How accurately can we estimate the proton lifetime in an SU(5) grand unified model?

Nuclear Physics B171 (1980) 273-300 © North-Holland Publishing C o m p a n y

H O W ACCURATELY CAN WE E S T I M A T E T H E P R O T O N L I F E T I M E IN AN SU(5) GRAND UNIFIED M O D E L ? *

T. G O L D M A N 1 and D.A. ROSS

California Institute of Technology, Pasadena, California 91125, USA Received 14 February 1980

We reduce the theoretical uncertainties in the prediction of the proton lifetime in the simplest SU(5) model to an uncertainty reflecting our lack of precise knowledge of the Q C D coupling constant, of the mass of color Higgs' bosons, and of the proton wave function at the origin. This enables us to put a confident upper b o u n d on the lifetime in this model. Our analysis includes all renormalization prescription dependence effects and phase-space effects. Branching ratios to experimentally significant decay modes are also discussed.

I. Introduction

The SU(5) [1] grand unified theory of strong weak and electromagnetic interactions has an appealing practical advantage over all other currently proposed models: there are no new parameters beyond those already measured apart from the masses of the Higgs scalar particles (and that of the t-quark which is yet to be discovered). The theory is unique in that precise predictions may be made for all new processes beyond those already implied by SU(3)~ × [SU(2)× U(1)]~, [2]. Of these, the most exciting (and perhaps the most easily testable) is the prediction of proton decay**. With the advent of sufficiently sensitive detectors [3] to observe this process at a rate in the range suggested by the approximate calculations using an SU(5) theory, it has become urgent to predict this rate as accurately as possible. At least, we must specify exactly where our uncertainties lie and set an upper bound on the proton lifetime so that we can state at what level the failure to observe this decay rules out the theory. We have previously published [4] such estimates on the basis of a field theoretic calculation which took into account two-loop effects. In this paper we present further details of that calculation, and also include some additional features which * Work supported in part by the US Department of Energy under contract no. DE-AC-03-79ER0068. i Permanent address after September 1980: Theoretical Division, LASL, Los Alamos, N M 87545, USA. ** For brevity, we shall write "proton (neutron) lifetime" where we m e a n the (appropriate) nucleon lifetime for b a r y o n - n u m b e r violating decays. 273

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T. Goldman, D.A. Ross / Proton lifetime

were previously estimated to have an effect of much less than an order of magnitude in the rate. The results are essentially unchanged, but the uncertainties have been reduced considerably. The remaining reducible uncertainties consist primarily of the uncertainty in the QCD coupling constant at some reference momentum and the value at the origin of the proton wave function in terms of quark fields, ~/(0). All other uncertainties, including the effects of yet higher order corrections are smaller than the combination of these two. Below, we shall work with the minimal SU(5) model in which there are precisely six flavors (u, d, s, c, b, t) of quarks and the Higgs sector consists of one 5-plet and one 24-plet of scalar fields, whose vacuum expectation values break SU(5) to SU(3)c X U ( 1 ) e m (through SU(3)c × [SU(2) × U(1)w]). We shall comment occasionally on the effect of adding extra flavors or changing the system of Higgs scalars. Additional flavors of quarks tend to reduce the predicted value of sin 2 0 w (where 0 w is the Glashow-Weinberg angle [2]), which, as we shall see below, is already slightly (about one standard deviation) low compared with the experimentally determined value; i.e., the discovery of additional quark flavors (apart from the t-quark) could possibly rule out the SU(5) model without any reference to proton decay. Moreover, the successful prediction for the b-quark mass [5] is spoiled from higher-order corrections by the introduction of extra fermions [6]. On the other hand, light Higgs particles tend to reduce the proton lifetime. This would affect our optimal estimates below, but obviously cannot invalidate our upper bound. We shall explain this mechanism in more detail below. In sect. 2 we describe in detail the one- and two-loop renormalization group equation (RGE) calculations of the grand unified scale M x including our analysis of residual theoretical uncertainties and the effect of input data uncertainties. Our analysis includes the effects of thresholds, renormalization prescription dependence, Higgs particles and the (small) effect of running the mass M x from the grand unification point (U = 2 M x) to the proton mass. In sect. 3 we calculate the (baryon-number violating) nucleon decay rates. We also include, in an approximate way, estimates of phase-space effects and of the branching ratios into particular modes (or classes of modes). In sect. 4 we analyze the scale corrections to the four-fermion interactions involved in the decays. Most of these corrections can be described in terms of an operator product expansion, but we also discuss the uncertainties associated with the prescription-independent definition of I~k(0)]. In sect. 5 we present our best estimates and upper bounds for the nucleon lifetimes and discuss the results.

2. The value of the SU(5) grand unification mass M x

In this section we determine M x by following the variation with momentum scale, Q, of the separate coupling constants ~s(Q2), ~w(Q 2) and ~B(Q 2) (for the

T. Goldman, D.A. Ross / Proton lifetime

SU(3)c, SU(2) and U(I) subgroups of SU(5) respectively, to the value defined by f s ( 4 M x2 ) = f w ( 4 M x2 ) = ffa(4M 2) ~ a o u M .

275 Q2= 4 M 2

(2.1)

There are, of course, several corrections to this naive description which we describe below. First, however, let us define a convenient reference point, by using the Appelquist-Carazzone (AC) decoupling theorem [7]. In lowest order the variation of coupling constants is given by the renormalization group equation

d C ,(Q2) =

fl%/4~r,

(2.2)

where t---ln(Q2//~2),# is the arbitrary scale at which renormalization has been performed and flj is a rational number, characteristic of the degrees of freedom of the theory. The AC theorem shows that fl% need only include the contribution from effective degrees of freedom whose mass scale m is smaller than Q. Thus in the crudest approximation, for Q > 2m, f j ( Q 2 ) obeys eq. (2.2) with fl% = b + b,,, where b is the value of fl% computed as if the degree of freedom with mass scale m did not exist, and bm is the value due to this degree of freedom only. For Q < 2m, fj(t) obeys eq. (2.2) with fl% = b only. Thus in leading order a graph of f f I(Q2) against In Q2 is a straight line with kinks at the thresholds of the various degrees of freedom. This is called the 0-function approximation. More precisely, it should be noted that the mass scales have an effective variation with Q and one should deal with coupled equations as did Buras et al. [5]. One of the present authors [8] has examined the deviation from the 0-function approximation for threshold effects by following the Q2 dependence of diagrams involving degrees of freedom with mass scale m in the internal loops, in the threshold region Q 2 m2. F r o m these results it has been shown that very accurate results can be obtained using the 0-function approximation and applying a small correction to the effective coupling constants. At the moment, we are only interested in noting that AC decoupling implies that the variation of fw below a mass scale of the order of M w (the mass of the weak intermediate vector boson) is frozen (flw = 0) for Q2 < M 2 : there are no graphs for flw which do not involve the virtual excitations of the W-boson degrees of freedom, A similar statement is true for fiB, with the exception that a Q2 dependence survives for the combinations of fw and fB which forms the electromagnetic coupling constant, fern" It is therefore convenient to follow the variations of fern from its experimentally measured value in the Thomson limit ( Q 2 = 0 ) up to Q2 = 4 M ~ . From then on we follow fw and fin separately.

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2.1. THE VALUE OF &,m(4M2) AND CONVERSION TO tiB(4M2) T h e variations of Otemwith Q 2 are determined f r o m the one-loop Q E D renormalization group equation

d ,- 2-) = &m ~-~ffe-m(Q ~,

(2.3)

where

43 fl~m(Q 2 ) = _ __

q / 2f 1 +

E leptons and quarks

2m2 ¢Q4 + 4m2Q2

CQ, + 4m2Q2 _ Qz In

¢Q4 + 4m2Q2 + Q2 '

qi is the charge of the ith f e r m i o n a n d m i is its mass. The quantity in curly brackets is the exact function for the threshold behavior. It has the properties that it vanishes as Q2--~0, a n d tends to 1 as Q2___~oo (its value is 0.38 at Q2 = 4m/2). In the 0-function a p p r o x i m a t i o n this function would be replaced b y O(Q2 - 4m2). W h e n integrated over ln Q 2 from Q 2 = 0 to Q2 >>4m/2 we get ln(Q2/4m2i)+ l n 4 - ~ , whereas the 0-function a p p r o x i m a t i o n just gives ln(Q2/4m2). Integrating eq. (2.3) and using for the integration constant the fact that for Q 2 = 0 (the T h o m s o n limit) O/em - i ~-- 137.04 (this selects a particular renormalization prescription and this will be i m p o r t a n t later w h e n we m a t c h ~em with ffw a n d as) we find* 1

q2[ln(M2)

ff~-d(4m2) -- 137.04 = -- - -

leptons and quarks

[ \ m~

+ln4--~

+

O( mZi I] kM2lJ (2.4)

W e take the lepton masses as m e a s u r e d a n d the following quark masses m u = m a = 0.3 G e V , m t = 25 G e V ,

m s = 0.5 G e V ,

mc=

1.5 G e V ,

m b = 5.0 G e V , (2.5)

although the results are not very sensitive to the precise values. As can be seen from eq. (2.4) most of the change in 8em comes f r o m the light fermions since it is for these fermions that the logarithm is quite large. T h e lighter quark masses are the constituent values which we have taken since semilocal duality tells us that these reflect the effective thresholds. T h e only truly u n k n o w n quantity is the mass of the t-quark, which we choose near the top end of the range of recent predictions. * This calculation has also been performed using slightly different techniques by Marciano [9] and Paschos ll0].

T. Goldman, D,A. Ross / Proton lifetime

277

Using M w = 82 GeV (consistent with our final results) we find a ; - / ( 4 M ~ ) = 128.5.

(2.6)

Reducing m t to 15 GeV would reduce this value by about 0.2. About the same effect would be found by introducing another generation of quarks between the t-quark and the W-boson. Compared with this uncertainty, the effect of higher-order effects is estimated to be negligible (they would be down by O ( a / ~ r ) on the above corrections and would only have a single logarithm). This result is independent of grand unification and will vary (very slightly) with the experimental determination of M w. If this variation of 0(era had not been included (i.e., if ff~-mZ(4M2w)= 137.04 had been used), M x would have come out about I0 × larger than the value actually obtained. The uncertainty of 0.2 from the quark masses is reflected in an error of 6% in M x. The conversion to aa now just involves the SU(5) relation

aB =

5~¢1711

,

(2.7)

3 cos 2 t~w where/~w is the ("running") value of the Glashow-Weinberg angle at Q 2 = 4M2w" In the numerical calculation to find Mx, 0w(4M2w) is a parameter which is adjusted so that the three running coupling constants arrive at the same value at some point. Thus 0 w is also a prediction of the theory as originally noted by Georgi, Quinn, and Weinberg [11]. 2.2. RENORMALIZATIONPRESCRIPTION DEPENDENCE AND Ks It is a recently realized property of all quantum field theories that the definition of the parameters in the theory depends not only upon the scale at which renormalization is implemented, but also on the precise manner in which the subtractions have been performed. In QED the natural choice of the Thomson limit to define the electromagnetic coupling constant obscures this point, but in general there is a prescription dependence which is unavoidable in non-abelian theories. It is of interest here that in the usual discussion of the analysis of data (from e + e - --~ hadrons or scaling violations in deep inelastic lepton scattering), ffs is not taken directly from any experiment but is chosen in terms of a dimensionful parameter A which is related to a by fl0 ffsl(Q2)= ~ ln(-~)+

(~)ln(ln(-~))+

1

O( ln(Q2/A 2) )"

(2.8)

The value of A depends on the subtraction scheme (renormalization prescription); the usual one employed is called the "truncated minimal subtraction scheme" [12]

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T. Goldman,D.A. Ross / Protonlifetime

(M--g) in which divergent diagrams are calculated using dimensional regularization [13] and the renormalization is effected by removing the pole as d---~4, where d is the number of dimensions, and the accompanying l n ( 4 c r ) - 7r. (The latter is an artifact of the regularization method and serves to make the higher order terms artificially large.) However, the calculations may be performed in any scheme. In our discussion above, the electromagnetic coupling constant has been calculated using the definition of Otem at the Thomson limit to perform the renormalization. In QED, the Ward-Takahashi identity Z~ -= Z 2 tells us that only diagrams with a fermion loop insertion on the photon line contribute to the renormalization of aer=. However, this is only true if the fermion lines are taken on-mass-shell. In ref. [8] the analysis of the detailed effects of AC decoupling in the region of the vector boson thresholds has been done in the "symmetric momentum" subtraction scheme in which a divergent Green function is subtracted with all its external legs put equal to the same spacelike momentum. This prescription, although often used, suffers from the disadvantage that it requires further specification. Firstly, it depends on the process being considered, e.g., the finite parts of the one-loop triple gauge boson vertex are different from the finite parts of the one-loop gauge boson-fermion-fermion vertex. Secondly, if we choose the latter, we are faced with the problem that although the leading order vertex is proportional to 7 ~, in one loop we have the general form (for ingoing fermion momentum p", and vector boson momentum qt, p2= q 2 = _Q2)

F~(p,q)="/"

½floln ~

+C1(Q2'Mi2)

+ q"C2(Q 2, Mi2) + qpa"~C3(Q2, Mi2) + ' "

,

(2.9)

(where M i are the masses of the particles in the internal loops) and we must choose how much of the functions Ci to include in our definition of the effective coupling constant. Even if we specify that we look for the coefficient of 7 ~, the result is not unique since by manipulating "/-matrices the form of eq. (2.9) can be changed in such a way that CI(Q 2, Mi2) is altered. In ref. [8] an unambiguous (although not unique) prescription for defining the coefficient of "/~ was given. It consisted of squaring the amplitude and summing over spins and projecting away the linear combination of "off-shell structure functions" (i.e., coefficients of g ~ and P~,Pv) which vanish in the tree diagram approximation (see ref. [8] for further details). Furthermore, since we are dealing with off-shell Green functions and only on-shell S-matrix elements are gauge invariant, the functions Ci(Q 2, Mg2) are gauge dependent; we work in the Feynman gauge. All this arbitrariness in the specification of the exact renormalization prescription merely reflects the fact that an effective coupling constant is not a physically measurable quantity and should only be used as a parameter to compare different

T. Goldman, D.A. Ross / Proton lifetime

279

physical processes. This comparison is valid provided the same renormalization prescription is used in the calculation of each process. Thus in our case we note that the unification point can be found correctly provided we use the same definition for the three coupling constants fiB, ffw, and ffs. We thus have to translate ffem(4M2 ) from the Thomson limit definition and ffs(4M~) from its usually quoted A ~ into the symmetric momentum subtraction scheme defined above so that they are compatible with ffw whose threshold effects have been calculated in ref. [8] in this scheme. For ffem we must calculate the diagrams of fig. 1 (which cancel for on-shell renormalization) in the symmetric momentum scheme. We find for these diagrams a contribution to ff~-m~ (2.10)

A(ff~-m1) -----~-~(4I + 1), where fo' I = -

In(x) = 2 ~] sin(]nTr) d x (x 2 - x + 1) ~/3 n = l rt 2 ~ 1.17.

This has the effect of changing the ff~-m~(4M~) which we should use in our calculation to 129.1 (this change increases M x by 20%). For ffs we must calculate the gluon-quark-quark vertex at the one-loop level in the M.S. scheme. The diagrams of fig. 1 re-interpreted for this process now give us 1

At(ff~-') =-3-~ [(4I +

1)Cv-(21-¼)CA].

(2.11)

To this we must add the contributions from the diagrams of fig. 2. Fig. 2a gives

=-~--(I

+

.

(2.12)

Fig. 2b gives A2b(ff~-l)=

-31G 367r

'

(2.13)

Fig. 1. One-loop vertex correction graphs which m u s t be calculated to relate the electromagnetic coupling in the T h o m s o n limit to the symmetric m o m e n t u m renormalization prescription value.

280

T. Goldman, D.A. Ross / Proton lifetime

(o)

(b)

(c)

Fig. 2. One-loop vertex correction graphs which, together with those in fig. 1, must be calculated to relate a non-abelian coupling constant defined in one subtraction scheme to its value as defined in any other subtraction scheme. and f r o m fig. 2c we have* 5TR

(2.14) 4

A d d i n g these contributions a n d using the values C v - 5 , C A = 3 a n d T R = 3 f r o m SU(3) with six flavors, we find that ff in the symmetric m o m e n t u m scheme is related to ~ in the M.S. scheme b y * 2 )=-ct~.s.(Q 2 )[ 1 + -5~3 ffw~.s.(Q )]. Otsym(P . 2.-I

(2.15)

This is a surprisingly large correction. At Q2 = 4 M 2 it increases the ~ which must be used by 22%, a n d this increases M x by a factor of 3. Large corrections necessarily introduce large uncertainties since the next-order corrections m a y be relevant. F o r instance, if we h a d written eq. (2.15) a~.s.(Q z ) -

z

,

1-

(2.15a)

)

~r

which is identical to eq. (2.15) to the order to which we are working, we could express eq. (2.15a) b y the statement Asym - e 31'8/25 = AM.S.

3.6,

(2.15b)

provided A~2.s" was determined in the energy region in which there were four 'active' flavors (see below). Since M x is almost exactly proportional to A (see below) we would have an increase in M x of a factor of 3.6 (instead of 3). Thus we have an uncertainty of 20% in M x arising f r o m the translation of Ks f r o m the M.S. scheme to the symmetric m o m e n t u m subtraction scheme at the two-loop level. To calculate ~s, we begin with the one-loop approximation. W e assume that the experiments from which AM~.s" is calculated have typical Q2 >>m 2 a n d Q2 <
T. Goldman, D.A. Ross / Proton lifetime

281

that there are four 'active' quarks (the rest are below threshold). Thus for some typical Qo2 in this range we have ff~-t(Q0Z) =

25 ( Q2

~

k~

)"

(2.16)

Now we use the R G E to project tiffs up to 07 in the region m 2 << Q~ <
25

= ~ln

( Q~ ) - __1 ~ln/ Q 2 ) 6~r~ ~4m~ + l n 4

-- } ~ ,

~A~z.s.~

(2.17)

where the I n 4 - 5 5 is a fermion threshold effect identical to those discussed in the treatment of ffcm" Finally at Q2 = 4M~v we have ff~_,(4MZw)= 25 l n / 4 M 2 ) 1 [. ~-~-~[m/~t2]+ln

[M~v~ (M__~)+ 1 n 1 6 - ~ . ]

(2.18)

Putting in the value AM~s.= 400 MeV (we discuss the sensitivity of the result on A~-~ later) we find a value of 7.6 for ff~-l(4M~) in the ~-~.S. scheme and 6.2 when converted the symmetric scheme.

to

momentum

2.3. T H E T W O - L O O P C A L C U L A T I O N

The AC theorem is true to all orders, and so the statements above apply to a calculation taken to two loops. Between the W-boson mass and the grand unification mass there are no thresholds, and the two-loop R G E for the running coupling constants may be simply expressed as follows: 3

..

d f f i ' = fl~ I- ~ fl't' dt 49 j=l

O~j

(2.19)

( 4 'n" ) 2 '

i,j

where run over the U(I), SU(2) and SU(3) subgroups respectively. The off diagonal terms in ,8[j arise from diagrams with a fermion loop inside a gauge boson line and a gauge boson from a different subgroup inside the fermion loop. fit is thus a 3 × 3 matrix* 0 fll j

=

0 136

-30

0

~3

]9

~

3

44

0

1

-~

49

~-

4

11

3

102

- no

~6

* This matrix was misprinted in ref. [4]; this is the correct version,

1-3 ,

(2.20)

T. Goldman, D.A. Ross / Proton lifetime

282

where n 6 is the number of generations of fermions [each generation consists of a and a 10 representation of SU(5)]. Eq. (2.19) can be solved by iteration; namely, one solves the one-loop RGE for aj and then substitutes the expression in the second term of eq. (2.19). Neglecting momentarily the off diagonal contributions, which in fact have a small effect, in/3 l, we find that at the two-loop level, eq. (2.16) is replaced by

-

(.')

8s-~(Q2) = 1-~-~ in ~

,,4 c : )

+ ~

lnln

.

(2.21)

Eq. (2.17) becomes 25 in( Q~ ) 1 [ln( Q~ ) + l n 4 - 3 s ] Az --~[ ~4m~,

[

+ ~116 In 1

e0 : 1 ,

(2.22)

with a similar expression for 8~-l(4Maw). Numerically we find that 8~-l(4M~v) is now 7.3 (in the symmetric momentum subtraction scheme). This has the effect of decreasing M x by a factor of 2.3. The effects of off-diagonal terms and of including /~ when following the effective coupling constants from the W boson threshold to unification enhance this factor to 4.3. This effect is of the same order as (but of opposite sense to) the effect of converting 8, from the M.S. scheme to the symmetric momentum subtractions scheme, and one may estimate that, as in that case, the next-order effect may introduce uncertainties of order 20%. However, it must be remembered that A M-~.s. was determined from experiment using the two-loop formula for if,. If we were to calculate to three loops one would have to change the parameterization of if, in terms of AM-~s.and the change in AM-~g:would almost exactly cancel the three-loop effects. More precisely, M x is obtained from a formula of the form (neglecting threshold effects) Mx = g e x p [ A (

1

--

1)+B+C(ffs(Q~)--aGUM)] 2

(2.23)

where OtGUM is the common coupling constant at unification and A, B, and C are constants of order 1. But 2 is in principle fixed from experiment and only its parameterization with A changes as one includes more loops. Thus it can be absorbed into g and we have

as(Qo)

M x=g'exp

[A

----CaGu O~GUM

]

M+B

,

(2.24)

T. Goldman,D.A. Ross / Protonlifetime

283

SO the effect of putting C = 0 is of order e ÷~oU~' where the grand unification coupling constant, aGUM, is about 0.02; these corrections are truly negligible. 2.4. VECTOR BOSON THRESHOLDS Having converted all effective coupling constants to the symmetric m o m e n t u m subtraction scheme, we can use the results of ref. [8] to include corrections for the thresholds at Q2~4M~v and Q 2 ~ 4 M 2 . F r o m fig. 9 of ref. [8] it can be seen that the threshold at Q 2 ~ 4 M 2w increases the effective ffw by 3.7%. This means that the full threshold effect may be simulated by introducing a discontinuity at Q2 = 4M2w" ffw(4M 2 + e) = ffw(4M 2 - e). 1.037.

(2.25)

Thus when projecting the coupling constants from 4 M 2 to 4 M x2 we use for ffw(4M 2 ) the relation ffw(4MZw) = l'037aem(4M2) sin2 0w

(2.26)

At M x, the effect of vector boson thresholds appears differently. The numerical calculation in the 0-function approximation finds the value at which the coupling constants meet. In fact, the coupling constants are delayed from meeting by the effects of passing through the thresholds for the X and Y particles. All three f l ' s pick up new contributions f r o m graphs involving these particles inside the loops. The trajectories of the coupling constants only actually meet at infinite Q2. Again we can use the calculations of ref. [8]. F r o m fig. 17 of ref. [8] we see that for ln(Q2/4M 2) = - 5, ffs(Q 2) = 0.0252, ffw(Q 2) = 0.0242 and ffa(Q 2) = 0.0221 so that in the 0-function approximation the effective couplings would meet at 1

1

0.0252

0.0221

=

1._1_1 In

-

5

.

(2.27)

4~r

This gives Q2= 3.8Mx2 so we see that to correct for this threshold effect, the uncorrected value of M x must be reduced by a factor of 1.95. Since these threshold effects only affect coupling constants whose values are less than 0.04 we once again estimate that the combination of mass thresholds and the two-loop effects will be negligibly small (of order 1%). 2.5. EFFECT OF HIGGS SCALARS The introduction of an irreducible SU(5) representation of Higgs scalar whose masses are equal has the effect of altering the slope of the coupling constant trajectories for all coupling constants in such a way that the unification point remains unchanged. However, if the components of the multiplet which couple to

284

T. Goldman, D.A. Ross / Proton lifetime

flavor but not color gauge bosons have a lower (higher) mass than the rest, this will have the effect of reducing the slope of the trajectory of ffw before (after) it reduces the slope of the trajectory of et s and thus reduces (increases) the value of the grand unification scale. For example, in the model with which we are working let us assume that the 24-plet all have the same mass whereas for the 5-plet the two components which carry flavor only have an effective threshold at Q2= M2w, but the three components which carry color have masses larger than or equal to M x. This changes the fl for the SU(2) and U(l) subgroups by 1/24~r which has the effect of reducing the unification point (since it slows down the decrease of fiw without changing the rate of decrease of ffs) by a factor of 1.8. It has been shown [16] that the colored Higgs scalars cannot have a mass of less than 10 ~° GeV, otherwise proton decay via the exchange of scalar particles would be too rapid. In this case the reduction factor for M x takes the smaller value of 1.45. We use the intermediate value of 1.65"* and allow a 10% uncertainty in M x from these Higgs effects. Recently Cook, Mahanthappa and Sher [17] have shown that varying the masses of the colored Higgs bosons within a 45-plet can have the effect of enhancing M x by as much as 2.7. A similar effect can occur within the 24-plet, but in this case the masses of the physical Higgs particles must be greater than 1013 GeV since they pick up masses in higher order from their interaction with X and Y particles. Thus the maximum uncertainty which can be introduced due to different masses for the colored and flavored components of the 24-plet*** is 15%. Finally we point out that the introduction of any further Higgs scalar multiplets for which the scalars that carry flavor only have a mass smaller than M w, whereas the others have a mass of at least 10 l° GeV would lower M x by at least a factor of 1.45 (reducing the proton lifetime by at least a factor of 4). 2.6. THE RUNNING MASS The calculation of the unification mass determines the physical mass of the X and Y particles [18], M x ( 4 M ~ ) , whereas the mass which should be used in the propagator of the X or Y in proton decay is Mx(4m2). Typical diagrams contributing to the renormalization of M x are shown in fig. 3. Diagrams of the type shown in fig. 3a involve X or Y particles in the internal loops and so by the AC decoupling theorem, their contribution decouples as we take Q2 below M2x so that these diagrams can only give a negligible contribution of order •GUM" Thus we must concentrate on diagrams involving only internal Higgs particles or a Higgs *Even if the mass of the physical Higgs is much smaller than Mw, the mass term in the W-propagator provides an effective threshold at QI= M2 in the contributions of Higgs scalar loops to ft. **This corresponds to a mass of 1012GeV for the Higgs scalars which according to ref. [17] is probably the theoretical lower limit. * * * The components of the 24-plet that carry both color and flavor are the Goldstone bosons for the X and Y particles and have a mass Mx in the Feynman-'t Hooft gauge.

285

T. Goldman, D.A. Ross / Protonlifetime G X

\.

J

X

/ P--,

S ""

X

X

X

t

' ,

X

(a)

X

t

G

X

X

(b)

(c)

Fig. 3. Typical diagrams contributing to the variation with scale of the mass of the X-boson. particle and a gluon since it is possible that these diagrams have a logarithmic "lever" from m H (which can be as low as 10 l° GeV) to M x. The diagrams involving only internal scalar particles (fig. 3b) are purely transverse and therefore do not contribute to the mass term. This leaves us with diagrams of the type shown in fig. 3c. For such diagrams we have found

M~(Q 2<
5 [ m~ 1 = 8--'~aGUM ln~ ~ x x ] "

(2.28)

Taking the minimum value of m H this can reduce Mx(4m2p) by 4%. We note that this uncertainty is coupled to the uncertainty in the effect of the Higgs particles on rio; when the running mass effect is maximal, the effect of the Higgs 5-plet on the unification point is minimal and vice versa. Thus we conclude that the total effect of Higgs' particles (i.e., the running mass effect and the effect discussed in subsect. 2.5) is to reduce M x by a factor of 1.65 with 10% uncertainty arising from the 5-plet and another 15% uncertainty arising from the 24-plet. 2.7. DEPENDENCE ON Algt.~. It can be shown that M x is almost exactly proportional to A~-Tg~. In fact the results of our numerical analysis show Mx

c c A 0"98 .

(2.29)

This dependence is correct in the range 0.05 G e V < AM~.S.< 1 GeV. Machacek [19] has previously noted that our analysis shows an almost linear relation between M x and A M ~ as would be expected if A ~ . s " were the only mass scale in the theory. However, M w provides a second scale and it is quite reasonable to find an anomalous dimension of order aGUM relating these two scales. Unfortunately, the proton lifetime calculation involves the value of M x raised to the fourth power, and ffs, or equivalently A ~ is not well-known. Scaling violation studies in lepton deep inelastic scattering [20] suggest A ~ s ~ < 0.6 GeV, but a precise value is difficult to obtain due to the difficulty in isolating the effects of higher-twist operators. Perhaps the best opportunity for determining ffs resides in the analysis of 3-jet to 2-jet ratios in e + e scattering into hadrons. However, these experiments [21] are still not accurate below the 30% level a n d the requisite higher order calculations have not yet been carried out.

286

T. Goldman, D.A. Ross / Proton lifetime

We have taken A ~ = 0.4 G e V as a nominal value and we allow a 50% error in A M-~.s.to cover the probable range. This corresponds to a factor of 5 (either way) uncertainty in the proton lifetime. We wish to emphasize that this is not a necessary theoretical uncertainty. A sufficiently accurate (5%) experimental determination of ffs at Q 2 20 - 40 GeV2 can reduce this error to about 50% in the rate. 2.8. RESULT AND TOTAL UNCERTAINTY We conclude that, with the above-mentioned value of A ~.s., our best estimate for the mass of M x in the simplest SU(5) grand unified model is M x = 4.2 × 1014 G e V .

(2.30)

This number has a 50% experimental uncertainty arising from the probable range of A M - . The remaining theoretical uncertainty, combining the uncertainties in all our corrections (thresholds, higher order effects, conversion of renormalization prescription, effects of Higgs particles, uncertainties in some of the fermion masses) we estimate to be less than 55%. We emphasize that the upper limit of 55% is extremely pessimistic and would only obtain if all the uncertainties added up in the same direction. Byproducts of our analysis are the values of a r u u and the Glashow-Weinberg angle for which we obtain a ~ u M = 0.0244 ___0.0002, sin 2 0 w = 0.21,

(2.31a) (2.31b)

with an uncertainty in the latter of about 0.01. This has been evaluated on a scale of the order of the W-boson mass. A more detailed analysis of the value of this angle at different scales, the predicted W-boson mass and the relation to the mass of the (light) Higgs boson of the Glashow-Weinberg-Salam model has been undertaken by M a h a n t h a p p a and Sher [22]. We note that this value is only about one standard deviation away from the current average experimental value of 0.23 ___0.02. Furthermore, the effects of increasing A ~ . s " or the number of flavors is to reduce the predicted value of sin 2 0 w (by about 0.005 per 100 MeV in A~-~.s. and 0.01 per extra generation of fermions). Thus this experimental result will constrain attempts to increase the proton lifetime in SU(5) by adjusting the assumed values of A ~ s " and of n G (the n u m b e r of generations).

3. The proton decay rate We now turn explicitly to the calculations of the proton (and otherwise stable bound neutron) lifetime ~'p within the SU(5) model described above. We just remind the reader of the relevant couplings in the (broken) SU(5) lagrangian. The

287

T. Goldman, D.A. Ross / Proton lifetime

baryon-number violating vertices are ff-~1 gX/ tt [ e- + ~, d Ri +UO,.U j ~k R eUk -- e- + y . d ci ]

"~ V ~Ig Y /

/t

IC "~/zdRi [/~e

"1 "~-dJL'~p.t,tCkREij k + e IX ]''uLI

(3.1)

where t~L(R ) -~- (½(l(W)ys)~p, tp c = iY2tp* - Ctp and everything is repeated for the #, s, c, and r, b, t generations. These couplings produce, in the limit q 2 << M x2, v, the effective baryon-number violating 4-fermion interaction g2 2M 2 __

-ck j i [(eiJ kuR gituLl(e+y~de.

+

+

-e

-+

i

Y. dL)

Cr,

(3.2)

where M x ~ M y has been used. The weak isospin splitting between M x and M v is O ( M w) and so clearly negligible. These effective 4-fermion interactions lead to quark fusion graphs relevant to nucleon decay, which are displayed in figs. 4 to 7. The graphs are constructed by considering all possible combinations of quarks in the nucleon wave function, and allowing all possible real (not virtual) final states as determined by the energy release available in nucleon decay. These points are reflected in the fact that figs. 4, 5, and 7a and b, and figs. 6a and c are the same except for interchange of a pair of incoming and outgoing quarks, and that there is no fig. 6d, which would have a /~CR final state, which could never be realized as hadrons since the minimum energy required for such a final state is 1.9 GeV. In constructing these graphs we have ignored Cabibbo (or more general Kobayashi-Maskawa [23]) mixing and assumed that its effect is as small for these interactions as for the known weak interactions. Jarlskog [24] has raised the disturbing point that this need not be the case. To avoid this we appeal to the argument of Mohapatra and of others [5, 25] that all mixings must be of the same order if there is only one 5-plet of Higgs. .

+ ÷

UL

UR

(a)

UR

UL

(b)

(c)

(d)

Fig. 4. Tree graphs relevant to nucleon decay showing two-quark to anfiquark-lepton scattering mediated by X-boson exchange and producing negative helicity charged antileptons.

288

T. Goldman, D.A. Ross / Proton lifetime dR

e~

dR

e~ eR

uL

UL

UR

UL

~R

UR

(Q)

(b)

(c)

(d)

Fig. 5. As in fig. 4 but producing positive helicity charged antileptons. + eL

UL

UR

dL

+ eL

UL

OL

~

uR

(o)

(b)

dL (c)

Fig. 6. As in fig. 4 but mediated by Y-boson exchange. E x a m i n a t i o n of figs. 4a - c a n d 6 a - c shows t h a t e a c h a, b, c p a i r has the s a m e initial a n d final states ( i n c l u d i n g the s a m e chirality) a n d so these g r a p h s will interfere. This i n t e r f e r e n c e on the basis of q u a r k states is necessarily a s s u m e d in the G I M [26] c a n c e l l a t i o n m e c h a n i s m a n d m u s t be t a k e n j u s t as seriously here. T h e s e g r a p h s m a y be p u t in a c o m m o n f o r m b y F i e r z t r a n s f o r m a t i o n s . I n d e e d , it is c o n v e n i e n t for the SU(6) a n a l y s i s t h a t follows, to p u t all g r a p h s in the f o r m of s - c h a n n e l exchanges. W h e n this is done, the effect of i n t e r c h a n g i n g c o l o r indices in the eijk t e n s o r to a c o m m o n f o r m p r o d u c e s a m i n u s sign t h a t cancels the o n e in eq.

OR

c VeR

CJR

c VeR

UR

dL

dL

U~

(a)

c

(b)

lv c

(c)

(d)

Fig. 7. As in fig. 6 but producing antineutrinos. + eR

UL

dR

~ (a)

uR

dR (b)

+ eR

UL

~

uR

+ eR

% (c)

+ fir

UL

UR (d)

Fig. 8, Graphs of fig. 5 as Fierz-transformed to the form of s-channel exchanges in the approximation of four-fermion interaction.

289

T. Goldman, D.A. Ross / Proton lifetime +

it_ e~_ 2 x~-.-..(~ eL

÷

+

uR

eL

UL

eL

u

d~

~

%

~-~

uR

~

/

~

(b)

(o)

(c)

L

\ s-~

(d)

Fig. 9. Graphs of figs. 4 and 6 as Fierz-transformed to the form of s-channel exchanges. The factors of 2 x indicate the constructive interference between corresponding graphs in figs. 4 a n d 6.

(3.2) so that the sets of graphs in figs. 4 and 6 interfere constructively. In the Fierz transformed form, the graphs may be made to reappear as shown in figs. 8-10, where figs. 9a to c have a weighting factor of 2 for the corresponding amplitudes arising from this constructive interference. The wavy lines in figs. 8 - 1 0 represent J = 1 s-channel exchanges while the dotted lines represent J = 0 s-channel exchanges. Apart from the different couplings of the proton to J = 1 and J = 0 s-channel exchanges arising from the SU(6) proton wave function, the two-quark to lepton + antiquark scattering cross section in the relativistic limit is 3 times larger for J = 0 than for J = 1. This reflects the different angular dependences for the differential cross sections and is the same as the 3 to 1 ratio of neutrino to antineutrino scattering cross sections on isoscalar targets. Although it is inconsistent to perform the SU(6) analysis below and yet to use the relativistic cross sections, we go through the exercise in an attempt to gauge the sensitivity of the calculation to phase-space effects. The results below suggest that the uncertainty related to this question of phase space is less than a factor of 2. The SU(6) analysis simply involves writing down the proton (and neutron) wave function(s), appropriately symmetrized and extracting the J = 1 and J -- 0 combinations of 2 quarks appropriate to each graph. For the spin-up proton wave function we have

15 >

-

1 {21u >,lu,>jld,>,_lu,>,ld,>jlu*> *

I

-

Id*> lu*)jlu*>, + permutations of i,j,k),

(3.3)

where i,j, k refer to all quantum numbers other than spin or flavor (including position). The neutron wave function is obtained by the interchange u.-->d and the other polarization by the interchange 1'~--~+. Because it is the static quark model JR

J~

(a)

c ~eR

dR

dL

dL

(b)

c VeR

UR

u-.-

d~

(c)

c VeR

UR

~

dL

V,uCR

(d)

Fig. 10. G r a p h s of fig. 7 as Fierz-transformed to the form of s-channel exchanges.

290

T. Goldman, D.A. Ross / Proton lifetime

TABLE 1 Non-relativistic rate weight factors Graph no..

Sub.

(a)

(b)

8

3

9

9 l0

4.6 × 9 9

4.6 × 3 12

Wp = 178.5

(c) 12 4.6 × 12 3 Wn = 93.3

(d) ~) 8.4 1.14 X 8.4 2.1

a)Due to the higher average hadronic mass, a phase-space suppression factor of 0.7 has been applied to strangeness non-zero modes.

t h a t is consistent with SU(6) we c a n m a k e n o a s s o c i a t i o n b e t w e e n the spin d e c o m p o s i t i o n a n d q u a r k chiralities. (The small l e p t o n masses k e e p l e p t o n chirality a n a l m o s t c o n s e r v e d q u a n t u m n u m b e r a n d severely suppress b y O ( r n e / M w ) a n y further i n t e r f e r e n c e of graphs.) N o t e that fig. 10b o b v i o u s l y vanishes (neglecting c o n t r i b u t i o n s f r o m sea q u a r k s ) for p r o t o n d e c a y w h e r e a s figs. 8c a n d d a n d figs. 9c a n d d vanish for n e u t r o n d e c a y . A n overall SU(6) e n h a n c e m e n t occurs b e c a u s e m o r e t h a n one t e r m in (3.3) often contributes, with c o n s t r u c t i v e interference. T h e resulting SU(6) (rate) weight factors for e a c h g r a p h are shown in t a b l e 1. T a b l e 2 shows the s a m e weights b u t with relativistic k i n e m a t i c s . E a c h t a b l e also shows the total weight f a c t o r W N due to SU(6) a n d the n u m b e r o f graphs, which multiplies the c o m m o n non-relativistic cross section, o0, for 2 - q u a r k f u s i o n i n t o a l e p t o n a n d a n a n t i q u a r k . T a b l e s 3 a n d 4 show the c o r r e s p o n d i n g b r a n c h i n g ratios o b t a i n e d f r o m a p p r o p r i a t e sums of ratios of e l e m e n t s of tables 1 a n d 2 to this total weight. These results c o r r e s p o n d closely to those of M a c h a c e k [19]. T h e m u o n b r a n c h i n g ratios are for direct m u o n s , that is, n o t i n c l u d i n g those due to the d e c a y of final-state mesons. T h e l a t t e r effect will i n c r e a s e the a p p a r e n t m u o n b r a n c h i n g ratio seen in a n e x p e r i m e n t b u t will d e p e n d on the details of the final state a n d even the e x p e r i m e n t a l c o n d i t i o n s . TABL~2 Relativistic rate weight factors Graph no.

Sub.

(a)

Oa)

8

1

9

9 l0

4.6X9 9

4.6× 1 4

Wp =98

(c) 4

4.6×4 l Wn= 71

(d) a) 4

1.14×4 1

a)The relativistic analysis represented in fig. 11 suggests that the phase-space suppression factor, used in table 1, may be compensated for by quark momentum distribution effects and so it is omitted here.

291

T. Goldman, D.A. Ross / Proton lifetime

TABLE3 Non-relativistic inclusive branching ratios (for N--~A + X)

Ax e v bt g

P

n

0.81 0.08 0.11 0.12

0.72 0.28 0 0.02

TABLE 4

Relativistic inclusive branching ratios (for N-~ A + X)

e v # g

p

n

0,80 0,11 0.09 0.10

0.79 0.21 0 0.01

I n c o n s t r u c t i n g all of these tables we have i n c l u d e d a relative rate e n h a n c e m e n t factor of 1.14 for graphs in fig. 9, a n t i c i p a t i n g the results of e n h a n c e m e n t factors in the o p e r a t o r p r o d u c t analysis, discussed in sect. 4. The complete expression for the total n u c l e o n decay rate is Fp(n ) ~.

2AsAwBWp<.)Ool@(O)I2,

(3.4)

where the factor of 2 is due to the color s u m ( e i j k e i j t ) , A s a n d Awa are the e n h a n c e m e n t factors due to the strong a n d electroweak i n t e r a c t i o n s respectively, (calculated in sect. 4), Wp(n) are the weight factors from tables 1 or 2 as described above, I~b(0)l2 is the square of the q u a r k wave f u n c t i o n inside the n u c l e o n at the origin*, a n d describes the effective l u m i n o s i t y (flux) available to the basic cross section %. A value for Iff(0)l 2 is estimated in sect. 4. The structure of o0 itself is 1

O~2 GUM

/

00 = ~ 7 t ' ~ , r r l q q ?

1,1X

2

\

,

(3.5)

where (mq2q) is the average total c e n t e r - o f - m o m e n t u m energy squared p r o v i d e d b y the two quarks which fuse. * Alternatively in the language of the operator product expansion this can be viewed as the sum of the squares of the three-quark operators between a proton state and all possible final states.

292

T. Goldman, D.A. Ross / Proton lifetime

In the static quark approximation (for which the SU(6) analysis is relevant), = (3 raN) 2" We have attempted to estimate the sensitivity of this value to the assumptions of the static quark model. To this end, we have constructed a graph of (m2qq > as a function of the mass of the final state q~ pair, using effective two-body (lepton plus q~l pair) kinematics. The average here is taken over the momentum and direction of the spectator quark in the initial nucleon (we have used P ( x ) = 4(1 - x) 3 as a convenient rough approximation for the momentum distribution of the spectator quark). The resulting curve is shown in fig. 11. According to semilocal duality, summing over all possible final states is equivalent to taking the mean value of this curve over the range of the nucleon mass. This gives

mqq z

< >--~(0.6 -- 0 . 7 ) ~4m N2,

(3.6)

where the variation depends on whether the average is taken over m N or m 2. Furthermore, if one takes the relative weights of hadrons with a given total mass to be as given by the curve, then the implied mean pion multiplicity agrees well with that found experimentally for a total hadronic center-of-momentum energy squared FIG II

co

~.

.7

g z ~

.-~

(:3

average : .7 x

m

0

9

.2

d i

"6 g L~

.5

K/

~

Pair M o s s of A n t i - q u a r k

p~,ua

9 4 = end point

and S p e c t a t o r in GeV

Fig. 1I. Mean energy-squared of the quark pair that fuses from the nucleon undergoing decay, as a function of the pair mass of the outgoing antiquark and spectator quark from the nucleon, with the constraints of phase space (only) applied.

T. Goldman, D.A. Ross / Proton lifetime

293

value given in eq. (3.6). The closeness of the results of these various calculations reassures us that a parton-model like calculation such as is done here is not likely to be wrong by more than a factor of 2. Independent of detailed calculation, it is interesting to note that the q~l final-state mass is easily influenced by gluon exchange between the initial and final states. The transmission of a set of soft gluons whose invariant mass vanishes and whose total energy is between 200 and 300 MeV is all that is required to shift the q~ mass from the region of the p mass to that of a single pion (and the couplings of such soft gluons are large). It is therefore surely very difficult to predict the partial rates to specific hadronic modes*. Nevertheless the parton model may allow a significant statement to be made on the basis of the mean pion multiplicity, (n~ }. In the static quark model (which as we have shown is not very different from other approaches) no more than 0.6 GeV of hadronic energy is liberated at rest (i.e., in its own center-of-momentum frame). One m a y reasonably ask for the mean pion multiplicity of a hadronic system at or below this energy and with enough spread so as not to be proscribed to a particular resonance. D a t a on (n~> in e + e - annihilation at low energies (or extrapolations down from high energies) [28] must have one unit subtracted as this system is kinematically constrained to have at least two pions in the final state, whereas the nucleon decay need only have one. Similarly, extrapolations for (n,~ } pions from p-p scattering [28] must have one unit added. These give similar results, namely that one m a y expect (n= } ~ 1.5,

(3.7)

in proton decay. Simple application of Poisson statistics then produces the conclusion that more than 70% of the time, the hadronic final state consists of a single pion. When combined with the positron branching ratios f r o m tables 3 and 4, we conclude that the branching ratio into a positron and a single pion is likely to be in excess of 50%. This particular point may be of some experimental significance [3].

4. Enhancement effects from the operator product expansion We now calculate the enhancement effects that arise due to the difference in scale between M x and the nucleon mass. These may be thought of as the summation (and exponentiation) of leading logarithms of M x / m p. T h e matrix elements for the proton decay processes shown in figs. 8 to l0 can be expressed as

eiq.x d4x (q2 _ M xz ) < P [ J i ( x ) J J ( O ) l X ) '

(4.1)

* Donoghue [27] has attempted to calculate directly the individual hadronic modes using the bag model to calculate the wave-function overlaps. This calculation does not allow for the corrections due to soft gluons propagated between initial and final states.

294

T. Goldman, D.A. Ross / Proton lifetime

where p is a proton state and X is some final state. It can be seen from eq. (4.1) that the product of the two currents Ji(x)JJ(O) is dominated by the short distance region ] x l ~ 1 / M x and thus we can make a short-distance expansion

o(,) --~x '

~z~=f d4xeiq'xE C~(x)(Pl~lX)8(q2- M~)

(4.2)

where t9,4 is the set of spin-zero leading twist operators with baryon number - 1 and lepton number + 1. The Fourier transform of the coefficient functions CA(q 2) obey a Callan-Symanzik equation

k= ~1,2,3 flk-~ak +

31nq2

)

½7A G ( q 2)

=

0

(4.3)

,

where 7A is the anomalous dimension of the operator tgA:

~,.~=

Y~

k= 1,2,3

yo , , , , ~ak -g

1

+ ....

(O~[G [j~)

d (alt~Alfl) dln~,

.

(4.4)

The couplings of the W, B, and G (gluons) to the fermions are needed for this calculation and so for completeness, we note them here:

g(UL'dL)(W~'I~T]'Yt~(%LI"}-g(eR,PeR)(W~'2~tlL q')~z/ J

g

+

dRBd R +

g

ffL]JUL +

2v'i3 +

3g

2Vi~

g

(e+)

1

t~

~LBd L -- - -2g U L , ~ U- cL

3g U~Be~ +

x/~ ~eR,~ Vep.

-c

3g

2vq3

R pcg

c

e_+ Rie + R

g dR'Y#(~a" I~.) dR +gdL]/tX(~rt~'½~)dL

-~ gUR"/'a(~ • l)k) UR -'l- g/dL'y'a(~ • / X) UL ,

(4.5)

(where ~- are the Pauli spin matrices and ~ are the matrices of SU(3)), plus analogous repetitions for the other generations; sums over color are understood in the W and B quark terms. For strong interactions the contributions to YA are calculated in leading order from the graphs of fig. 12, where the gluon corrections may couple to any pair of the three hadron legs. Since the gluons do not distinguish between the chirality of the fermions to which they couple, the strong interaction contribution to the

T. Goldman, D.A. Ross / Proton lifetime

295

Fig. 12. Graphs contributing to the leading-order calculation of the strong interaction induced anomalous dimension of the operators associated with the graphs in figs. 8-10. a n o m a l o u s dimension is the same for all operators and therefore we suppress the suffix A. F r o m the solution to eq. (4.3) we see that the effect of changing scale f r o m ft 2 to Q2 due to the strong interactions is given by

(4.6)

where ~,o has been calculated in ref. [5] and f o u n d to be equal to four. Since the value of fl0, changes when a new flavor threshold is crossed, it is slightly more accur.ate to take this into account. We therefore write the strong interactions e n h a n c e m e n t effect on the rate

,

As=

ffs(4m2)

as(4m~)

ffs(4mt2)

(4.7)

O~GUM

1

where m N is the nucleon mass. We take ffs(m~) = 5 which is slightly lower than the value for spacelike Q 2 ~ r n 2 due to the M o o r e h o u s e - P e n n i n g t o n - R o s s [29] effect. The net size of A s is approximately five. The weak interactions* do d e p e n d on chirality and yet it turns out that when the analogous graphs (some of which are shown in fig. 13) are calculated the value of yo is the same for all operators corresponding to the graphs of figs. 8 - 10. In each graph, only two lines have the correct chirality to participate in the weak interactions and so we have v° = d -

2 G,

(4.8)

where d is the contribution f r o m fig. 13a a n d d , is the contribution from fig. 13b or c. In the F e y n m a n gauge** we find d = 6 a n d d ~ = ¼ . Since ffw 'freezes' at Q 2 ~ 4 M 2 by the A C theorem and n o thresholds are crossed above Q 2 ~ 4 M ~ v we * The effect of weak and electromagnetic interactions on the enhancement factor has also been calculated by Wilczek and Zee [30]. , ,,yo is of course gauge independent but its decomposition into contributions from different graphs depends on the gauge.

296

T. Goldman,D.A. Ross / Protonlifetime

qR

qc

qR

qc

qR

qL

Fig. 13. Typical graphs contributing to the leading-order calculation of the weak interaction induced anomalous dimension of the operators involvedin nucleon decay, may write the weak interaction enhancement

A w = ( a w (,4aMG~UvM ) ) 271/9

(4.9)

where we have included one Higgs doublet in the calculation of fl~. The contribution to 7A from the interactions with the B field must also be taken into consideration. Here we do find a slightly different value for operators corresponding to different graphs in figs. 8-10. When we apply B boson corrections to the operator graphs corresponding to fig. 9 we find d - - ~ while those for figs. 8 and 10 give d = ~-. Subtracting d¢. for each external line (in this case all four external lines couple to B) with their appropriate couplings we find To -_ T6 23 for fig. 9 operators and ~II for fig. 8 and fig. 10 operators. Note that the fact that these quantities are positive gives us a further enhancement due to the B interactions. In the analogous equation to (4.6) fl0I has the opposite sign from fl03but now ~B(M2w) is smaller than aGO M SO we again get an enhancement

A,=

aGUM

B(4M v)

aGUM

or

AB=

B(4M )

,

(4.10)

in the two cases described above. Below Q 2 ~ 4 M ~ v , ~a also freezes but there remains one linear combination of fib and ~w, the electromagnetic coupling constant, which continues to vary with Q2. This variation, however, is extremely small (of order (aem/~")ln(Mw/mN)~ 1%) so we neglect it. If the product of A B and A w is denoted by A wa [the notation used in eq. (3.4)] we find that A w B ~ 2 * . In obtaining this, we have used the smaller value of A B leaving an additional amplitude enhancement factor of 1.07 [the square root of the ratio of the two values in (4.10)] to multiply the graphs of fig. 9. This means that the amplitude weight factor 2 coming from the constructive interference for figs. 9a to c should be replaced by 2.14 and fig. 9d should have an amplitude weight factor * It is not so surprising that weak interaction corrections are capable of producing a factor of 2 enhancement in rate w h e n one remembers the large lever arm coming from the ratio of the mass scales, so that the expansion parameter is actually (aw/Z,) ln(Mx/Mw)~0.4.

T. Goldman, D.A. Ross / Proton lifetime

297

1.07. We do this just for convenience so that the enhancement effect m a y be written as an overall factor in eq. (3.3). We now come to the estimate of the matrix element of the operators between a proton and any final state, or alternatively to the estimate of the quark wave function inside the nucleon at the origin. This has been estimated by Finjord [31] in a study of weak hyperon decay rates in which the SU(6) baryon wave functions and the Q C D enhancement effects were taken into consideration in the same way as in the above analysis. It is therefore consistent for us to use his fit to the hyperon decay rates. This gives us the value ]~b(0)[ 2 = 1.1 ×

(4.11)

10 -3 GeV 3 .

F r o m the discussions in ref. [31] it appears that we should allow a possible 30% uncertainty in this estimate. A further uncertainty can arise from higher-order Q C D corrections. The estimate of [~p(0)[2 in ref. [31] arises from the baryonic matrix element of a four-quark operator, whereas for proton decay we are interested in the matrix element of a three-quark operator between a proton and any final state. In the tree diagram approximation these are the same. The leading Q C D corrections are exactly the leading-order enhancement factors that have correctly been taken into account. In the next order we expect corrections which are of the form

A 3

°

2(/3o) 2

~_Ba

_

__+BA,

2/3 °

2(/3o) 2

,

(4.12)

where A and A' are three and four-quark operators, respectively, and B~(BA, ) are the finite parts of the one-loop correction to the coefficient function for the operators A and A', respectively. We have not calculated these quantities but previous experience [32] has led us to the approximate relation

yo --/3o

(4.13)

(to within 50%), so that there is a partial cancellation and eq. (4.12) is dominated by B A - B ~ , . These quantities by themselves are renormalization-prescription dependent, but it is in the minimal subtraction scheme (or truncated minimal subtraction scheme) that eq. (4.13) is valid. We have calculated BA and find BA =

1 4 + 81n2 3 ~6.5,

(4.14)

0 2 ~ 1. Thus we estimate that the uncertainty on compared with which 3,°fl 3 31/ 2 (/33) the amplitude for proton decay arising from such effects is no greater than 25% (a

298

T. Goldman, D.A. Ross / Proton lifetime

50% error in the rate). Thus we allow a total uncertainty of a factor of two in our value for I q~(0)l 2. 5. Discussion and conclusions

Combining our results from sects. 2 - 4 , we find our optimal values for the nucleon lifetime of • p = 1.6 ×

10 30

years,

% = 2.6 x

10 30

years.

(5.1)

There is an uncertainty in these values of about a factor of 100 which is composed of the following contributions: (i) A factor of 5 from our estimate of the uncertainty in A ~ [we have taken 400 MeV in eq. (5.1)]. Here we note that if A ~ s < 200 MeV the SU(5) model would be already ruled out by the experimental lower limit on the proton lifetime. This uncertainty can be reduced by more accurate experimental determination of the strong coupling constant at some fixed Q2. (ii) A factor of 6 due to the possibility of the concatenation of uncertainties due to changes in the estimated t-quark mass, the masses of the Higgs scalars and the higher-order effects in the translation of the strong interaction coupling constant from the M.S. scheme to symmetric m o m e n t u m subtraction, and a very small estimated uncertainty due to the effects of /3 in three loops. Each one of these uncertainties cannot introduce a correction of more than a factor of 2 in the rate, but in the unlikely event of them all having their m a x i m u m estimated possible value and the same sign we must allow a full factor of 6. (iii) A factor of 2 uncertainty from phase-space effects. This includes the discrepancy between the non-relativistic and the extreme relativistic parton model calculations and a study of what is the expected invariant mass of the two quarks which fuse in the fundamental process. (iv) A factor of 2 for I~b(0)l2 which includes a rough estimate of the effect of higher-order Q C D corrections to the matrix elements of the three-quark operator used in proton decay relative to the matrix elements of the four-quark operators used in the analysis of the decay rate of hyperons, from which Iq~(0)l 2 was taken. If all these uncertainties were resolved in a manner that reduced the proton lifetime, the SU(5) theory would already be ruled out [33]. Allowing all of the uncertainties to act to increase the proton lifetime we arrive at the (conservative) upper bounds of Zp < 2

x 10 32

years,

%< 3

x 10 32

years.

(5.2)

T. Goldman, D.A. Ross / Proton lifetime

299

F a i l u r e to observe d e c a y s at least as fast as the a b o v e rates s h o u l d be v i e w e d as an i r r e d e e m a b l e failure of the SU(5) g r a n d unified t h e o r y since a d d i n g m o r e f e r m i o n s (which does i n c r e a s e the p r o t o n lifetime b y a b o u t a n o r d e r of m a g n i t u d e for e a c h g e n e r a t i o n of f e r m i o n s d u e to the sensitivity o n the n u m b e r of f e r m i o n s of the t w o - l o o p fl effect) will surely ruin the successful p r e d i c t i o n of the b o t t o m q u a r k m a s s a n d p r o b a b l y also the p r e d i c t i o n for sin 2 0 w. Such a n a b s e n c e of p r o t o n d e c a y w o u l d e l i m i n a t e the use of this effect to d e t e r m i n e the c o r r e c t g r a n d u n i f i e d t h e o r y (if such a t h e o r y exists) a n d leave us in the p o s i t i o n of l o o k i n g a r o u n d r a n d o m l y for w h a t e v e r new degrees of f r e e d o m m a y a p p e a r (e.g., m o r e fermions, m o r e interactions, q u a r k substructure, etc.) or, as we h a v e suggested b e f o r e [34] l o o k i n g for specific effects at specific, b u t very large, e n e r g y scales, w h i c h w o u l d u n d o u b t e d l y be e x p e r i m e n t a l l y very difficult, if at all possible. W e are grateful to the f o l l o w i n g for useful c o n v e r s a t i o n s a n d e n c o u r a g e m e n t : J. D o n o g h u e , J. Ellis, S. H a w k i n g , W . M a r c i a n o , M. M a c h a c e k , E.A. Paschos, D. Politzer, P. R a m o n d , S.P. R o s e n , M. Sher a n d last b u t n e v e r least S. W o l f r a m .

Note added in proof A f t e r this w o r k was c o m p l e t e d , we received the p r e p r i n t s given in ref. [35]. These p a p e r s all agree with o u r results a n d with each o t h e r to within a n o r d e r of m a g n i t u d e for the p r o t o n lifetime. H o w e v e r , there is wide d i s a g r e e m e n t o n specific b r a n c h i n g ratios.

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