Radiative corrections to β decay and the possibility of a fourth...

VoLUME 56, NUMaER 1 PHYSICAL REVIEW LETTERS 6 JAGUAR+ 1986

Radiative Corrections to P Decay and the Possibility of a Fourth Generation

~. J. Marciano

Department ofPhysics, Brookhaven Ãational Laboratory, Upton, /Vew York II973

and

A. Sirlin

Department of Physics, /V'ew York University, /Vew York, /Vew York I0003(Received 1 October 1985)

Leading-logarithmic radiative corrections to P decay are summed via the renormalization groupand structure-dependent O(n) effects are estimated by a form-factor analysis. These refinementsreduce the Kobayashi-Maskawa quark-mixing-matrix element I Vzl by 0.13% to 0.9729+0.0012.Combined with IC,3-, hyperon-, and bde-cay constraints, this implies I v~l'+ I v I'+ v„, l

=0.9954+0.0025. Although consistent ~ith unitarity at the 2o. level, our result leaves open thepossibility of a fourth fermion generation which at 90/0 confidence level may have mixing I V I as

large as 0.088.

f

~~ ~~ ~ub

VdV, VbS~~ ~is ~~b

' '

(2)

That would constitute a clear violation of unitaritywhich, for example, requires in the three-generationcase

I v~1'+ I v I'+I v~ I' =1. (3)

PACS numbers: 13.40.Ks, 12.15.Ff, 23.40.8~

The comparison of 0+ 0+ superallowed Fermi Ptransitions and muon decay has played an importantrole in the development of the standard model ofstrong and electroweak interactions. The conserved-vector-current hypothesis, current algebra, Cabibbouniversality, and a host of other theoretical advanceshave their roots in such studies. More recently, it hasbeen shown' that experimental measurements of & tvalues in superallowed Fermi P decays, when com-bined with muon and K,3 (or hyperon) decay rates,provide a precision test of the standard SU (2) L8 U(1) model at the level of its quantum corrections.

Indeed, neglecting radiative corrections, one finds

I v~l'+I

V I'=1.032

(without radiative corrections) for the first two ele-ments of the Kobayashi-Maskawa2 quark mixing ma-trix

Fortunately, the O(a) radiative corrections which arefinite and calculable in the SU(2)L 8 U(1) modelhave the right sign and magnitude to render I

V~lz+ I V I & 1 and restore unitarity. ' This quantum-loop triumph for the standard model is, in a sense, anelectroweak counterpart to the many O(a) successesof quantum electrodynamics (g —2, hyperfine split-

tings, etc.).In this Letter we wish to refine still further the cal-

culation of radiative corrections to P decay. Our goalis to push this O(n) test of the standard model to thelevel of a few tenths of a percent. Then, scrutinizingthe three-generation unitarity constraint in Eq. (3), wewill be able to place restrictions on new physics such asfourth-generation mixing or glean a signal of its pres-ence. To that end, we will employ a recent detailedanalysis of K,3 and hyperon decays by Leutwyler andRoos3 that found

Iv I=0.221+0.002 (4)

and combined measurements of the b-quark lifetimeand bounds on I'(b u)/1 (b c) which imply3 4

I v„, I & o.oo8. (5)We begin by summarizing the results of Ref. l.

There it was shown that after normalization of P-decayamplitudes in terms of G„=1.16635&&10 s GeVthe muon decay constant, O(n) radiative correctionsto superallowed 0+ 0+ Fermi transition rates gaverise to an overall factor'5

1+ g(E ) 1+ 41n +ln +2C+ Mg2 tr™ 27r mp mg

which reduces ~t values and I V~I2 derived from them. The first factor in Eq. (6), sometimes called the outercorrection, s represents a spectrum-averaged effect that depends on the end-point p energy E . The remaining(inner) corrections are dominated by the large short-distance term 4 ln(mz/mp) (mz = 93.2 GeV,mp = 0.938 GeV). Also present is an axial-vector-induced structure-dependent contribution ln(m~/m„) +2C,

1985 The American Physical Society

VoLUME 56, NUMBER 1 PHYSICAL REVIEW LETTERS 6 JANU&Rv 1986

where m& is a low-energy cutoff applied to the short-distance part of the y H'box diagram and 2C representsthe remaining long-distance (low-frequency) correc-tion. This term depends on the details of strong-interaction structure at low energy and is therefore themain source of uncertainty in the radiative corrections.Finally, M ~ denotes small perturbative 0(n, ) QCDcorrections that can be rather reliably calcUlated. '

In previous extractions of ~ V~t from measured ~ r

values, the inner correction factor in Eq. (6) was gen-erally taken to be 1.0210. From that value, a recentup-to-date analysis by Towner and Hardy which ob-tained a recommended Xr = 3080.1 + 2.4 sec leads to'

t V„d [= 0.9742 + 0.0004, (7)

where + 0.0004 represents the error arising only fromexperimental and nuclear theory uncertainties. (Nu-clear effects stem from isospin-breaking Coulombcorrections to transition matrix elements. s)

To refine further the Towner and Hardy result andestimate the theoretical uncertainty in t V~t due to ra-diative corrections, we have carried out the followingreanalysis of Eq. (6).

(I) The small perturbative QCD correction M~,which was estimated to be —0.37 in Ref. I, has beenreduced slightly in magnitude to

gf g= —0.34.

This change is due to a shift in sin &w and n, (1 «»

ln +2C =0.0012 +0.0018.27r mg

(9)

(3) Since the (2n/m )In(mz/m~) contribution is thelargest O(n) correction in Eq. (6), we have approxi-mated the effect of higher orders by summing allleading-logarithmic corrections of 0 (n" ln "mz), n= I, 2. . . , via a renormalization-group analysis. Sucha summation replaces Eq. (6) with'2

from 0.35 and 0.5 (used in Ref. 1) to more contem-porary values of 0.22 and 0.3, respectively. 9

(2) The structure-dependent term ln(m~/mz ) + 2Carising from the axial-vector current was estimated bya procedure similar to the approach used by Marcianoand Sirlin. 9 For the low-energy cutoff m„, we allowedfor a range 400 MeV«mz «1600 MeV, while thelow-energy part 2C was approximated by the Born con-tribution in two models of nuclear decay. In the first,following Dicus and Norton, ' we considered the0+ 0+ decay of the nucleus as a whole in whichcase C = 0. In the second, we employed an inde-pendent-particle model of the process, in which casethe Born contribution was analyzed by a new calcula-tion involving nucleon electromagnetic and axial-vector dipole form factors. " We found C= 3g„(0.266) (p~+ p, „)= 0.885 where p, + p, „=0.88is the nucleon isoscalar magnetic moment. Allowingfor the above ranges, and using a symmetric error, weestimate

tlap n(mp)1+ In +2C + [g(E )+A ] S(m, m ),27F m

(10)

where

S(m~, mz) = n(m, )' ' n(m, )

'n(mb) n(m, ) n(mg ) n(mz)

n(mp) n(m, ) n(m, ) n(mq) n(m) n(m~)

is a QED short-distance enhancement factor and n(p, )is a running QED coupling defined by MS (modifiedminimal subtraction) which satisfies'2

n(p, ) = bon2(p, ) + higher orders,dp,

(12a)

0 XOge(p, ——m~) — «—p —m~) (12b7

3~ 2m

(sum over all elementary fermions),

n ' (0) = 137.036+ — = 137.089.I(12c)

The values of n(p, ) to be used in Eq. (11) are found

n '(mz = 93 2 GeV) = 127 66

n '(mg =82.2 GeV) =127.73,

n '( m, = 40 GeV) = 128.95,

n '(mp=4. 5 GeV) =132.04,

n '(m, = 1.78 GeV) = 133.29,

n ' ( m, = 1.25 GeV) = 133.69,

n '(m~=0. 938 GeV) =133.93,

which when inserted into Eq. (11) yield

S(mp, mz) =1.02256. (14)

VoLUME 56, NUMBER 1 PHYSICAL REVIEW LETTERS 6 JANU&Rv 1986

Including the above refinements, we find in the caseof '40 (for example) where g(E ) =11.107 that Eq.(10) gives 1.0369 + 0.0019 to be compared with1.03417 used in the Towner and Hardy analysis.(The relative increase is essentially the same for alleight 0+ 0+ transitions analyzed in Ref. 7.) Rescal-ing their result for l V~l by (1.03417/1.0369)'t2 andtaking the conservative approach'3 of doubling thequoted uncertainty in Eq. (7), we find

l V~l =0.9729+0.0008+0.0009. (15)

The central value of l V~l has been reduced by 0.13%and an estimate of the theoretical uncertainty,+0.0009, in the inner corrections has been deter-

mined. Using this value along with the constraints inEqs. (4) and (5) and combining errors in quadrature„we f"md

I v~l'+ Iv I'+

Iv~l'=0. 9954+0.0025. (16)

The result in Eq. (16) is consistent with three-generation unitarity at the 2-standard-deviation leveland must continue to be viewed as a quantum-loop tri-umph for the SU(2)L 8 U(1) model. Nevertheless, itis interesting to speculate as to what effect new physicswould have on our analysis. In particular, the ex-istence of a fourth fermion generation could manifestitself as a deviation from three-generation unitarity.Employing Eq. (16), we find that at 90'/0 C.L. thefourth-generation mixing parameter l V&, l is not very

stringently bounded:

I V„& I & 0.088 (90% c.L.). (17)

In fact, the central value in Eq. (16) corresponds tolV, l

=0.068 [compare with Eq. (5)]; but the errorwould have to be significantly reduced before onecould argue that evidence for fourth-generation mix-ing had been found. Regarding the likelihood of afourth generation, we remind the reader that such ascenario is not in conflict with any experiment. Infact, some grand unified theories actually predict'~ thatthere should be at least four relatively light genera-tions. It is also interesting to note that recent papersby Anselm et al. and He and Pakvasa'5 have demon-strated how relatively large fourth-generation mixingcould resolve problems with accommodating the CJ'-nonconservation parameters ~ and ~' in the three-gcneration theory. It may also lead to detectable—1% Do-Do oscillations'5 and an enhancement inK+ m. + v v. Clearly, a fourth generation withl V, l

—6% would be an exciting development with

many interesting experimental implications.If a fourth generation is not responsible for the

small deviation from unity in Eq. (16), what is. Thesimplest explanation is that I V~l, I V l„or I V„, I (ormore than one) is actually larger than the range of

values we have allowed. In the case of I V„b I, there is atendency for the experimental b lifetime to be gettingshorter than the value 1 x 10 '2 sec used in obtainingEq. (5). If, in addition, I (b u)/I (b c) actuallyexceeds the experimental bound 0.04, then l V„t, l

could increase beyond the bound in Eq. (5). Howev-er, it seems unlikely that l V„t, l can be as large as 0.068which is needed to bring the central value in Eq. (16)up to 1. Nevertheless, 7b, I'(b u)/I (b c), andtheir mutual dependence on l V„bl need continued ex-perimental and theoretical scrutiny.

A second (perhaps more likely) possibility is thatl V~l is actually as large as 0.231 rather than in therange of Eq. (4). Indeed, hyperon P decays do favor ahigher value. 3'6 Unfortunately, SU(3)-breaking ef-fects in hyperon P decays are potentially large and notwell under control. 3 More theoretical work is calledfor.

Finally, there may be additional uncertainties in thenuclear 0+ 0+% t values. We have allowed some-what for such a possibility by doubling the error quot-ed by Towner and Hardy. However, it is our beliefthat the O(Za2) corrections (Z= charge of daughternucleus) and the nuclear Coulomb corrections stillneed to be critically reexamined. '3 On the side of ex-periment, it seems unlikely that the precise data'7 em-ployed by Towner and Hardy will shift very much. Itwould, however, be useful if comparable measure-ments could be carried out for 0+ 0+ superallowedtransitions in 'OC which has a (small) Z = 5 daughterand hence relatively small Z-dependent effects. Ofcourse, all Z-dependent corrections as well as much ofthe theoretical uncertainty can be eliminated by a pre-cise high-statistics experiment on pion P decay n+

m 0+ e+ + v, .' " (The small branching ratio,1&&10 8, makes such a measurement difficult. ) A re-cent Los Alamos experiment'8 found lv„ql=0. 962+0.018 from pion beta decay. Although that result is

in good agreement with Eq. (15), the experimental er-ror is too large for it to be meaningful at the level ofour analysis. However, it should be clear from ourdiscussion that a new higher-statistics experiment isimperative and should be undertaken. '9 20

In summary, our results demonstrate the impor-tance of radiative corrections and precise experimentalmeasurements. Superallowed 0+ 0+ nuclear beta-decay ~ t values remain a quantum-loop triumph forthe SU(2)t 8 U(l) theory even at the several tenthsof a percent level we have analyzed and provide astringent constraint on new-physics appendages orcompetitors to the standard model. Future experi-ments and continued theoretical scrutiny will deter-mine whether the small deviation from unity in Eq.(15) is evidence for new physics, such as a fourth gen-eration, or merely the effect of additional theoreticaluncertainties. To help resolve this issue and provide

VOLUME 56, NUMBER 1 PHYSICAL REVIEW LETTERS 6 JAGUAR+ 1986

as precise a determination of I V~l as possible, a newhigh-statistics pion beta-decay experiment ~ould beparticularly useful and we strongly urge such an under-taking. '9 2o Finally, we note that it is extremely impor-tant to lower the errors on I V,dl=0. 24+0.03 andI V„I =0.85 +0.15 by precise measurements of charmproduction and decay. Used in the unitarity con-straints IV„dl2+IV,dl'»I and IV I'+I V„l'~1,they can provide useful checks on I V~l and I

V I.More importantly, on the assumption that a reliabledetermination of I V,&l from b decay is forthcoming,the three-generation unitarity constraint I Vd I2

+ I V„I'+ I V,al2=1 may also be used as a probe offourth-generation mixing. Unitarity of the Koba-yashi-Maskawa matrix is a powerful constraint that weare only beginning to exploit.

We thank D. J. Millener for useful discussion onnuclear uncertainties in P decay. We thank the AspenCenter for Physics (W.J.M.) and Brookhaven NationalLaboratory (A.S.) for their hospitality during the sum-mer of 1985 while this work was in progress. Thisresearch was partially supported by the U.S. Depart-ment of Energy under Contract No. DE-AC02-76CH00016 and the National Science Foundationthrough Grant No. PHY-8116102.

iA Sirlin, Nucl. Phys. B71, 29 (1974), and Rev. Mod.Phys. 50, 573 (1978). We follow the analysis of radiativecorrections to beta decay given in these articles and refer thereader to the second paper for a detailed discussion of stronginteraction uncertainties.

2M. Kobayashi and K. Maskawa, Prog. Theor. Phys. 49,652 (1983).

3H. Leutwyler and M. Roos, Z. Phys. C 25, 91 (1984).4The bound in Eq. (4) is based on ri, = I X 10 '2 sec,

I (b u)/I (b c) & 0.04. See, for example, G. Barbiel-lini and C. Santori, CERN Report No. EP/85-117 (to bepublished) .

sA. Sirlin, Phys. Rev. 164, 1767 (1967).6For a comprehensive review of superallowed Fermi beta

decays see D. H. Wilkinson, in Nuclear Physics ~i' HeavyIons and Mesons, edited by R. Balian, M. Rho, and G. Ripka(North-Holland, Amsterdam, 1978), p. 877.

71. S. Towner and J. C. Hardy, in Proceedings of the SeventhInternational Conference on Atomic Masses and Fundamental Constants, edited by O. Klepper (Gesellschaft furSchwerionenforschung, Darmstadt, 1984), p. 564.

aEarlier estimates of I V~l, e.g. , R. E. Shrock and L.-L.Wang, Phys. Rev. Lett. 41, 1692 (1978); D. H. Wilkinson,Ref. 6; M. A. B. Beg and A. Sirlin, Phys. Rep. SS, I (1982);E. Paschos and U. Turke, Phys. Lett. 116B, 360 (1982),found a somewhat lower value than Eq. (7) mainly becausethey employed smaller nuclear Coulombic corrections. For a

discussion see Ref. 6 and I. S. Towner and J. C. Hardy,Phys. Lett. 73B, 20 (1978).

9W. J. Marciano and A. Sirlin, Phys. Rev D 29, 75 (1984).'oD. Dicus and R. Norton, Phys. Rev. D I, 1360 (1970).~~A free-nucleon calculation by S. M. Herman and A. Sir-

lin, Ann. Phys. (N.Y.) 20, 20 (1962), gave C=—T.. In the

case of pion beta decay, 6 =0 in the Born approximation,but axial-vector-meson intermediate states are expected tocontribute at the level of C —0.1; see Ref. 1. Improvementof this estimate may, in the future, become an importantchallenge to lattice gauge theorists.

i25(m~, mz) is obtained by the solving of therenormalization-group equation

T

iA, +bon ——n S(mp, p, )=08Qp. $0! TT

and using S(m~, m~) =1. See W. Marciano and A. Sirlin,Phys. Rev. Lett. 46, 163 (1981).

'3We have doubled the error quoted in Eq. (7) because a

recent analysis of ' Cl by %. E. Orrnand and B. A, Brown,Nucl. Phys. A440, 274 (1985), suggests that the nuclearCoulombic corrections are actually smaller than the valuesemployed in Ref. 7. In addition, we feel that certain aspectsof the O(Za') corrections [see W. Jaus and G. Rasche,Nucl. Phys. A143, 202 (1970)] need clarification.

&4See, for example, J. Bagger et a/. , Phys. Rev. Lett. 54,2199 (1985).

i5A. A. Anselm et al. , Phys. Lett. 156B, 102 (1985); X.-G.He and S. Pakvasa, Phys. Lett. 156B, 236 (1985).

'6M. Bourquin er al. (WA2 Collaboration), Z. Phys. C 12,307 (1982), obtained I V 1=0.231+0.003 from hyperonbeta decay, More recently, A. Bohm and M. Kmiecik, Phys.Rev. D 31, 3005 (1985), included new Fermilab hyperondata and found I V~ I

= 0.225 +0.002. Unfortunately,SU(3)-breaking effects are potentially large and not reliablycalculable (see Ref. 3); so it is difficult to assess their effecton these results.

i7V. T. Koslowsky ei al. , in Proceedings of the Seventh International Conference on Atomic Masses and Fundamental Con-stants, edited by O. Klepper (Gesellschaft fur Schwerionen-forschung, Darmstadt, 1984), p. 572.

isW. K. McFarlane ei al. , Phys. Rev. Lett. 51, 249 (1983),and Phys. Rev. D 32, 547 (1985).

'9The uncertainty in m + —m o presently gives rise to

+0.0020 uncertainty in I V~l. Presumably, that could belowered if a high-precision pion beta-decay experiment wereundertaken.

20A precise measurement of AL —E -+ e+'I ' could also beused to determine I V~l with little theoretical uncertainty.The expected branching ratio =0.5X10 8 makes such anexperiment feasible at future high-flux kaon factories; butthe clean identification of such decays should be difficult.See V. Highland, in "Physics with LAMPF II,"Los AlamosNational Laboratory Report No. LA-9798-P, 1983 (unpub-lished), p. 302. We thank M. Goldhaber and L. Littenbergfor discus'sions on this decay mode.