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Parity violating longitudinal muon polarization in K+ → π+μ+μ− beyond leading logarithms
22 September1994 PHYSICS
ELSEWIER
LETTERS
B
Physics Letters B 336 (1994) 263-268
Parity violating longitudinal muon polarization in K+ + T$$beyond leading logarithms * Gerhard Buchallaa,
Andrzej J. Buras b
a Max-Planck-Insntutfitr Physlk - Werner-Hersenberg-lnstnid - Fohnnger Rmg 6, D-80805 Munchen. Germany b Technlsche Unwersltat Munchen, Physlk Department, D-85748 Garchmg, Germany Received 14 July 1994 Editor PV Landshoff
Abstract
We generalize the extstmg analyses of the panty violating muon polanzahon asymmetry ALRm ti -+ ~‘~+cL’,u-beyond the leading loganthmic approximation. The mcluslon of next-to-leading QCD corrections reduces the residual dependence on the renormahzahon scales, which IS quite pronounced m the leadmg order. This leads to a considerably improved accuracy m the perturbative calculation of the short distance dommated quantity ALR,Accordmgly this will also allow to obtain better constraints on the Wolfenstem parameter e from future measurements of ALR For -0 25 5 e 5 0 25, &b = 0 040 f 0 004 and m, = (170f 20) GeV we find 3 0 10m3 5 ~ALR]5 9 6 10m3
It has been pointed out by Savage and Wise [ 1 ] that measurements of muon polarization in K+ + T+,u~,udecay can give valuable information on the weak rmxmg angles and m particular on the parameter Q m the Wolfenstem parametrization [ 21 Indeed as shown m [ 1,3], the parity-violating asymmetry
(1) is dommated by the short distance contributions of Z-penguin and W-box diagrams with internal charm and top quark exchanges, while the total rate is completely determmed by the one-photon exchange amplitude The interference of this leading amplitude with the small short distance piece 1s the source of the asymmetry ALR. Here Ia (IL) ts the rate to produce right- (left-) handed p +, that is p”+ with spin along (opposite to) its three-momentum direction The factor r arises from phase space integrations It depends only on the particle masses mK, m, and m,, on the form factors of the matrix element (T+ I(&f) “-_A1K+), as well as on the form factor of the K+ --f T + y * transition relevant for the one-photon amplitude In addrtron r depends on a possible cut which may be imposed on 0, ;he angle between the three-momenta of the p- and the pion m the rest frame of the ,X+,X- pair. Without any cuts one has r = 2 3 [ 31. If cos 6 is restricted to he m the region * Supported by the German Bundesmnustenum SCl-CT91-0729
fur Forschung
und Technologe
0370-2693/94/$07 00 @ 1994 Elsevler Science B V All nghts reserved SSDIO370-2693(94)00997-X
under contract 06 TM 732 and by the CEC science project
G Buchalla. A J Buras / Physics Letters B 336 (1994) 263-268
264
-0 5 5 cos 8 5 1 0, this factor 1s Increased to I = 4 1 As discussed m [ 31, such a cut m cos 0 could be useful since it enhances ALR by 80% with only a 22% decrease m the total number of events. Ret is a purely short dtstance function depending only on CKM parameters, the QCD scale Am and the quark masses m, and m, We will discuss tt m detail below ALR as given m ( 1) has also been considered by B&anger et al [4], who emphasized its close relation to the short distance part of the decay amplitude KL + ,ucL+,u- Unfortunately the authors of Ref [4] did not include the internal charm contnbutton to A L,R As we will show explicitly below the charm contrtbutton cannot be neglected as its presence increases the extracted value of e by roughly A@ = 0 2 Let us briefly summarize the theoretical situation of A LR. The “kinemattcal” factor r can be essentially obtamed from experimental mput on the particle masses and the form factors The form factor f describing the K+ + ~+y* vertex has been dtscussed m detatl m [5] within choral perturbation theory. While the tmagmary part can be reliably predicted, the real part 1s only determmed up to a constant to be fitted from experiment On the other hand, data on Ki -+ T + e + e - decay [ 61 allow to extract the absolute value of this form factor directly We will adopt this approach, following [ 31. Smce the imaginary part of f is qmte small [ 3,5], we then also have the real part Re f In prmctple Im f could yield an extra contrtbutton m ( 1) proporttonal to Im ,.$ We have checked, based on the approach of [ 31 that this contrtbutton is below 1% of the dominant part shown m ( 1) and can therefore be safely neglected Clearly the factor r involves some uncertamty due to the experimental errors m the form factors, which can however be further reduced by future improved measurements For the present discusston we will assume fixed numerical values for r Besides the short distance part of A LR there are also potential long distance contributtons commg from two-photon exchanges, which have also been discussed by the authors of [3]. These are difficult to calculate m a reliable manner, but the estimates given m [ 31 indicate that this contnbutton is substantially smaller than the short distance part, although tt cannot be fully neglected Therefore the short distance effects are expected to safely dominate the quantity ALR and we shall concentrate our dtscusston on this part, keeping m mmd the possible existence of non-negligible long dtstance corrections The short distance physics leadmg to ALR is generally considered to be very clean, as it can be treated wtthm a perturbattve framework However this does not mean that tt 1s free of theorettcal uncertamttes. An mdtcatton of the mvolved error due to the necessary truncation of the perturbation series m the strong couplmg constant cy, can be obtained by studymg the senstttvtty of a phystcal quantity to the relevant renormahzatton scales on which rt should not depend m prmctple The existmg short distance calculations of ALR [ 1,3,4] mclude QCD correcttons m the leading logarrthrmc approxtmation (LLA) [ 71 As tt turns out, they suffer from sizable theoretical uncertamties due to the restdual scale dependences. The mam purpose of our letter is to extend the analyses of [ 1,3,4] beyond the leading logartthmtc approxtmatton thereby reducing considerably the theoretrcal uncertamties m question To this end we will use our next-to-leadmg order analysts of KL + ,xfpu- presented m i&91 Our dtscusston of the Cabtbbo-Kobayasht-Maskawa matrix will be based on the standard parametrization [lo], which can equtvalently be rewritten m terms of the Wolfenstem parameters (A, A, p, r]) through the defimttons [ 111 ~23 G AA2
s13e -”
=
AA3(p - IT)
The umtartty structure of the CKM matrix 1s conventtonally represented complex plane with coordmates (0, 0) , ( 1, 0) and ( 0, 75) where
(2)
through the umtarrty
triangle
m the
(3) To better than 0.1% accuracy
0 = e( 1 - A2/2) and 7~= T( 1 - A2/2)
G Buchda,
[ 8,9] it 1s stratghtforward
Followmg We find
A J Bums / Physm
to generahze
Letters B 336 (1994) 263-268
the expression
265
for Re 5 of [ 31 beyond leading logarithms.
Re(=tc.
(4)
A4
K=
= 1.66 * 1o-3
(5)
Here A = IV,,l = 0 22, en*@ w = 0.23, xI = rnf/Mk and ,$ = yzK;d The function contribution, is given by
x 4-x Xl(x) = s l_x [
+
(6)
3x (1 -x)2
lnx
and 4x + 16x2 + 4x3 3(1-x)2
+ 8~~‘(~) JX
where xP = ,u*/M$
for the top
ZYt (n)
Y(x) = X)(x) +
Yi(x) =
Y, relevant
1
(7)
4x - ;lxt;);s
- x4 lnx + 2x - 14x2 + x3 - x4
-
2(1-x)3
ln*x+
2x+x3 (1 _x)2~2(l
lnx P
-x)
(8)
with ,U = put = O(m,)
and
(9) The QCD correction
Yt has been calculated
m [ 81 Next
(10) where Y,, represents rithmic approximation YNL = CNL
the renormalization group expression for the charm contribution (NLLA) calculated in [9] It reads
=
x(m) -KF 32
loga-
(--l/2)
- BNL
(11)
where CNL is the Zpengum part and BhL”*) is the box contribution, with weak isospm T3 = -l/2. We have CNL
m next-to-leading
24
48 ?K+
+ ;K-
- ?K,,
[(
relevant for the case of final state leptons
15212 4?r -+m(l-K?) >( %(P)
+(l-ln$j(l6K+-8K_)-~K+-~K_+~Kss _ 81448 ZZK-+
4563698 pK33
144375
>I
(12)
where (13)
266
G Buchalla, A J Buras / Physics Letters B 336 (1994) 263-268
Table 1 The function PO for vanous A=
AK
0 0 0 0
NL
m, [GeV]
lGev1
20 25 30 35
&=I&
pm
and m,
--I*
K_=KF
&
1 25
1 30
1 35
0 132 0 135 0139 0 142
0 141 0 145 0 148 0152
0 0 0 0
150 154 158 162
= Kg
(14)
=FK+-K2)(--&+~(1-K;,))
329 P2 - In 2 - 12 + 152r2K2 625
30581 + -KK2
(15) I
Here K2 = K-‘/25, m = m,, x = m2/M$ In ( 12) - ( 15) the two-loop expression has to be used for cy,( pu> and p = pu, = O(m,). The explicit ,u-dependences m the next-to-leadmg order terms (8), (12) and (15) cancel the scale ambtguity of the leading contributions to the considered order m CY~The consequences of this feature will be discussed later on Numerical values of PO are given m Table 1 where m, E ti, (m,) It is evident from ( 1) and (4) that, given IALR], one can extract Re At
(16) Furthermore,
using the standard parametrization
of the CKM matrix we obtam from the definition
1 + 4slzct2Re &IS& - (2s12ci21mht/s&)2
- 1 + 2$,
2c223 s212
(3)
(17)
Up to the very accurate approximations that &V,*b IS real (error below 0 1%) and cts = 1 (error less than 10P5) ( 17) is an exact relation. Using the excellent approximatton Im hr = vA2A5 [ 111, we see that a measured value of Re Ar determmes by means of (17) a curve m the (e, T)-plane Since the dependence on Im hr is however very small, this curve will be almost parallel to the ~-axis Thus, knowledge of (ALR], hence Re A,, implies a value for Q (or 8) almost independently of Im ht. For simphcity we shall neglect Im At m ( 17) completely, which introduces a change m 0 of at most 0.01 It is evident that the more general treatment can be easily restored tf desired Note that the charm sector contributes to Ej the non-negligible portion A &harm M Z’o/(A2Y(x,)) In order to demonstrate we consider the followmg ALR = (6 0 f 0.6)
(18)
M0 2
briefly the phenomenological consequences of the next-to-leading order calculation scenario We assume that the asymmetry ALR is known to wtthm flO% 1O-3
(19)
where a cut on cos 0, -0 5 5 cos 0 5 1 0, is understood m,=(170&5)GeV
m,=(130&005)GeV
Next we take (m, E rii, (m,) )
V,b=O040fOOOl
(20)
G Buchalkz, A .I Buras / Phystcs Letters B 336 (1994) 263-268
261
Table 2 0 determmed from ALR for the scenano described m the text together with the uncertamhes related to various mput parameters
-0 06
0
A==
UALR)
A(m)
A(b)
A(m)
A($&
f0 13
f0 05
zt0 06
fool
fOO0
(030fO05)GeV
(21)
The errors quoted here seem quote reasonable if one keeps in mmd that it will take at least ten years to achieve the accuracy assumed m ( 19) The value of m, m (20) 1s m the ball park of the most recent results of the CDF collaboration [ 121. In Table 2 we have displayed the central value for 0 as it is extracted from A,x (( 16) and (17)) m our example, along with the uncertamties due to the parameters mvolved. This is intended to indicate the sensitivity of 0 to the relevant input The combined errors due to a simultaneous variatton of several parameters may be obtamed to a good approximation by simply adding the errors from Table 2. It is interesting to compare these numbers with the renormahzation scale ambiguities, which mevttably limit the accuracy of the shortdistance calculation. If we vary the scale m the charm- and in the top sector as 1 GeV 5 pu, < 3 GeV and 100GeV I ,u~ I: 300GeV, respectively, keeping all other parameters at their central values, we obtam the following range for e -015
(NLLA) (LLA)
(22) (23)
We would like to emphasize the followmg points. - The error m 0 from (22)) which illustrates the theoretical uncertamty of the short distance piece alone, is not negligible It seems however moderate when compared to the errors shown m Table 2 We stress that (22) is based on the complete next-to-leading order result for Ret If only the leading log approximatton is used instead, the range obtained for e is by almost a factor of 3 larger (23) - The error m (22) is almost entirely due to the charm sector Indeed, if we vary only pc, keeping p, = m, fixed, the correspondmg interval for g reads (-0 14, -0 03) This illustrates once more, that the charm sector, being the dommant source of theoretical error m the short distance contribution to A,x, should not be neglected - In the case x << 1, which is relevant for the charm contribution, the function Y has a very special structure Expanding the renormalization group result YNLto first order in CX,one finds (here x = mz/M$)
We observe that the leadmg logarithms N x In X, present m the Z-pengum- and the box part, have canceled m YNL, leaving the subleadmg term x/2 as the only contributton m the limit cry, = 0. On the other hand QCD effects generate an cy,xln2 x “correction”, which is of the order c3( x In x), hence a leading logarithmic term’ As a first consequence the charm function Y is enhanced considerably (by a factor of N 2.5) through strong mteraction correcttons, compared to the non-QCD result (This feature is m a sense similar to the case of the rare decay b + sy.) A second point is that the x/2 term, though formally subleadmg, is important numerically Workmg within LLA one is then faced with the problem of how to deal with this term since it should strictly speakmg be omitted m this approximation. Let us illustrate this issue m terms of the c determmation m our above example We find c = -0 12 if we use the LLA formulae (with ,u~ = m,, pt = mt) and stmply add the x/2 piece By contrast, omittmg this term and using the strict leading log result
268
G Buchalla, A J Buras /Physrcs
L.&ten B 336 (1994) 263-268
we obtam @= -0 20 The scale ambrgumes are very similar m both cases, roughly three times as big as m the next-to-leadmg order drscussed above. For defimteness we have included the x/2 part to obtain (23) The problem of the x/2 term is naturally removed m the next-to-leadmg logarithmic approximation ( YNL) where this contributron 1s consistently taken mto account. Finally we give the standard model expectation for ALR, based on the short distance contribution m (4), for the Wolfenstem parameter @ m the range -0 25 I @ I 0 25, Vcb = 0.040 f 0 004 and m, = (170 f 20) GeV. Including the uncertamtres due to m,, Am, pu,and ,q and rmposmg the cut -0.5 I cos 0 < 1, we find 30
lop3 5 ]ALR/ 5 9 6
lop3
(25)
employing next-to-leading order formulae Antrcrpating improvements in Vcb, m, and e we also consider a future scenario m which e = O.OOfO 02, V& = 0 040f0.001 and m, = (170 f5) GeV The very precise determmation of Q used here should be achieved through measuring CP asymmetries m B decays in the LHC era [ 131. Then (25) reduces to 4.8
low3 < IALR] 5 6 6
lop3
In both of the scenarios the lower (upper) limit for ALR would be smaller by 0 6 10v3 (1 3 10e3) rf the charm contribution was omitted In this letter we have generalized the short distance calculation of the muon polarization asymmetry ALR m the decay Kf -+ rr+,u”+pu- to next-to-leading order m QCD We furthermore discussed the theoretical uncertamtres involved m thus analysis We have demonstrated that the complete next-to-leading order calculatron achieves a reductron of the rather large scale ambigumes m leading order by a factor of N 3 and is necessary to provide a satisfactory treatment of ALR This is particularly important since long distance contributtons to ALR seem to be small, though perhaps not fully negligible [ 31. In any case a measurement of ALR would yreld a very mterestmg and useful piece of mformation on short distance flavordynamics and the umtarity triangle which is worth pursumg m future experiments References [ 11 M Savage and M Wise, Phys Lett B 250 ( 1990) 151 [2] L Wolfenstem, Phys Rev Lett 51 (1983) 1945 [ 31 M Lu, M Wise and M Savage, Phys Rev D 46 (1992) 5026 [4] G B&anger, C Q Geng and P Turcotte, Nucl Phys B 390 (1993) 253 [5] G Ecker, A Plch and E de Rafael, Nucl Phys B 291 (1987) 692 [6] C Alhegro et al, Phys Rev Lett 68 (1992) 278 [7] J Elhs and J S Hagehn, Nucl Phys B 217 (1983) 189, VA Novlkov, M A Shlfman, A I Vamshtem and VI Zakharov, Phys Rev D 16 ( 1977) 223 [8] G Buchalla and A J Buras, Nucl Phys B 400 ( 1993) 225 [9] G Buchalla and A J Buras, Nucl Phys B 412 (1994) 106 [ 101 Partxle Data Group, Phys Rev D 45 (1992), No 11 part II [ 111 A J Bums, ME Lautenbacher and G Ostermruer, MPI-Ph/94-14, TUM-T31-57/94 [ 121 F Abe et al (CDF Collaboration) FERMI-PUB-94/097-E [ 131 A J Buras, MPI-PhT/94-25, TUM-T31-61/94
ELSEWIER
LETTERS
B
Physics Letters B 336 (1994) 263-268
Parity violating longitudinal muon polarization in K+ + T$$beyond leading logarithms * Gerhard Buchallaa,
Andrzej J. Buras b
a Max-Planck-Insntutfitr Physlk - Werner-Hersenberg-lnstnid - Fohnnger Rmg 6, D-80805 Munchen. Germany b Technlsche Unwersltat Munchen, Physlk Department, D-85748 Garchmg, Germany Received 14 July 1994 Editor PV Landshoff
Abstract
We generalize the extstmg analyses of the panty violating muon polanzahon asymmetry ALRm ti -+ ~‘~+cL’,u-beyond the leading loganthmic approximation. The mcluslon of next-to-leading QCD corrections reduces the residual dependence on the renormahzahon scales, which IS quite pronounced m the leadmg order. This leads to a considerably improved accuracy m the perturbative calculation of the short distance dommated quantity ALR,Accordmgly this will also allow to obtain better constraints on the Wolfenstem parameter e from future measurements of ALR For -0 25 5 e 5 0 25, &b = 0 040 f 0 004 and m, = (170f 20) GeV we find 3 0 10m3 5 ~ALR]5 9 6 10m3
It has been pointed out by Savage and Wise [ 1 ] that measurements of muon polarization in K+ + T+,u~,udecay can give valuable information on the weak rmxmg angles and m particular on the parameter Q m the Wolfenstem parametrization [ 21 Indeed as shown m [ 1,3], the parity-violating asymmetry
(1) is dommated by the short distance contributions of Z-penguin and W-box diagrams with internal charm and top quark exchanges, while the total rate is completely determmed by the one-photon exchange amplitude The interference of this leading amplitude with the small short distance piece 1s the source of the asymmetry ALR. Here Ia (IL) ts the rate to produce right- (left-) handed p +, that is p”+ with spin along (opposite to) its three-momentum direction The factor r arises from phase space integrations It depends only on the particle masses mK, m, and m,, on the form factors of the matrix element (T+ I(&f) “-_A1K+), as well as on the form factor of the K+ --f T + y * transition relevant for the one-photon amplitude In addrtron r depends on a possible cut which may be imposed on 0, ;he angle between the three-momenta of the p- and the pion m the rest frame of the ,X+,X- pair. Without any cuts one has r = 2 3 [ 31. If cos 6 is restricted to he m the region * Supported by the German Bundesmnustenum SCl-CT91-0729
fur Forschung
und Technologe
0370-2693/94/$07 00 @ 1994 Elsevler Science B V All nghts reserved SSDIO370-2693(94)00997-X
under contract 06 TM 732 and by the CEC science project
G Buchalla. A J Buras / Physics Letters B 336 (1994) 263-268
264
-0 5 5 cos 8 5 1 0, this factor 1s Increased to I = 4 1 As discussed m [ 31, such a cut m cos 0 could be useful since it enhances ALR by 80% with only a 22% decrease m the total number of events. Ret is a purely short dtstance function depending only on CKM parameters, the QCD scale Am and the quark masses m, and m, We will discuss tt m detail below ALR as given m ( 1) has also been considered by B&anger et al [4], who emphasized its close relation to the short distance part of the decay amplitude KL + ,ucL+,u- Unfortunately the authors of Ref [4] did not include the internal charm contnbutton to A L,R As we will show explicitly below the charm contrtbutton cannot be neglected as its presence increases the extracted value of e by roughly A@ = 0 2 Let us briefly summarize the theoretical situation of A LR. The “kinemattcal” factor r can be essentially obtamed from experimental mput on the particle masses and the form factors The form factor f describing the K+ + ~+y* vertex has been dtscussed m detatl m [5] within choral perturbation theory. While the tmagmary part can be reliably predicted, the real part 1s only determmed up to a constant to be fitted from experiment On the other hand, data on Ki -+ T + e + e - decay [ 61 allow to extract the absolute value of this form factor directly We will adopt this approach, following [ 31. Smce the imaginary part of f is qmte small [ 3,5], we then also have the real part Re f In prmctple Im f could yield an extra contrtbutton m ( 1) proporttonal to Im ,.$ We have checked, based on the approach of [ 31 that this contrtbutton is below 1% of the dominant part shown m ( 1) and can therefore be safely neglected Clearly the factor r involves some uncertamty due to the experimental errors m the form factors, which can however be further reduced by future improved measurements For the present discusston we will assume fixed numerical values for r Besides the short distance part of A LR there are also potential long distance contributtons commg from two-photon exchanges, which have also been discussed by the authors of [3]. These are difficult to calculate m a reliable manner, but the estimates given m [ 31 indicate that this contnbutton is substantially smaller than the short distance part, although tt cannot be fully neglected Therefore the short distance effects are expected to safely dominate the quantity ALR and we shall concentrate our dtscusston on this part, keeping m mmd the possible existence of non-negligible long dtstance corrections The short distance physics leadmg to ALR is generally considered to be very clean, as it can be treated wtthm a perturbattve framework However this does not mean that tt 1s free of theorettcal uncertamttes. An mdtcatton of the mvolved error due to the necessary truncation of the perturbation series m the strong couplmg constant cy, can be obtained by studymg the senstttvtty of a phystcal quantity to the relevant renormahzatton scales on which rt should not depend m prmctple The existmg short distance calculations of ALR [ 1,3,4] mclude QCD correcttons m the leading logarrthrmc approxtmation (LLA) [ 71 As tt turns out, they suffer from sizable theoretical uncertamties due to the restdual scale dependences. The mam purpose of our letter is to extend the analyses of [ 1,3,4] beyond the leading logartthmtc approxtmatton thereby reducing considerably the theoretrcal uncertamties m question To this end we will use our next-to-leadmg order analysts of KL + ,xfpu- presented m i&91 Our dtscusston of the Cabtbbo-Kobayasht-Maskawa matrix will be based on the standard parametrization [lo], which can equtvalently be rewritten m terms of the Wolfenstem parameters (A, A, p, r]) through the defimttons [ 111 ~23 G AA2
s13e -”
=
AA3(p - IT)
The umtartty structure of the CKM matrix 1s conventtonally represented complex plane with coordmates (0, 0) , ( 1, 0) and ( 0, 75) where
(2)
through the umtarrty
triangle
m the
(3) To better than 0.1% accuracy
0 = e( 1 - A2/2) and 7~= T( 1 - A2/2)
G Buchda,
[ 8,9] it 1s stratghtforward
Followmg We find
A J Bums / Physm
to generahze
Letters B 336 (1994) 263-268
the expression
265
for Re 5 of [ 31 beyond leading logarithms.
Re(=tc.
(4)
A4
K=
= 1.66 * 1o-3
(5)
Here A = IV,,l = 0 22, en*@ w = 0.23, xI = rnf/Mk and ,$ = yzK;d The function contribution, is given by
x 4-x Xl(x) = s l_x [
+
(6)
3x (1 -x)2
lnx
and 4x + 16x2 + 4x3 3(1-x)2
+ 8~~‘(~) JX
where xP = ,u*/M$
for the top
ZYt (n)
Y(x) = X)(x) +
Yi(x) =
Y, relevant
1
(7)
4x - ;lxt;);s
- x4 lnx + 2x - 14x2 + x3 - x4
-
2(1-x)3
ln*x+
2x+x3 (1 _x)2~2(l
lnx P
-x)
(8)
with ,U = put = O(m,)
and
(9) The QCD correction
Yt has been calculated
m [ 81 Next
(10) where Y,, represents rithmic approximation YNL = CNL
the renormalization group expression for the charm contribution (NLLA) calculated in [9] It reads
=
x(m) -KF 32
loga-
(--l/2)
- BNL
(11)
where CNL is the Zpengum part and BhL”*) is the box contribution, with weak isospm T3 = -l/2. We have CNL
m next-to-leading
24
48 ?K+
+ ;K-
- ?K,,
[(
relevant for the case of final state leptons
15212 4?r -+m(l-K?) >( %(P)
+(l-ln$j(l6K+-8K_)-~K+-~K_+~Kss _ 81448 ZZK-+
4563698 pK33
144375
>I
(12)
where (13)
266
G Buchalla, A J Buras / Physics Letters B 336 (1994) 263-268
Table 1 The function PO for vanous A=
AK
0 0 0 0
NL
m, [GeV]
lGev1
20 25 30 35
&=I&
pm
and m,
--I*
K_=KF
&
1 25
1 30
1 35
0 132 0 135 0139 0 142
0 141 0 145 0 148 0152
0 0 0 0
150 154 158 162
= Kg
(14)
=FK+-K2)(--&+~(1-K;,))
329 P2 - In 2 - 12 + 152r2K2 625
30581 + -KK2
(15) I
Here K2 = K-‘/25, m = m,, x = m2/M$ In ( 12) - ( 15) the two-loop expression has to be used for cy,( pu> and p = pu, = O(m,). The explicit ,u-dependences m the next-to-leadmg order terms (8), (12) and (15) cancel the scale ambtguity of the leading contributions to the considered order m CY~The consequences of this feature will be discussed later on Numerical values of PO are given m Table 1 where m, E ti, (m,) It is evident from ( 1) and (4) that, given IALR], one can extract Re At
(16) Furthermore,
using the standard parametrization
of the CKM matrix we obtam from the definition
1 + 4slzct2Re &IS& - (2s12ci21mht/s&)2
- 1 + 2$,
2c223 s212
(3)
(17)
Up to the very accurate approximations that &V,*b IS real (error below 0 1%) and cts = 1 (error less than 10P5) ( 17) is an exact relation. Using the excellent approximatton Im hr = vA2A5 [ 111, we see that a measured value of Re Ar determmes by means of (17) a curve m the (e, T)-plane Since the dependence on Im hr is however very small, this curve will be almost parallel to the ~-axis Thus, knowledge of (ALR], hence Re A,, implies a value for Q (or 8) almost independently of Im ht. For simphcity we shall neglect Im At m ( 17) completely, which introduces a change m 0 of at most 0.01 It is evident that the more general treatment can be easily restored tf desired Note that the charm sector contributes to Ej the non-negligible portion A &harm M Z’o/(A2Y(x,)) In order to demonstrate we consider the followmg ALR = (6 0 f 0.6)
(18)
M0 2
briefly the phenomenological consequences of the next-to-leading order calculation scenario We assume that the asymmetry ALR is known to wtthm flO% 1O-3
(19)
where a cut on cos 0, -0 5 5 cos 0 5 1 0, is understood m,=(170&5)GeV
m,=(130&005)GeV
Next we take (m, E rii, (m,) )
V,b=O040fOOOl
(20)
G Buchalkz, A .I Buras / Phystcs Letters B 336 (1994) 263-268
261
Table 2 0 determmed from ALR for the scenano described m the text together with the uncertamhes related to various mput parameters
-0 06
0
A==
UALR)
A(m)
A(b)
A(m)
A($&
f0 13
f0 05
zt0 06
fool
fOO0
(030fO05)GeV
(21)
The errors quoted here seem quote reasonable if one keeps in mmd that it will take at least ten years to achieve the accuracy assumed m ( 19) The value of m, m (20) 1s m the ball park of the most recent results of the CDF collaboration [ 121. In Table 2 we have displayed the central value for 0 as it is extracted from A,x (( 16) and (17)) m our example, along with the uncertamties due to the parameters mvolved. This is intended to indicate the sensitivity of 0 to the relevant input The combined errors due to a simultaneous variatton of several parameters may be obtamed to a good approximation by simply adding the errors from Table 2. It is interesting to compare these numbers with the renormahzation scale ambiguities, which mevttably limit the accuracy of the shortdistance calculation. If we vary the scale m the charm- and in the top sector as 1 GeV 5 pu, < 3 GeV and 100GeV I ,u~ I: 300GeV, respectively, keeping all other parameters at their central values, we obtam the following range for e -015
(NLLA) (LLA)
(22) (23)
We would like to emphasize the followmg points. - The error m 0 from (22)) which illustrates the theoretical uncertamty of the short distance piece alone, is not negligible It seems however moderate when compared to the errors shown m Table 2 We stress that (22) is based on the complete next-to-leading order result for Ret If only the leading log approximatton is used instead, the range obtained for e is by almost a factor of 3 larger (23) - The error m (22) is almost entirely due to the charm sector Indeed, if we vary only pc, keeping p, = m, fixed, the correspondmg interval for g reads (-0 14, -0 03) This illustrates once more, that the charm sector, being the dommant source of theoretical error m the short distance contribution to A,x, should not be neglected - In the case x << 1, which is relevant for the charm contribution, the function Y has a very special structure Expanding the renormalization group result YNLto first order in CX,one finds (here x = mz/M$)
We observe that the leadmg logarithms N x In X, present m the Z-pengum- and the box part, have canceled m YNL, leaving the subleadmg term x/2 as the only contributton m the limit cry, = 0. On the other hand QCD effects generate an cy,xln2 x “correction”, which is of the order c3( x In x), hence a leading logarithmic term’ As a first consequence the charm function Y is enhanced considerably (by a factor of N 2.5) through strong mteraction correcttons, compared to the non-QCD result (This feature is m a sense similar to the case of the rare decay b + sy.) A second point is that the x/2 term, though formally subleadmg, is important numerically Workmg within LLA one is then faced with the problem of how to deal with this term since it should strictly speakmg be omitted m this approximation. Let us illustrate this issue m terms of the c determmation m our above example We find c = -0 12 if we use the LLA formulae (with ,u~ = m,, pt = mt) and stmply add the x/2 piece By contrast, omittmg this term and using the strict leading log result
268
G Buchalla, A J Buras /Physrcs
L.&ten B 336 (1994) 263-268
we obtam @= -0 20 The scale ambrgumes are very similar m both cases, roughly three times as big as m the next-to-leadmg order drscussed above. For defimteness we have included the x/2 part to obtain (23) The problem of the x/2 term is naturally removed m the next-to-leadmg logarithmic approximation ( YNL) where this contributron 1s consistently taken mto account. Finally we give the standard model expectation for ALR, based on the short distance contribution m (4), for the Wolfenstem parameter @ m the range -0 25 I @ I 0 25, Vcb = 0.040 f 0 004 and m, = (170 f 20) GeV. Including the uncertamtres due to m,, Am, pu,and ,q and rmposmg the cut -0.5 I cos 0 < 1, we find 30
lop3 5 ]ALR/ 5 9 6
lop3
(25)
employing next-to-leading order formulae Antrcrpating improvements in Vcb, m, and e we also consider a future scenario m which e = O.OOfO 02, V& = 0 040f0.001 and m, = (170 f5) GeV The very precise determmation of Q used here should be achieved through measuring CP asymmetries m B decays in the LHC era [ 131. Then (25) reduces to 4.8
low3 < IALR] 5 6 6
lop3
In both of the scenarios the lower (upper) limit for ALR would be smaller by 0 6 10v3 (1 3 10e3) rf the charm contribution was omitted In this letter we have generalized the short distance calculation of the muon polarization asymmetry ALR m the decay Kf -+ rr+,u”+pu- to next-to-leading order m QCD We furthermore discussed the theoretical uncertamtres involved m thus analysis We have demonstrated that the complete next-to-leading order calculatron achieves a reductron of the rather large scale ambigumes m leading order by a factor of N 3 and is necessary to provide a satisfactory treatment of ALR This is particularly important since long distance contributtons to ALR seem to be small, though perhaps not fully negligible [ 31. In any case a measurement of ALR would yreld a very mterestmg and useful piece of mformation on short distance flavordynamics and the umtarity triangle which is worth pursumg m future experiments References [ 11 M Savage and M Wise, Phys Lett B 250 ( 1990) 151 [2] L Wolfenstem, Phys Rev Lett 51 (1983) 1945 [ 31 M Lu, M Wise and M Savage, Phys Rev D 46 (1992) 5026 [4] G B&anger, C Q Geng and P Turcotte, Nucl Phys B 390 (1993) 253 [5] G Ecker, A Plch and E de Rafael, Nucl Phys B 291 (1987) 692 [6] C Alhegro et al, Phys Rev Lett 68 (1992) 278 [7] J Elhs and J S Hagehn, Nucl Phys B 217 (1983) 189, VA Novlkov, M A Shlfman, A I Vamshtem and VI Zakharov, Phys Rev D 16 ( 1977) 223 [8] G Buchalla and A J Buras, Nucl Phys B 400 ( 1993) 225 [9] G Buchalla and A J Buras, Nucl Phys B 412 (1994) 106 [ 101 Partxle Data Group, Phys Rev D 45 (1992), No 11 part II [ 111 A J Bums, ME Lautenbacher and G Ostermruer, MPI-Ph/94-14, TUM-T31-57/94 [ 121 F Abe et al (CDF Collaboration) FERMI-PUB-94/097-E [ 131 A J Buras, MPI-PhT/94-25, TUM-T31-61/94