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Studies a propos {μ⁻p → \nu n ϒ / π → e \nu bar ϒ} experiments Beder, D. S. 1975

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TR IUMFSTUDIES A PROPOSjj p ■+ vny tt -> evyEXPERIMENTSD.S. Beder University of British ColumbiaMESON FACILITY OF:UNIVERSITY OF ALBERTA SIMON FRASER UNIVERSITY UNIVERSITY OF VICTORIA UNIVERSITY OF BRITISH COLUMBIATRI-75-1TR IUMFSTUDIES A PROPOSp p -*■ vny tt -*■ evyEXPERIMENTSD.S. Beder University of British ColumbiaPostal address:TRIUMFUniversity of British Columbia Vancouver, B.C.Canada V6T 1W5 December 1975Introduct ionThe purpose of this report is to examine the reactions1) y”p -* vny2) it -*■ evywith an aim to clarifying the feasibility and most interesting kinematic domains for possible experiments.The main interest in the y" radiative capture is the possibility of studying the 'induced pseudoscalar' weak interaction in a kinematic domain where it is relatively enhanced compared to other weak couplings. This will be seen in subsequent discussion to occur for the largest possible photon energies; our aim is to see to what extent the experimental sensitivity to this coupling varies with photon energy. We also shall explore the theore­tical uncertainties in constructing a gauge invariant amplitude for this process: an ambiguity will be elucidated which seems to have escaped con­sideration in the literature.For the radiative ST decay, interest has generally focused on determin­ing so-called 'structure dependent' vector and axial vector weak couplings which cannot be observed in the non-radiative decay. Here we wish to explore other possible 'aberrations of nature' to which experiment might be sensitive, particularly the presence of off-shell effects in the weak interaction and of tensor currents.- 2 -li p •> vny616 Ki nemat icsWe shall be concerned with the radiative decay rate for p-p -* vny at fixed photon (y) and neutron (n) angles, as a function of photon energy k. This rate we denote byd5rdfiyd^ndk ( 1 . 1)it will be calculated for the initial p having zero three momentum (in an atomic state).The 4-momenta of the n,y,v and initial state are denoted bymuon q mass pneutron p' mass M (energy E f, momentum p')photon kneutr i no KQ.+1P- PThe angle between the neutron and photon will be called 0p'k;and sine we denote by x and y, respectively.By making use of the fact that (suppressing Lorentz indices)K2 = (P-p'-k) 2 = 0we obtain, setting W = M + p,2kp'x = W2+M2-2Wk - 2E'(W-k).This can be solved for E':1(1 .2)(1 .3)E'(k,x) = (W-k)(W2+M2-2kW)2((W-k) 2 - k2x2)+ kx((W2-M2-2kW) 2 - 4M2k2y2} (1.4)One can show that for 0^  <  tt/2, only the - sign gives a valid solu­tion. However, for 0^  > tt/2 and for sufficiently large k, both solutions are valid. This is an important point! (The reader can easily see that this- 3 -is so by considering a special case, namely, 0p rk = ir, k large, K small.)Now that we have E'(k,x) we trivially obtain K by energy conserva­tion:K = W - E' - k. (1.5)All momenta are now fully specified; relevant angles are easily obtained using momentum conservation.To find the maximum k (= kmax) at given x, we maximise k as given by Eq. (1 .3) . After some tedious algebra, we obtain the result0p'k $ *Snax =v k > , / 1  ° - 6)For this latter value of k, the ( in Eq.(I .4) vanishes, and bothkinematic solutions are identical. In this case, we obtainr ,„ N M(W2+M2+2WMy)E (kmax’x) =  7— “ (1.7a)2WM + y(W2+M2)-xM(W2-M2)2WM + y(W2+M2) ’P  ^max,x  ^ “ T- “ “  77“ . (1.7b)(Physically, for 0p /k < TTUDI the photon energy is maximised when the neutronis at rest, with the y and v sharing the remaining energy = y.)Finally, it is useful to look at the relevant momentum transfers in this problem; we list the results for the various amplitudes for y emission:y emission Q2 = (p' - pp ro ton ) 2 = 2M(M-E ') < 0n,P emission Q2 = (p' - pproton + k ) 2E (k - q ) 2 = y(y-2k) £ 0 . (1.8)Thus the hadron emission occurs with a relatively enhanced induced pseudo­scalar coupling, since this coupling has a pole at Q2 = +m2.7T- A -1 • 2 Phase spaceIf the initial state of y_p (or pup in liquid H ) has wave function <f>(ry"rp)» then the decay rate is given byr = | MO) I2 d_3k d 3K d V  ) fj_ n , . ^(2ir)5 AyM . 8kKE' K ’ A f |T| (K9)spinsand T is the usual Lorentz invariant transition amplitude.Thus we obtaind5rdftk d£2p' dk unpol.I M P )  I2 k(p ( .  2EDSTA 5 32yM X i  I m 2spins( 1.10)whereX = KE '= KE'p ’ p '+kx——  +  ----E' KP '^7 - coseKp'= Kp' + E'(p'+kx)(1 .11a)(1 .lib) (1 -lie)= p f(W-k) + E'kx . (1 .lid)Straightforward algebra then reveals that (except for x=-l)when 6p ,k > ir/2M W * )  = 0 . (1.12a)This singularity of the k-spectrum no longer remains if we perform an ave­raging over x as does any finite experimental detector. We examine this now in more detail, by considering X (k^*o},x) for x near x . Clearly, as(, X )x _1> kmax increases; this has a corollary for ^ tt/2 :Phase space for k^*o) vanishes for x > xn .max (1 .12b)To proceed further, we consider3X9xk2xM2(1 -13)-  5 -This result is obtained by using kinematics Eq.(1•3) for p', phase space Eq.(l.lld) for X, and simply differentiating. It follows that at constant k,Thus for k = k(x°}maxX(x,k) = kM /x2+a2, a constant. (1.14)(1.15)We now consider a detector which measures (with equal weight) a range of x denoted by 2A, centred at x0 . The counting rate (for 100% efficiency)at energy k^xo* is then given by maxfX 0 - Ar . I * M I 2 g . 2,(2ir) 5 32pMI2 (p ' ) 2 TR*02kmax,xo(p ' ) 2 |T|2 dx Siax M ^ - X q 2EDTTA 5 32 pM2 /2x^2 ITx0-Adx/x-> (1.16)The integral is simply Z/R\ we used Eq.(1.12b) to eliminate half the range of integration. The 'averaged dr1 is then given byd5r R M o ) | 2 (p ' ) 2 XJdft dft dk 2ST • 2A (2it) 5 32yM2 UDPR O (1.17)which is finite for finite angular resolution.1.3 Matrix elementsFor capture from an atomic state 4>Crpp) » we have a matrix elementT ~  <j)(0) ^nvk|T| y(q=0), p(p=0)^ . (1.18)We now need to look at various contributions to the momentum space amplitude above. First we write the np weak current:weakj (a)np "  Un °y  1 ( (1.19)6y (Q) = YyFy(Q2) - pvEn2) + YyY59A(Q2) + Y5Qy9p(Q2)where (1.20)a) 61 2 0 is the left-handed helicity projection operatorb) are the isovector nucleon charge and magnetic moment formfactors with F^(0) = 1 F^(0) = 3-7c) gA (0) = 1.2d) gp(Q.2) is the induced pseudoscalar coupling constant, expected to be dominated by the TT pole. We take2MgA (0),(Q2) = o ~~o—  x  r.’P "  ' m 2 - Q 2e) Q, = p'-p and where r = 1 according to theory.The amplitude for y external emission is then (Fig. 1.1)( 1 .21 )whereTx = U(v) Yy (l+Y5)($"K+y)^ U(y)/2q«kweak Jnp (p,’p> ye = y polarization 4-vector 1a = 137G = 10"5/M2 .(1.22)The amplitudes for n,p emission are then (Figs. 1.2, 1.3):T2 (n) =  (Ly Un ^ J1 ^ 7  M p '+k- p )up)x I UM (p '+ k -p )2 =F. ) ( 2 Ly un Oy (p '+k -p )  I ^ ^  + ^ p j u p  ( ’ *23)-  7 -whereLy = U(v) Y y (1+Y 5) U(y), (1.24)and F are the anomalous moment form factors of n.p, with n,pFn (0) = -1.91 Fp (0) = +1.79.Note thatpyEt DA - p>Et 2) - F^(a2).E M  MWe shall assume that Fy ’ , Fn>p, gp are constant in the range of moment transfers occurring in this reaction. If we writeurnv A--- ^T = I Tj = /4ira —  e*J, i=1 /2then the divergence of this current is explicitly k-J = L■y e X E M XNM A yJ ^ ak(p'+k-P)(1.25)= Ly U(n)MfF!_X-,2M (ky-Yy^) + 9p ((p '"P)2)Ys(p'~p) ;- 9p((p '+k-p)2)Y 5 (p ,+k-p)p|u (p ) | . (1.26)We now consider the construction of a 'counter-current1 to be added to J, such that its divergence cancel.s the divergence of Eq. (1.26) . The total current-conserving amplitude desired will then beT = e*J + e*Jcounter (1.27)First, the weak magnetism counter-current gives a counter term47rarMn  —=.fa 2M ^  Un Yy/-£y(1.28)unambiguously (for Fv = constant). However, for the induced pseudoscalar counter term there is an important ambiguity not previously noted in the literature: we may take this counter term either asUnY 5Up[‘-  8 -  9p((Q+k)2) ~ gp (Q2)k‘Q Q = p'-pe*Q(Q+k)y + gp (Q2)ey Lp (1.29a)■ W p (te((Wt)i? ' 9p(Q2> e-Qtft,) ♦ gp ((Q+k)2)£y (1.29b)Both choices have the same divergence. The first is the conventional choice, resuiting from treating the induced pseudoscalar term as due to emission from an internal ir plus a contact termat the ir-lepton vertex. The second choice corresponds to interpreting the derivative coupling at the ir-lepton vertex as a derivative of the hadron fields, with the resulting contact term at the hadron vertex. The first approach is clearly implied by a microscope Lagrangian field theory. We leave this question open for now and explore it numerically later. We do not present trace expressions forI 11" I 2spinsas we prefer to simply let the computer do our Dirac matrix algebra in con­structing each independent spin amplitude.1.A Presentation of resultsWe shall present below our calculated d5a/dftdftdk rather than inte­grations w.r.t. variables £2^  or k, in the event that it might be experi­mentally useful to detect the final neutron in coincidence with the y in order to reduce experimental background noise. To avoid difficulties in the visual presentation, the kmax point in the spectrum (but only this point) is calculated for an angular resolution of2A = 2 sin0 A0 with AG = 0.05 rad (1.30)according to the prescription of Eq.(1.18).We shall separately present results for the cases in which the initial state is eithera) single, preferred in the y“p atom because of the hyperfine i nteraction;b) statistically averaged singlet, triplet, appropriate to a y"- 9 -bound to a nucleus described as a fermi gas of uncorrelated nucleons; orc) doublet p-y-p molecule.For case d) we use the molecular ground stateIf we do not observe the spectator proton for case c) [and this is almost always the case since it emerges with almost no KE], then the capture rate is the incoherent sum of the capture rate for each proton. We use previous estimates forOn a more trivial kinematic level one also needs to keep in mind the rela­tion between kmax and Sp^; since one prefers to look at high energy y to avoid background confusion with y -> evv y. From Eq. (1.6) we see thatWe shall therefore present our most detailed results onlyfor 0 p f ^  >164 deg.In Fig. 1.2 we indicate the various momentum transfers as a function of photon energy, at fixed In Fig. 1.3a, b, c we present the actualphoton spectra for Op/^ = 3 rad. Note the numerical sensitivity to the counter-term construction. The singlet and (p-y- p ) ^  results are|0^ =  | space; antisymm. in Tpj, ?p2^ © | s p i n ;  symm. in s\, (1.31)whereThus= *+++++)protons ' 2y“+ (++) protons * ^ y ' ^d3 r2 lz*(ry*rp1 = rp2) I = 0.505 x (y"p atom case). (1.32)The following questions seem a priori interesting to us:1) Are the results sensitive to how we construct the counter term,i.e. to the choice of (1.29a) or (1.29b).2) For what kinematic region is the spectrum maximally sensitiveto the induced-pseudoscalar coupling constant.kmax = 80 MeV when Opf^ = 16^ deg.- 10 -considerably smaller than the triplet case [incidentally, d5o for (p-y-p)^ can be shown to equal 1/4 d5a)triplet + 3/** c'5<7)singlet^ anc*’ as exPerience has proven to us, therefore sensitive to any details of the amplitude. We show calculations in which the constant r which specifies the strength of the induced pseudoscalar coupling gp is either 1 (as theoretically pre­dicted) or 2. Also, for the (p-y-p) case, we increased the strength of gp by taking2MgA9P " y2-Q2f2”gA = c)y? (1.33))5 ^irThis gives a result quite similar to merely doubling r, as shown in Fig. 1.3c. The main point to notice is: the (p-y-p)1y 2 and singlet ratesare indeed increasingly sensitive to gp as k increases.Our calculation differs from the earlier one by Opat in that we have chosen to look at a differential rate for fixed gamma-neutron opening angle. This choice was motivated by the possible desirability of detecting the neutron as well as the gamma, in order to decrease background noise experimentally. A typical anticipated counting rate is presented below; we assumea) (d5a) 'standard' = 10-3/(sec - 10 MeV - sr2).Thus, fraction of stopped y's radiatively decaying (per sr2 - 10 MeV) equalsIQ' 31/(2 .2 x lo-6)Next we assume= 2 .2 x 10-b) 10 y-detectors, each 10 x 10 cm at 2/3 m from the target, i.e.(Afl)y = 9/40 src) Neutron detector (large enough to encompass much of the cone of neutrons associated with high energy y's) has (Afi)n =0.1 srd) 100% efficient detectorse) 10 MeV 'bin' for y'sf) 107 stopped y's per secondThis gives R = 43 events/day.Thus, for 70-90 MeV y's (i.e. 20 MeV bin), according to Fig. 1.3c for r(gp) = 1, R ** 40 events/day.11y~p vny BibliographyThe most recent y“p -> vny calculation that the author knows of isGeoffrey I. Opat, Phys. Rev. Vik_, Bk28 (196 k).The present work overlaps very much with this calculation but hope­fully presents some material more explicitly useful to experimental design est imates.AcknowledqementsI thank Dr. B.M.K. Nefkens (UCLA), J-M Poutissou and M. Hasinoff (UBC) for arousing my interest in these radiative processes; I also thank Dr. D. Smith (UBC) for assistance in checking the (y-p ->■ ...) calculations.a) b) c)Fig. 1.1. External emission diagrams.- 12MOMENTUMFig. 1.2Momentum transfers vs photon energy.a)Fig. 1.2Photon spectra. In these figures solid lines refer to 1dhe conventional counter term (Eq. 1.29a) while dotted lines refer to the alternative (Eq. 1.29b).In c) the crosses are for the conventional counter term, but with gp as described in Eq. 1.33.-  14 -2 . tt evy (at rest)2.1 Ki nemat i csWe denote the various 4-momenta byir,e,v,Y *-* q ,p,K,k .We also denote by 0 the angle between the electron and photon momenta. Fol­lowing convention, we define x and y in terms of 3-momentum magnitude:x = —  (2 .1a)%y = —  (2.1b)mTTwhere m^ is the ir mass. From 4-momentum conservation, a third variableX = sin2 (0/2) (2.1c)is easily related to x,y, ignoring the electron mass me :Xxy = x + y - 1 . (2.2)It is then easily shown thatm2- 0 ,  (2.3a)(2.3b)22.2 Phase spaceWe might be interested in several differential decay rates. We shall always use p as the z-axis. Then (ignoring me):mTTk-p = -JL (x +2m2k-K = -E- (1 -r =It follows thatf d3k d3K d3p L . ~ . 12 i—  —  6^ (k+K+p-q) I | T | 2 —lom-jjkKp spins .DTT(2.4)}) dr _ p(m1T-2p) IlT ] 2dfie dfiy dp 32 (m^ - p(l-cos0 ) ) 2 32tt52) If we measure electron and photon momenta, then- 1 5 -dr = TT2m7r y IT I2 1dx dy 8 32tt5 ’The fact that the weight multiplying [T |2 is constant is well known to users of Dalitz plots; our x,y variables are just the energy variables used in these plots.Recall our discussion of double kinematic solutions in radiative y decay (y“p -> vny):The double-valued solutions to kinematics for fixed 0 and k only apply for k > (m2 - here. Thus only for k within of kmaxdo we need worry about these; in practice this is of no consequence evidently.2.3 Matrix elementsWe start with the irev interaction Lagrangian in momentum space^ w e a k  " F U(e)rf(l+y5) V(v) , (2.7)1 -1 -1 1 in 2 x 2 notation, as the pro-where our convention gives (I+Y5) = jection operator for right handed \T.It follows that the sum of electron emission, pion emission and contact term due to the ir-momentum dependence in Eq.(2.7) gives, after us­ing the Dirac equation to simplify the diagrams of Fig. 2.1,( A O - 1 T = U ( p )e-p /Xje + m_-----+ m. q*e - m„— -  - £ q*kek*p e2k*p electron emission it emission contact( 1+Y5 ) V ( K )  F= U f— - + —  - fl+Yslv Fme in lab frame. (2.8)lk -p  2k -p J l  IbJ eNote that this amplitude^me , in agreement with the present experimental observation. Any attempt to delete internal emission (contact) would give an amplitude of order instead of me , in gross disagreement with experi­ment.The vector internal emission of Fig. 2.2 contributes a gauge invari­ant ampli tude-  16 -( A ™ ) ' 1 TVector " u Yy (l+Ys)V 9pV -p 2 evaax ea.ko<*T 9V <2 -9)where P = q - k and we have pretended that a I- boson contributes. For very large boson mass M we write this as(A ira) 1 Tvector_  gv FA (1+Ys)Vfrom which = epa0T eakaqT . (2 .10)By using the CVC hypothesis the ryv coupling is related to TTBd d r and one estimates that9v = / rTT0/TTI%.am^2m^ g2 21 x (8F2m2).Similarly, an axial internal emission amplitude can be considered as due to an axial vector boson as in Fig. 2.3. This contributes(AirctJ _1 Taxial = gA idM ESsd-Ao ea fgctu - (QgQu)/^2) (2. 11)Q2 - M2plus a counter term for gauge invariance; the net result for large M is(Aim } ’ 1 Taxia] = g A U (A-q - Xe*q)(l+Y5)v (2 .12)with cf^  model dependent, a priori unknown, and of order (me).Without changing the accepted currents we can still speculate about possible off-shell weak interactions, which we write as^ o f f  = F U (X-me)q Ys V (2.13a)axial m,,r^ o f f  = F — f- U ( A m e) V. (2.13b)m -j.scalar 1These contribute (together with associated counter terms) to radiative decay( A ^ ) - 1 T ' x , i = F ^ U # Y 5 »  (2.14a)IMirTgcalar= ® c*ue to current conservation. (2 .lAb)- 17 -Finally, we speculate about an on-shell tensor weak interaction, which can­not be manifested in non-radiative )5 decay: it contributes a radiativeampli tude( A W T '  ^ t e n s o r  -  ^  U l+ Y s )  VirFern& U A(l+Y5) V • (2.15)2.4 SummaryNext we tabulate the rates ensuing from these matrix elements and their likely significant interferences only. In Table I we showf-n2 m^ m2 F2 »47Tg32tt-■i v drspins dx~dy^ <2 *16)Note that our expression (m^ gA)/(2me f) is usually called y, the ratio of axial to vector (internal) couplings. The expressions in Table II are obtained from amplitudes (1.8), (1.10), (1.12), (1.14a), (1.15) by standard trace algebra.Note thatso that_ %  m 2 F2TT+ev ,fTT- dr ] = —  r x Tab,e 1dx dV J external 2tt 7r+'>ev entryi.e. in Eq. (1.16),( ) - rTr+-*-ev "2tt- 18 -Table INature of Term1) External2) Off-shell axial3) Interference of 1,24) Vector (internal)(r —  21 from CVC)5) 1,4 Interference6) Axial (internal)7) 1,6 Interference8) A , 6 interference9) Tensor10) 1,9 interference2^2 (1_y)(x+y-l)b(l-y)/xr(l-x)((x+y-l) 2 + (1-y)2)1 —x (x+y-l) 2 + (1-y) 2 '9Am|'2meF(l-y)x2Fm(ejmu9A'2m*»F(x+y-l)/r • x(l-x)(x-2+2y)'(x+y-l)(1-x) T <DECM1 x2 2j . 1% .(l/2)c2 (x+y-l)(1-y) : looks like 2 c (1-y)x : looks 1ike 3To maximize sensitivity to b (or c) type terms (i.e. terms 2, 3, 9, 10) it is preferable to look at large X, which also minimizes the large internal vector term k. Small y then maximizes sensitivity to these terms (6).In Table II we indicate the evaluation of external, vector internal and b terms of Table I, for judiciously selected (x,y). We omit evaluation of axial terms for now in view of the current experimental indications that these are small (Stetz et at.).- 19 -Table IIX II o Co y = 0.9 0.7 0.5 0.3Externa 1 0.156 0.625 1.55 4.4Vector internal 1.36 0.9** 0.84 1.1b term (b=l) 0.11 0.33 0.55 0.8b2 term (b=l) 0.04 0.09 0.2 0.07X = 0.8 V = 0.9 0.7 0.5 0.3External 0.23 1 .0 2.7 11.4Vector internal 2.1 1.47 1.43 2.1b term (b=l) 0.22 0.375 0.65 0.875b2 term (b=l) 0.035 0.075 0.075 0.075We see that large X, y~>0.5 maximizes sensitivity to terms such as off- shell axial, or (on-shell) tensor.- 20 -nevy Bibliography (with thanks to J-M Poutissou, University of BritishColumbia)The following papers of 'Group 1' are in error in the calculation of the CVC estimate for internal vector emission, being too small by /I this amplitude; 'Group 2' is correct in this respect.Group 1 :1. V.G. Vaks and B.L. Ioffe, Nuovo Cimento X^, 342 (1958)2. V.F. Muller, Zeitschrift fur Physlk 173, 438 (1963)3. F. Scheck and A. Wu11schleger, Nucl. Phys. B67, 504 (1973)Group 2 :1. S. Bludman and J. Young, Phys. Rev. 118, 602 (1960)2. D.E. Neville, Phys. Rev. JZ4, 2037 (1961)Recent results:A.W. Stetz et oil., Phys. Rev. Letters 33_, 1455 (1974)Fig. 2.2. Internal ’vector’Fig. 2.1 Fig. 2.3. Internal 'axial'- 21 -3. Conclusi ons y~p -> vny1) The 'counter current' necessary to maintain current conservation is not uniquely prescribed; numerically this ambiguity is of the order of 20% at large photon energies.2) Ignoring the above feature, one sees that photon spectra are most sensitive to gp at large k (>70 MeV, say) so that an experiment concen­trating on this kinematic domain would be most welcome.3) Estimated counting rates indicate that this experiment is only marginally feasible with even the full anticipated flux of stopped y's at TRIUMF. Further work to be reported elsewhere indicates a much larger cross-section for y 3He -* 3Hvy, which appears to therefore be a better candidate for experimental work.TT -» evyTo enhance contributions of tensor currents (or off-shel1 axial interaction) one should look for high-energy photons and medium-energy electrons. This contrasts slightly with the preferred domain for sensitivity to internal axial couplings, namely high-energy electrons and photons.BEST-PRINTER CO. LTD VANCOUVER. B.C.mV - i  f ' a .

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