Physics with beams of virtual pions

Nuclear Physics A577 (1994) 471c--478c North-Holland, Amsterdam

NUCLEAR PHYSICS A

Physics with beams of virtual plons T E O Ericson CERN, CH-1211 Geneva 23, Switzerland An alternative apploach to spln-lsospln excitations in various charge exchange reactions is that the projectile is equivalent to a beam of virtual pmns Although equivalent to the standard approaches this view emphasizes the role of the plon and how it is generated inside the nuclear by Its source It clarifies why virtual plons are so much more effective in the exploration of plon modes in nuclei and it gives n n a t u r a l explanation of coherent plon production as the pmmc counterpart to the transition radiation of charged particles in mhomogeneous dielectrics

There is a very close connection of the physics of spm-mospln excitations to that of nuclei1 pion physics This is apparent already from the famous Goldberger-Tleiman relation, which relates the axial couphng constant ga for the spln-lsospm operator of the nucleon to the ~rNN couphng constant to a plecislon of a few per cent[l] This connection m nuclei leads naturally to a renormahzatxon and quenching of the axial coupling lnslde of nuclei as was initially pointed out by Magda Ericson an an approach that now as called the A mechanism for quenching, one of the major ways by which the renormahzation occurs [2][3] The understanding of this connection for axial transitions both at low and at high excitations have been much refined since Once more much of this is due to the Lyon g~oup which has pioneered this A major reason for this close lank is now understood as being to the chlral symmetry constraints, which enter m term of the partially conserved axial current (PCAC) relations m nuclei[4] Once PCAC is assumed to be a locally exact aelatlon, it follows that the axial current, l e the spm-mospm physics, is given by the corresponding matrix elements for a pion even with meson exchange currents included[5] It is therefore not smprlsing that the longitudinal spln-lsospm response function is a basic tool for exploring the piomc n a t u r e of the nucleus and that it should be sensitive to a piomc mode m the nucleus All of this is incorporated m the standard approaches to tins ploblem, which rely heavily on the A hole model for the theoretical descllptlon of the response[6] and this approach is now routinely used by many authols In this sense we know well that spln-lsospm response is paon physics The connection to plons also concerns the probes we use m the forward chalge exchange reactions Just consider the situatmn for the prototype np--*pn charge exchange at small angles If we normalize the 0 ° cross section (which as nearly constant an the lab system anyway m a huge enelgy range from a few h u n d l e d MeV up to about 10 GeV), the cross sections have a universal forward peaked shape with m o m e n t u m transfer to percent accuracy, to such an extent that the 0375-9474/94/$07 00 © 1994 - Elsevier Science B V All rights reserved SSDI 0375-9474(94)00414-5

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T E 0 Ertcson/ Phystcs wtth beams of vtrtual ptons

I n f o r m a t i o n from widely different energies undastlngmshable[7] T h e scale is t h a t of the plon mass T h u s , also the c o m m o n probes are strongly linked to plon physics This is all known Here I ask a n o t h e r question and look at the s a m e physacs in a dafferent p e r s p e c t i v e Is t h e r e an 'easy' way to q u a h t a t l v e l y visuahze the physics in such processes 9 C a n we readily see t h a t spln-Isospin excitataons wath different charge e x c h a n g e probes probes will have u m v e r s a l features and in particular, can one see readily t h a t new features lake the recently observed coherent plon p r o d u c t i o n at Saturne[8] ? This a p p r o a c h is w h a t I refer to as ' t h e vartual plon b e a m ' , since it has physical features which we do not m e e t foi physical plons[9] T h e s e ale hidden f e a t m e s m t h e usual approach, which as m o r e suitable for q u a n t i t a t i v e l n v e s t l g a t m n s A plane wave b e a m e 'qz of physacal pmns has t h e energym o m e n t u m relataon w2 _ q2 = rn~2 O n e can c o n s t r u c t a plane wave b e a m of vartual paons t r o m the pmn cloud of a projectale, which vartually dissociates, such as p ~ pTc+, 3 H e ___,3 Hzc+ or # - --~ uTr- Such processes are d o m i n a n t an the long Lange m e s o n cloud, since the paon is hght Now, the e n e r g y - m o m e n t u m relataon daffers flora t h a t fol free plons ~ 2 q s ~ m rs T h e role of the p l o j e c t l l e is sunply t h a t of a source of the ~artual paon field, whach c a n n o t exist m its absence the v~rtual pmn 'p~ggybacks' on t h e p r o j e c t d e FoL the sake of concreteness consider the nucleon pion cloud an the Breit s y s t e m T h e plon is e m i t t e d at no cost m energy but with a m o m e n t u m k We are only concerned with the case t h a t t h e plon is e m i t t e d along the z-axas, since the transverse m o m e n t u m transfeL vanishes for 0 ° l e a c t l o n s T h e rest f l a m e e n e r g y - m o m e n t u m transfeL is w = 0, q = Let us now b o o s t the nucleon into tile l a b o r a t o r y h a m e by g w l n g it a ~elocaty 3 m the z-dHectaon T h e e n e r g y - m o m e n t u m loss m the charge e x c h a n g e r e a c t m n ~, q as now t h e vattual paon e n e r g y - m o m e n t u m w = LXE = 7 3 k , q = 7k, where 7 = (1 - 32) -L/~ T h e value foL k follows f r o m the relation 1,s = qS _ ~2 = w2/72 In practace k as r a t h e r small For energy losses c o r l e s p o n d m g to tile regaon of the ~h isobar e x c i t a t i o n k lS a b o u t 200 M e V / c for a 600 MeV incident nucleon One notes t h a t for c o n s t a n t energy transfer the Lest frame m o m e n t u m transfer k ~ 0 with increasing p r o j e c t i l e energy It is therefore a generic f e a t u r e of charge e x c h a n g e reactions t h a t any r e d u c t i o n an the cross sectmns due to t h e p r o j e c t d e f o r m factor dimmashes and becomes increasingly lrrevant as t h e projectile energy increases W~th the help of a pLojectde as a source at as therefore in principle s t r a i g h t f o r w a r d to c o n s t r u c t a plane wave b e a m of vHtual plons T h e r e are of course some p r o b l e m s T h e p r o j e c t i l e wall m general not be a plane wave It lS usually a strongly i n t e r a c t i n g partacle and ~t wall undergo a b s o r p t i o n by t h e t a r g e t as well as dlstorslons, whach m u s t be included an the description Let u~ for t h e m o m e n t neglect thl~ p o i n t and c o n c e n t r a t e on t h e specml characteL of a b e a m of viLtual plons Its c h a r a c t e r i s t i c feature is t h a t ~t c a n n o t exist unless ~t is constantl~ r e p l e m s h e d by a source In order to have an mcadent wave ¢,=~(z,w) = exp(~qz) t h e free pion wave e q u a t m n must have a source t e r m ( V s - m ~2 + w 2 ) ¢ ( z , w ) = (--q2 - m .2 + w2)exp(zqz)

(1)

w i t h (q2 + m ] - w ~) = (k s + m~) As we are used to, t h e raght h a n d s~de of the e q u a t i o n vanishes, w h e n t h e paon ~s a physacal one This simply m e a n s t h a t a physacal pmn needs not to be sustained by a source m order to p r o p a g a t e lake a free wave Let us n e x t consader w h a t h a p p e n s In an infimte nuclear m e d m m We can t h e n describe t h e p r o p a g a t i o n of a

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coherent plon wave in the medium in terms of a uniform optical potential U(q 2) dependent on the incident wave n u m b e r q Since we know that various processes attenuate the coherent wave, we will assume the potential to have an absorptive part, which corresponds to the physics We then solve the inhomogeneous wave equation

( v 2 - m2 + J

+ 2,.U(q))¢(z,03) = ( - q ~ - m~ + w~)exp(~qz),

(2)

which has the solution

¢(z,03)

q2

+

m ~2 - - 032

(3)

q2 + rn~ - w 2 -- 203U(q2) ezp(zqz)

In addition there wdl solutions to the dispersion equation with wave number K corresponding to the spontaneously propagating waves Since these are damped by absorption, we neglect them fol the moment Instead, there is now a wave with the znczdent wave number q generated throughout the medium This wave exmts only for virtual plons For real plons it is suppressed by the factor (q2 -4- m .2 - 032) = 0, whmh is the famous Ewald extinction of an incident plane wave within a medium (named after Hans Bethe's father-m-law) Because of the generated wave there now is an absorption probability per umt volume P

P = 2ImU(q)

I

qZ + 2 _ 032 m,~ q2 + m~ - w2 _ 203U(q))] 12

(4)

TMs means that one can immediately deduce the the reaction rate in the medium Since the absorption processes of the coherent wave of virtual plon is mainly due to either quasi-elastic scattering on nucleons or to real absorption which is a rather short ranged phenomenon for physical pions We have then excellent reasons to expect that the reaction products will be ~ery simdar to those for physical pions as long as the energy transfer stays comparable Since the denominator will be small ff there is a plon mode which appioxlmately fulfills the dispersion lelat,on m the medium we can also conclude at once that there wdl be a volume response of the medium to the v,rtual pion beam This is , m p o i t a n t , since the strong absorption of phys,cal plons in the A region makes them unsuitable for the exploration of the nuclei1 interior Here we can by-pass that problem Coherent productwn of physical pmns occurs in the interface regmn between the medium and flee space Cons,tier the v n t u a l pmn wave propagating through an mfimte half-plane to the left, emerg,ng into free space to the r,ght We recall once more that the medium is an absorptive one We must now take into account the reflectmn of the wave at the surface on the one hand and the fact that real pions will appear in the free region to the right Matching of the boundary conditmns for the problem (to keep notation short, we will not d,scuss the case of a m o m e n t u m dependent potentials but this is readily incorporated)

q2 + m~ 032 + K q(03) e_,t~ ) q2 + m2 _ 032 _ 203U(q) (e'q~ q + q(03) -

¢o~,(z,~) = e ' ~

K-

q@)e,~(=) L

K-q

-

(,5) (6)

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T E 0 Ericson / Physics with beams of virtual ptons

where K is the wave number m the m e d m m and q(w) is the vacuum wave n u m b e r at energy of a phymcal plon In addition to the wave of virtual plons there is now an additional wave of physical plons with wave number q(w) Thus, mmdarly to the electllC transition radiation generated when a charged partmle traverses the boundary between two media of different refractive indices, a coherent plon transition radiation is generated in the present case and the physical reason is basically the same one The missing m o m e n t u m is furmshed by the boundary, l e , it is furmshed by the nucleus and not by the plon m o m e n t u m distribution, such that it does not reflect m the energy-momentum loss of the projectile carrying the virtual pion beam Since the virtual beam is physically distinguished from the generated wave, we note that this wave has the a m p l i t u d e - ( I f - q ( w ) ) / ( K q) This means that m the limit of a physical pion the amphtude is -1, i e , there is then complete shadowing The emerging beam m the present case will instead correspond to an emission with the same amplitude but the diffractive effects are closely related accoldmg to Babinet's plmclple an absolblng disk and an emitting disk produce the same diffraction pattern We also note from the expression above that the amplitude for emission varies rathe1 weakly with off-shell nature of the scattering as is also found in detailed calculations[10] Suppose now that thele is a mode for the pion m the medium corresponding to the solution of the plon dmpermon relation, l e , that the energy-momentum transfer corresponds to that of a plon propagating spontaneously m the m e d m m Such a collective pion mode m the m e d m m was predicted already some time ago[6], it has been clearly realized that it is unsuitable to investigate such a mode using physical plons but that it would show up m investigations of lesponse functions using probes such as neutrinos, charge-exchange leactlons etc We may visualize this mode as the solution of the m o m e n t u m - d e p e n d e n t optical potential dlspelsion lelatlon, but the concept is far wider Tile wave number 1s in general a complex one K = Kr + zK,, since the wave will be attenuated From Eq (4) it follows that the unph:~ sical pion will have the interaction probability enhanced by a factor [ (q2 _ q(w)2)/(q2 _ i(2) [2 This plomc mode will show up as a peak m the energy loss s p e c t l u m of the projectile which genelates the v n t u a l paon beam Such a peak has indeed been observed [11],[12] and been interpreted in this fash~on[13],[14] It is clear in the approach we have used here, that the wave function is generated within the medium with an enhancement factor such that the unphysxcal plon in general will be absorbed or scattered This means that not only the enelgy loss spectrum, but also the spectrum of those reaction products whmh escape out of the nucleus should be sensitive to the collective mode and show a corresponding enhancement We may then ask whether the coherently ploduced pions will also will be enhanced by the mode The answer to this question is atfilmatlve \¥e have already seen that physical pions in scattering processes do not probe the pmn mode due to the mismatch in energy-momentum For coherently produced pions in our s~mple model the coherent folwmd amplitude is - ( K - q ( w ) ) / ( K - q) This amplitude is stlongly enhanced as the value of q satisfies the condition for the pionlc mode e~en though the physical plon itself cannot fulfill the resonant e o n d m o n The reason is that the enhancement of the wave in the medium survives the mismatch at the surface This indicates that the variation of the for~ard cross section for coherent pions can be used as

T E 0 Ertcson/ Phys,cs with beams of wrtual ptons

475c

an indicator for the plon mode Indeed, detailed calculations show such an enhancement [10] Such calculations must of course incorporate the attenuation and the dlstorslon of the incident beam in the nucleus, 1 e , of the plon source The physics of a virtual pion beam stands out particularly clearly in the case of forward neutrino reactions Consider a neutrino induced reaction u + N ~ l + X, where N is a nuclear or a hadromc target, 1 is a lepton (e or #) and X is the specific final reaction products in the leactlon, which can be anything from individual nuclear states, complicated emissions of nucleons or states with plons whether coherently of incoherently produced Any one of these specific processes can be related to the corresponding reactions 7r(q,w) + N --+ X induced by a virtual plane plon wave carrying the same energy and m o m e n t u m as the energy-momentum transfer More precisely Adler's theorem[15] states that the weak interaction matrix element M for such a forward n e u t n n o leaction is exactly proportional to the corresponding T-matrix element for the virtual pion-mduced reaction

M cx G W T ( r ( q ) + N ~ X),

(7)

q0 where Gw ~ 10-5M -2 is the usual weak interaction couphng constant Only weak and general assumptions go into this theorem It assumes on the one hand that the divergence of the axial current is proportional to the pxon field (PCAC)[16], which is natural an chiral models also in nuclear physics, and on the other hand that the lepton mass is negligible, an approximation that is increasingly accurate with increasing neutrino energy It is therefore evident that 0 ° lepton charge exchange is exactly equivalent to the ph)slcs of a v n t u M paon beam with w = q In this sense these neutrino reactions ale the best possible example of how such virtual beams arise the neutrino can be viewed as virtually decomposing into a p m n and lepton and at is this plon that is responsible for the virtual plon beam The trouble with the neutrino reactmns is the very small cross sections on the one hand and the poor resolution on the other one which presently make the neutrino a very exotic tool for exploring the physics of virtual plons This situation may change in the future with the advent of kaon factories There are still several aspects of the situation tol neutrinos that merit further leflectlon First, there as the surprising situation that neutrino reactions can be proportional to the physics of a strongly Interacting particle, which m the physical region has a cross section proportional to A 2/3, while neutrinos fleely penetrate even a heavy nucleus and has interactions proportional to the nuclear volume or A The explanation was given by Bell[18] As we have seen above the virtual beam generates a non-vanishing wave throughout a large uniform medium and this wave ploduces pmme processes proportional to the volume This means that there is no conflict between weakness of neutrino interactions and their plonlc nature The second interesting statement of the theorem is that the neutrino processes at small energy transfers are also described exphcltly by a plon off-the-mass-shell amplitude and that therefore those reactions are pure pion physics For small space-momentum transfers an such reactions the allowed impulse approximation is a good one The axial transltlon interaction is < 0 I M(+'-) I n >oc ~,,pto= < 0 I A(+'-) [ n >~- cq,~,o~ <

01 ~,~,(+'-) I n >,

(8)

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T E 0 Ertcson / Phystcs with beams of vwtual ptons

l e , the nuclear tranmtlon is gwen by the unretarded Gamow-Teller opelator It 1s of course well-known that the axial transition operator m nuclei is intimately hnked to the nuclear plon source function via the hypothesis of axial locality[5], but we see here this statement reahzed very transparently in a special case Whde the Adler theorem and its physics apphes to hlgh energy neutrino reactions, a somewhat reminiscent situation is associated with the physics of the nuclear capture of stopped muons from the ls orblt of a muomc atom This is the muomc counterpart of the K-capture of an electron, although for muons the energy avadable is 106 MeV The muomc wave function is determined by the extended Coulomb potential In practice that means that it is nearly constant over the nucleus, whmh is the only point of relevance for the present dascusslon The muon lifetime an thas state as long and at as m a heavy nucleus mostly determined by the absorptmn by the weak lnteractlon processes These are dormnated by capture due to the a×aal current coupling, strongly related to pmn physics and the excatataon of nuclear Gamow-Tellel states as has been known for a long tame The muon being a weakly interacting paltlcle penetrates freely throughout the nucleus and ats wave function as not dastmted by strong mteractaons However, the muon surrounds atself for a small ftactaon of the tame by a cloud of virtual plons for which at acts as a source We can vaew thas approxamately as produced by the virtual decay of the muon # - ---* uTr-, where the paon has an energ:y w corresponding to the energy transfer to the final nucleus Since the paon wave tunctmn as now generated by the constant muon wave functmn as a source wathout any attenuation, we have here a particularly clear case of offthe-mass shell physics The situation lesembles the previous one fol neutrinos, but there are some important daffelences In order to be precase we recall that the muon capture per u m t energy m the hmit of vamshmg neutrino energy is proportaonal to the amagmary part of the off-mass-shell r-nuclear scattering length a.(w) and scattering volume ap(w) The s wave c a p t m e as associated w~th the tame component of the axial culrent, the p wave capture wath its space component At thas partmular kmematmal point the free paon wave number as amagmaty wath t~ = (m 2 -w2)½ and w = mu and there as no external space m o m e n t u m transfer As before the lncadent wave generated an the off-mass-shell scattering as nolmalazed as exp0 q r), a e , to umty an the present case Consider first the case of a point nucleus Therefore m this case the value for the off-shell s wave scattering length Is a~ as(a~) -- 1 + t~a~'

Imam(w) -

Imas I 1 + ~a~ ]2,

(9)

which as a well known lesult, when the finite saze of the nucleus can be neglected Let us no~ see what happens for a large nucleus, the anteraor of whach can be treated as an infimte nuclear medmm[19] Although it is not really essential, I wall for specaficlty and t l a n s p a r e n c y of argument assume that the ~r-nucleus lnteractmn is sufficmntly well described an terms of a standard low energy optacal potentml of the schematm type 2~U=-V

XV+Q

(10)

wath the corresponding wave equatmn including the source term V

(1 - ~)V¢-4- O¢ + (w 2 - m~)¢ = J

- ,.~

(11)

T E 0 Ericson / Physics with beams of virtual ptons

477c

The basis for this potential is that the interaction In the medmm, whether for elastic scattering or for absorptive processes, are local s and p wave processes and the absorption is described by the imaginary part of X and Q As long as the unphyslcal plon has an energy w not to far below the physical plon mass and for m o m e n t a that are not large this description IS a good approximation We recall that the local interaction Q is repulsive, while the velocity dependent one m ~( is attractive In nuclear matter the generated wave function IS a constant Since V ( c o n s t a n t ) = 0, the generated wave is

¢o

022 - -

2

- -

m~

(12)

w 2 - m~ -t- Q

The remarkable restilt is therefore that the m o m e n t u m - d e p e n d e n t terms are irrelevant and that m this case there is no local p wave interaction in the m e d m m The local absorptmn late in the medium per unit volume is

P = 2 I m U ( q 2 = O) l ¢o 12- I m Q ] w

w~ -

m~ + Q

The corresponding discussion for the p wave scattering volume a v is immediate and follows nearly identical reasoning From these expressions we see at once that the Born approximation holds when the interaction Q is small compared to the characteristic squared w a v e n u m b e r m~r - w 2 The reason is that then the external driving force imposes itself completely The condition for the constant wave m the medium IS that the characteristic distance of the wa~e number in the medium should be smaller than the radius Thin means that the range of the Yukawa field about a point source in the medium should stay reside the medium \Ve therefore see clearly in the situation that the energy transfer to the nucleus is maximal that the problem of the muon capture in a large nucleus is exactly described by the strong interaction physics of off-mass-shell pIons What about #-capture into excited nuclear states In the first 10-20 MeV above the ground state ? This is the region to which the bulk of the capture takes place and the m o m e n t u m transfer by the neutrino is typically of the order of 100 MeV The reaction is dominated by the axial current capture The de~cuptmn of these processes is m a sense 'trivial' m so far that an impulse approximation on the individual nucleons is an excellent first approximation The axial current is therefore described by the s t a n d a l d spin-isospin operator with m o m e n t u m transfer and the predominant excitations are the spln-isospln analogs of the giant dipole resonance On the other hand, as for n e u t n n o reactions, the axial current can also here be linked dnectly to the plon source function The impulse approximation is just the approximation to viitual pion physics of rely low energy ~: in which the individual nucleons have the usual a V coupling The more sophisticated description in terms of axial locality to this situation generates naturally higher oMer many body terms to the nuclear axml current due to plon propagation in the nuclear environment[20] These are the terms usually lefened to as meson exchange currents, which become predominant in situations of large enelg~ transfer It is a generic feature of forward charge exchange reactions that their physics can be viewed as produced by a beam of v u t u a l pions The characteristic feature of such 'unphysical' plons is that a pion wave is genelated by the projectile also inside the nuclear

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T E 0 Ericson / Physics wuh beams of virtual plons

medium This IS a region that is poorly accessible to physical plons for large nuclei Contrary to physical plons virtual plons are sensitive to the nuclear collective paon mode both via the energy loss spectrum of the projectile and via the yield of reaction products These will be enhanced when energy-momentum condition of the plomc dispersion equation in the medium is fulfilled Coherently produced physical plons are a natural feature of such vntual plon beams They are generated by a mechamsm slmalar to the one which gives llse to the transition radiation which occurs when a charged particle tlaverses the boundary between two media of different refractive index In the same spirit, it is unnatural to treat the nuclear spln-lsOspln states excited by forward charge exchange in a different perspective fiom the higher excitations an the A region with the paon collective state and pion production as well as the associated disintegration phenomena The natural way of viewing all these exchange reactions, whether for nucleons, neutrinos or heavy ions, is in terms ot pion physics The special approaches in the literature should be viewed as approximations to this general picture In particular, while one can defend the use of DWBA in the analysis of of nuclear spln-asospm excitations, such an approach obscures the beauty and the generality of the physics The physics of this situation stands out with special clarity foi the neutI1no induced reactions Consequently, such reactions are of great fundamental interest in spite of the intrinsic experimental difficulties REFERENCES

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

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