 3 ) the dashed lines (σ 5 -σ 3 +π). The points indicate the experimentally measured linear polarization at 90˚and the best fit w(1/2), as calculated in the text.

(3. 8 )

 8 For the specific case of 207 Tl * decay, J 1 = 3/2 and consequently the angular correlation measurement must have a 4 = 0. This again simplifies the theoretical linear polarization formula; we start with * % & T ! . U < ! ? # % @ (3.9) However, from section 2 we can deduce that for a mixed E2/M1 transition F & . <! F * LMN % @

  two Compton scattering directions, since P(θ) = 0 when θ = 0°or 180°(where the reaction plane reduces to a straight line).

Figure 1 .

 1 Figure 1. Mathematically allowed w(1/2), w(3/2), w(5/2) for an aligned J 1 = 5/2 state mapped onto a 2 and a 4 assuming a subsequent pure E2 γ-decay to a J 2 = 9/2 state. The upper boundary lines are defined by the condition w(5/2) = 0, the lower boundary lines by the condition w(3/2) = 0 for any given w(1/2). The labelled points are determined by least squares fits to the data of, [point a] Hesselink [3] and [point b] Letessier et al [4] assuming aligned systems with population parameters as variables.