qcd corrections to cp violation in higgs exchange

Volume 237, number 2 PHYSICS LETTERS B 15 March 1990

QCD CORRECTIONS TO CP VIOLATION IN H IGGS EXCHANGE

Jin DAI and Hans DYKSTRA Theory Group. Department of Ph),sics, Universtty of Texas. Austin. TX 78712. USA

Received 27 December 1989

The dominant contribution to the neutron electric dipole moment from Higgs exchange comes from a three-gluon operator, produced by integrating out top quarks and neutral Higgs bosons. We calculate one-loop QCD corrections to this operator. Com- paring the result to current experimental limits on d, gives very strong constraints on (T violation in the Higgs sector.

1. Introduction

Recent measurements of the electric dipole of the neutron have encouraged new interest in the mechanisms of CP violation. An important class of such mechanisms involves CP violation by the exchange of heavy particles. For example, in a Higgs model with more than two doublets, or with two doublets which mix with scalar singlcts, CP will generically be vio- lated by charged and/or neutral Higgs exchange [ 1,2]. By integrating out top quarks and Higgs exchange we can produce a variety of low-energy CP violating operators. One such operator is the three- gluon operator [ 3,4 ]:

- - 3 " ~a (' -gs.f,,t,cG ,, "Gh.PG". ~ ( l )

Here .f.t,,-is the SU(3) structure constant, G~",.= 0,,A ,", - 0..4 ~", + gj.b,.A ,,.'- ~' ~ ". is the gluon field strength.

~ (" ~4, is the dual tensor, and g~ is the strong coupling constant.

This is a dimension-six operator. All other (7 ' violating operators are either suppressed by light quark masses or small mixing angles, or are of higher di- mensionality and suppressed by higher powers of the heavy particle masses in the denominator. One would expect this to be the dominant contribution to CP violation from this mechanism.

The coefficient of this operator can be calculated, but the operator will then be defined at a mass scale

~l near m, and m. . To use this to calculate low-energy quantities such as the neutron electric dipole moment, the operator must be renormalized at the appropriate scale.

In general the renormalization group mixes the operator (' with other CP violating operators. There arc no other independent dimension-six operators with the correct symmetries, but there is a dimension-five operator, the quark color electric dipole operator ~',:

, = Cltr~,.(~U" q . ( 2 )

This operator may also be produced by integrating out Higgs exchange, but it will bc suppressed by either light quark masses or small mixing angles. However. the renormalization group mixes U with this operator, raising the possibil ity that at low energies, it may make a significant contribution.

However, a quick calculation reveals that this operator mixing is also suppressed. To make the dimen- sionality correct, this operator must be multipl ied by a quark mass. To make a low energy operator with significant contribution to d,, this must bc a light quark mass, or small mixing angles will suppress it. Furthermore, the operator (2) does not itself mix with ((. The diagram which would produce such a mixing is finite, so the renormalization group equation for (~ will be unaffected. Thus the mixing with this operator may be neglected.

Work supported by the Robert A. Welch Foundation and NSF Grant PHY 8605978.

256 0370-2693/90/$ 03.50 Elsevier Science Publishers B.V. ( North-Holland )


2. One-loop divergences

To calculate the renormalization factors, we write the lagrangian at the scale ). as

ff)=fd4x( - 4 ~ ~, . . L ;a ('~" auv"l- N (A) C ) - - . (3) The second term in the operator ( 1 ) which is produced by integrating out top quarks and charged and neutral Higgs bosons. The coupling x has been calculated by Weinberg in the case where the lightest Higgs is much lighter than the top quark, and by Dicus in the general case [3,4]. We note that to one-loop order, quarks and ghosts contribute only to operator mixing and the wavefunction renormalization of G~",,. This merely converts the coupling constant gs at the scale 2 to the running coupling gs (~t). Apart from this, they have no effect and will henceforth be ig- nored. We use background field methods to calculate the one-loop contributions to the effective action. Write

a_ -a ,a ,a ,a vat , . , , l j ,-AI,+A ~,, D~,A ~=0~,A v-g~f , ,~ .A 'h -'"

where :~ is the background field and A' is the quantum field. We add the gauge-fixing term

.~, -= - j d 'x ( [), ,A '~" ) 2

to the action, and expand in powers of A'. This lagrangian is invariant under "'background gauge transformations", in which the background tl trans- forms as a gauge potential and the quantum field A' as a matter field in the adjoint representation. The derivative operator 0 is the background gauge co- variant derivative.

To one-loop order we need only the terms quadratic in A'. We can write this part of the lagrangian as

. C/,q uad I 4,a l ia ,.by = 2, ,~. t,~A (4)

The operator ~J is a function of the background field tl. The one-loop contribution to the effective action is

i F , (A )=- In det A (A)=- tr In A(A). (5)

To simplify the calculation of the determinant, we take et to be constant. This corresponds to taking all external momenta to be zero. Background gauge in-

variance can be resorted at the end of the calculation by interpreting h ,- gsf,,h,,'l j,A ~ = G,,~, as long as none of the intervening steps breaks the gauge invarianee. We expand the logarithm and look for the terms which are linear in K and proportional to ~6. These terms may be identified with the diagrams generated in a diagrammatic algorithm. Fig. 1 shows the diagrams which give non-zero contribution.

Calculation of the relevant terms can be carried out routinely. The sum of all one-loop contributions to the effective action which renormalize (' may then be written as

iF,(,~,,~ (-~n) 4 = f (k2_ie ~d4k I d4x K( t" . (6,

The integral may be regulated by introducing an ul- traviolet cutoffat the scale 2 where the lagrangian ( 3 ) is defined by integrating out the heavy fields. The in- frared divergence is regulated by cutting the integral off at the scale/~ where wc are calculating the matrix elements: 2

(k2_iE)2 =2n2i In . (7) H

The effective action is the sum of the lagrangian (3) plus the contribution from higher loop graphs. From eqs. ( 6 ) and ( 7 ) we may define the effective coupling:

x (#)= 1 + 47r2 g~ln x(2) . (8)

Fig, I. Diagrams contributing to the renormalization of the operator ( 1 ). Wavy lines represent background fields ,-1; solid lines are quantum fields ,4'. Vertices marked with a heavy dot come from the operator (1)" olher vertices are standard gluon interactions.

257


The loop expansion which leads to this result is sen- sible only iflt is close to 2, so we interpret this as a differential equation:

d 9 - -gs (B)x ( I .Z ) (9) 47t2

Using the fl-function equation for the strong coupling constant with five quark flavors [5], this equation may be integrated with the result

_ - - IO8/23

~c(2) -- \gs---~-) / " (10)

Following Weinberg [ 3 ] we evaluate this for g~(2) / 4n = 0.1, corresponding to taking ;t around 100 GeV. We take ~t such that g2 (/1) / 16rr 2 ~ ~. At this scale, the two-loop contribution to the strong coupling fl function is equal to the one-loop term, and the perturbation theory becomes untrustworthy. Using these values for 2 and/~ gives the result

~:(/.t) ~ 800K(,;.) . ( 1 1 )

We see that low energy matrix elements of C will be strongly enhanced by the renormalization group.

GFZ/ ( q2 + rn2 ) ,

then the coupling x of the operator 6 is proportional to Im Z. An estimate [3 ] of the contribution of ~


MEASURING THE TOP QUARK MASS USING RADIATIVE CORRECTIONS

Elizabeth JENKINS and Aneesh V. MANOHAR Department of Physics. B-019. University of California, San Diego. La Jolla. CA 92093, USA

Received 26 November 1989; revised manuscript received 2 January 1990

Recent LEP measurements indicate that there are only three light neutrinos, and thus presumably only three families. If this is true, the top quark has already been seen through its radiative corrections to electroweak parameters. We compute these corrections using a simple effective field theory method. Current experimental data places an upper bound on the top quark mass of ~ 225 GeV. Reduction of the error in the leptonic width Z--*f+~- by a factor of two will yield a lower bound on mr of ~ 205 GeV at the 90% CL (assuming the current central value). We also show that there are leading log corrections which are numerically important, but have not been included in the standard calculation of radiative corrections.

There have been numerous attempts to determine the top quark mass using radiative corrections. Most recently, Langacker [ l ] has performed a careful analysis of CDF and SLC results on weak gauge boson masses to put an upper limit on mr. In this paper, we extend the analysis to include LEP results. We describe a simple effective field theory calculation using the MS subtraction scheme that includes all the radiative corrections which are relevant for determining mt given the current experimental accuracy. In addition, we show how a modest improvement in the leptonic branching ratio of the Z will determine m,, not just give an upper bound. We also discuss how our calculation differs from the standard analysis of Marciano and Sirlin [2,3 ].

We begin by briefly describing the calculation of radiative corrections using minimal subtraction and an effective field theory. This method includes the most important renormalization effects needed to determine the implications of recent high energy electroweak measurements, and it ignores effects which complicate formulae but are very small. We will make a detailed comparison between MS and the renormalization scheme of Sirlin [ 2 ] at the end of this paper.

In the standard language of effective field theories,

one constructs a new effective theory each time one crosses a heavy mass threshold. Let us first assume that the top quark is heavier than the electroweak gauge bosons, which is almost certainly true given the lower limit of 78 GeV on mt from CDF [ 4 ] ~1. In such a scenario, one integrates out the top quark at/~= rot, and then scales the resulting effective theory down to the electroweak gauge boson mass scale, incurring logarithms of mt/M, where M is a gauge boson mass, in the process. At the electroweak scale, one integrates out the W and Z bosons, obtaining a new effective theory which can be scaled down to low energies. However, for a top quark which is not much heavier than Mw or Mz, logarithms of mt/M are not very important. We will therefore work in an approximation where we neglect terms which are of order l n (mJM) , as well as all finite radiative corrections not enhanced by large logarithms or by a large t-quark mass. Thus the only radiative corrections retained are those which grow quadratically with mt [6], and those which include large logarithms such as In (M/m~).

Instead of following the usual procedure of integrating out the t quark at lz= trlt, and then scaling the resulting lagrangian down to/~=Mw, we first scale the lagrangian down to/1 = Mw, and integrate out the t quark at this scale, since the difference in the two

On leave from the Department of Physics, Massachusetts In- stitute of Technology, Cambridge, MA 02139, USA.

~ The presence of a charged Higgs lighter than the top quark can evade this bound, see ref. [ 5 I.

0370-2693/90/$ 03.50 Elsevier Science Publishers B.V. (North-Holland) 259


procedures is of order In(m,/M) which we are neglecting. We can also integrate out the top quark at a scale p = Mw even if m, < Mw in our approximation. since we are neglecting logarithms of m,/Mw. Thus our effective lagrangian at the weak scale is the standard model lagrangian with all coupling constants renormalized at p=Mw, with additional finite terms obtained on integrating out the t quark. The procedure of integrating out a heavy top to get an effective thcory has been studied in detail in ref. [ 7 ]. The conclusion is that the only result ofintcgrating out a heavy top quark (in our approximation) is to change the charged Wboson mass term in the lagrangian:

4(s1'2"'2 ~ --~'I/!/+ J/V -,u_. ( lg2u2 + ~,~2 ) l/~,~,+ W -J' ( I )

where

&~2= 3a(Mw) m Z. (2) 16n sin 2Ow( M, , )

One then calculates the W and Z masses from the modified lagrangian by determining the poles in the gauge boson propagators. All one loop graphs will only involve logarithms of Mw/Mzor finite parts, both of which we are neglecting, so all results (including decay widths) can be obtained at p = M~ using the tree level effective lagrangian, which involves the unknown couplings g, g', v, and ~)M 2 (or m,) all renor- realized at p= Mw.

Two independent combinations of these couplings are determined by low energy measurements of GF and a. The effective operator (renormalized at l l=M, , ) responsible for p decay is obtained by integrating out the W boson:

G,: ( M .. ) 6- x/~ ~, ( l -ys )p ~(1 -75) u~. (3)

with GF(Mw). the Fermi constant renormalized at ll = Mw given by

Gv(MH-) g2(Mw) e2(Mw) x/2. - 8M~+m + 8M~vsin20w(Mw) ' (4)

where Mw is the physical W mass ( 1 )

,~vt ~. = ~g2 (m, . ) t;2 (;~../',.) + 6AI 2 . (5 )

Gv( m u), the Fermi constant renormalized at p= m,, is defined in p decay at low energy [ 8 ]

GZ(m~')rn~'( 1+ l ) F. 192n 3 2n [~(n2-25)1

[1 -8x+8x3-x4+12x21n( l ) ] , (6)

where 2 2 x=mJm, , . GF(m,) can be obtained from GF(Mw) by scaling the effective operator (3) from p = Mw to p = m,. However, the four-Fermi operator t~: responsible forp decay can be fierzed into a charge retention form. In this form, the charged leptons appear together as a conserved current which is not renormalized by electromagnetic effects. The anomalous dimension of the operator vanishes, and we conclude that [9]

GF(Mw) =Gv(m. )

= (1.166 37 +0.00002) 10 -s GeV-2 (7)

Thus we will simply refer to Gv(Mw) as Gv. The fine structure constant, however, does get renormalized

- ~ Q} In , (8)

where the sum runs over all fermions lighter than Mw. The fermions contribute to the running coupling constant as long as/z> rot; It is not obvious what the appropriate scale is for the light quark masses. Clearly the u and d quarks cannot contribute to the fl function at 10 MeV, since there are no charged hadrons with that low a mass. For comparison, we evaluate eq. (8) using constituent quark masses and using a cutoff of one GeV for the light quarks, which is equivalent to setting the light quark masses equal to one GeV. These two methods give

1 1 o~(Mw)- 128.7 and 129.1' (9)

respectively. This error of approximately 0.3% in a (Mw) will not be significant in our subsequent analysis, so we will simply use the value a(Mw) = I / 128.7.

We can express all relevant electrowcak experimental quantities in terms of Gv, a(M~) , &~2 and sin20~,(Mw). Since both Gv and a (Mw) are determined, measured values can be plotted in terms of the two remaining variables. If these measurements

260


are precise enough, they will determine sin 20w(Mw) and 8M 2, or mt 2 using eq. (2).

The W and Z boson masses are given by

e2(Mw)t, ,2 MS,= gZ(Mw)v2+ 6M 2= + 8M 2

4s2(Mw)

M2= ~ [g2 (M,,) +ga(Mw)]v 2

e2(Mw)b ,2 = 4s2(Mw)C2(M~.) , (10)

but using the connection between Mw and GF, eq. (4) , we get

na(Mw) 1 M2v= x/~G v s2 ( I I )

and

M~= na(Mw) 1 8M 2 GF S2C2 C2 , (12)

where s - sin Ow(Mw). CDF determines the ratio of Mwto Mzto be [10]

Mw = 0.880 + 0.007. ( 13 ) Mz

SLC and LEP determine the Z mass [ 1 I - 15 ] ~2

Mz=91.106 + 0.051 GeV. (14)

In addition, LEP has measured the partial widths of Z into e+e - and/,t+# - and the invisible width. In the MS scheme, the leptonic widths are calculated at tree level using the couplings renormalized at / t= Mu,,

Mza(Mw) ( l _4sZ+8s 4) . (15) F~+~_ - 24s2c2

The ALEPH Collaboration finds [ 12 ]

F~+~_ =85.9_+5.9 MeV, (16)

the L3 Collaboration finds [ 14]

F,,+,,_ = 88 + 11MeV, (17)

Fu+ ~_ =92_+6 MeV, (18)

while the OPAL Collaboration finds [ ! 5 ]

,2 We have made a weighted least squares fit to the five results. However, for the error on Mz, we have used the error of the most accurate LEP measurement since the systematic errors of the LEP measurements are correlated.

F~.~_ =88.1 +4.6 MeV. (19)

Averaging these values, and treating the errors as independent, we obtain

F~+~_ =88.5+ 3.0 MeV. (20)

From the width of the Z, one can conclude that the number of neutrinos is less than 3.8 at 99% CL ~3. If we assume three neutrino types and that there are no other invisible particles which couple to and are lighter than the Z, then

Mza(Mu.) Fi.vis = 3F~ = 8s2c 2 (21)

equals[14]

Pi.vis =0.567 + 0.080 GeV. (22)

The error on Pmvis is too large at present to be useful for determining m,, so we do not include it in our graphs.

Equating the experimentally measured quantities with the theoretical calculations gives us curves in the s2-mt plane. These are plotted in fig. 1. We have also graphed the allowed region at 90% CL in fig. 2. The existence of an allowed region is a non-trivial test of the standard model. The analytic expressions for the curves can easily be derived by expanding to first order in the correction 8M 2. The results are (for the quantities in parentheses)

~3 This conclusion follows from a least squares fit to the five experimental values.

0.26

0.25

0.24

% o .23

0.22

160 1~0 260 2~0 36o

,,,, (GeV)

Fig. 1. Allowed regions of the s2-mt plane. The + l e bands correspond to limits placed by Mz (solid black lines), Mw (solid gray lines), Mw/Mz (dashed lines), and F~.~_ (dotted lines).

261


%

0.234

0.232

0.23

0.228

0.226

0.224

0.222

i00 150 200 250 300

m, (Go, V)

Fig. 2. Plot of the allowed region in the s2-m, plane at the 90% CL limit. We have not included low energy uN scattering data which constrains m, to be less than about 225 GeV [ I ].

s2=s~ (Mw) ,

s~=sg sgc~ 5M 2 Co-Sg M~, (Mz) ,

,2 8M2 (Mw/Mz) , s2=sg +cO M--~w

-4So+8So)SoCo 8M 2 s2=sg_ (1 2 4 2 2 _3+lOs~_8s~+16s6 M2 w (F~.~_) ,

2 2 6M 2 2 ") SoCo

s =s~ 3(Co2-So 2)M~ (F, .... ) . (23)

Here So 2 is the value of the weak mixing angle obtained from eqs. (11), (12), (15), and (21), by setting 8M2=0, i.e. by using the tree level formulae and equating with the measured values ~4. In the evalua- tion of the widths, we have substituted eq. (12) into eqs. ( 15 ) and (21 ). The above quantities depend on different combinations of s 2 and m,, as can be seen by the different slopes with respect to 8M 2. Thus one can determine both s 2 and mt once the error bars on any two quantities are sufficiently reduced. In particular, an improved measurement of F~Q_ to + 1.6 MeV will restrict m, to the range 205 GeV < m t < 235

Note that each measurement yields a different value for So 2. The notation in eq. (23) is not meant to imply equal values of So 2 amongst the five curves.

GeV at the 90% CL, as shown in fig. 3 ,s. This preci- sion will be reached in the next few months at LEP. With the current released values, we can only extract an upper bound on the top quark mass of around 300 GeV. This upper bound can be reduced to about 225 GeV if one includes the low energy vN scattering data [ 1 ], which is the usual p parameter l imit on m,.

In the above expressions, we neglected some effects which we now justify. In particular there are Higgs boson contributions to the radiative corrections. One can also treat these using an effective field theory by integrating out the Higgs boson at a scale /~=rnH. Unlike the top quark, the Higgs boson contributions are all logarithmic; there is no term proportional to m ~t. This is the screening theorem [ 16 ]. The top quark contribution was large because the difference m,-mb violates the custodial SU(2) symmetry. However, m~, does not violate custodial SU(2) , and so all rn~ pieces can be absorbed into a redefinition of the coupling constants. Thus the Higgs corrections are much smaller than the top quark corrections. A detailed calculation of radiative corrections including Higgs effects has been performed by

*s The allowed range of values for m, depends on the central value for F, . , _. We have assumed the current central value of 88.5 MeV in fig. 3.

qb

0.228

0.227

0.226

0.225 180 1~,o 26o 21o z~o z~0 2,~o 2go

,m (G,,V)

Fig. 3. Plot of the allowed region in the s2-mt plane at the 90% CL limit, with the current error in the measurement of F~.~_ reduced from 3.0 MeV to 1.6 MeV leaving the central value un- changed at 88.5 MeV.

262


Sirlin [2], and Marciano and Sirlin [3]. They use a different renormalization scheme, in which

xa(me) 1 M2w= x/~ G ~ g2 (l + Ar) =M2e2 , (24)

where .~ is Sirlin's definition [ 2 ] of the weak mixing angle. Comparing with eqs. ( 11 ) and (12), we see that

8M2"~ a(Mw) g2 c2=C 2 1- M2w], l+Ar= a(rne) s 2' (25)

which allows us to convert between the two schemes. In fig. 4 we have plotted Ar as calculated by the expression given in Marciano and Sirlin [3] and compared it with our approximate calculation using eq. (25) ~6. One can see that the two calculations agree to within Ar~ 0.005, which translates into an error of about 0.25% in the gauge boson masses and widths. Equivalently, this corresponds to an error of about 17 GeV in m, for a top near 150 GeV. Variations in Ar due to a change in the Higgs mass are also of this order, as are radiative corrections to the decay widths which do not involve large logarithms [17]. There

Actually we have used our formula minus certain leading log contributions (see below) which were omitted in ref. [3]. In our estimates of rn,, we have included all the leading log contributions.

are several points to be noted in attempting to calculate radiative corrections at the 0.5% level. Eq. ( 8 ) is the leading log correction to the fine structure constant. The calculation of Marciano and Sirlin [2,3] includes the shift in a as a contribution to Ar. This contribution can be seen from eq. (25) to be

Arduetshiftina=2a(Mw)3zt ~ Q~ In (~) .

(26)

The value of Ar in ref. [2] has the contribution (26) with a(Mw) replaced by a(m~:). The difference in the two results is a leading log effect (through higher order in a) , and is numerically approximately 0.004 which is as important as the Higgs boson contributions. It is simple to modify the calculation of Marciano and Sirlin to include this effect. The coefficient of m, 2 in our calculation differs from that in ref. [ 3 ] by similar leading log effects which are higher order in a, but are summed by the renormalization group equations. The difference is again of order 0.004 in Ar. Another uncertainty in the radiative corrections is the renormalization of a between me and Mw. We have used the free field theory values for the quark and lepton contributions to the fl function for the electric charge. This is not accurate for the quarks at low energy because of the strong interactions which

k ,q

0.06"

0.04'

0.02

-0.02

-0.04"

50 i so 200 \ , < ' ' ' ' "" 30'0

\, m, (GeV) \

Fig. 4. A comparison of Ar using eq. (25) (solid line) with the expression for Ar given in ref. [3 ] for m,= 100 GeV (dashed line) and mn= 500 GeV (dotted line). Certain leading log contributions have been omitted from eq. (25), as mentioned in the text.

263


is reflected in the difference in the two values ofo~ in cq. (8). One can try and get a better est imate of the renormalizat ion of or by using dispersion rclations and the experimental value for the e+e - total cross sec- t ion [ 18]. This method, however, has an error of approximately 0.2%, comparable with our crude analysis in eq. (8) . The calculation presented in this paper is accurate to about 0.5%, which is adequate given the present experimental uncertainties. (Note that we need two independent measurements at the 0.5% level to determine s 2 and m, to this accuracy; at prcsent only M/ i s measured this accurately. ) Eventually, one can hope to determine the top quark mass more accurately, as well as determine the Higgs mass. Such a calculation has to include the leading log corrections ment ioned above, and also has to reduce the uncertainty in the renormal izat ion ofo~. In addit ion, three independent measurements are needed to determine the three unknown parameters s 2, m,, and mH. One can gct around thc problem with a by looking at ra- tios such as Mw/Mz, and F~~_/M 3 which do not depend on the renormal izat ion of a. However, Mw will not be measured to the required accuracy unti l LEPII.

Finally, we wish to make a few comments about the impl icat ions of a heavy top quark. For m,


STATIC CAVITY CALCULATION OF HEAVY MESON NEUTRAL MIXING WITH QCD CORRECTIONS

Lloyd C.L. HOLLENBERG and Bruce H.J. McKELLAR Research Centre for High Energy. Physics, School of Ph),sics, University of Melbourne Parkville, Victoria 3052, Australia

Received 20 June 1989; revised manuscript received 5 December 1989

The neutral mixing matrix elements describing D O and B mixing are calculated in a static cavity model previously fitted to the AI= ~ rule of kaon decay. Since the model is based on O(g) wavefunctions, computation of hadronic matrix elements includes O (g2) corrections. Specifically, we compute the quantitiesft, x/'Bo and fB ,~.,/-~B for basis sizes N8 = 1,2, 3 and 6. At NB= 6 we find the values fo ,v/-Bo = 90 MeV andfB ~,/-BB = 62 MeV. The ratio of these quantities agrees well with non-relativistic expectations.

1. Introduction

The analysis of CP violation in the standard model has long been hampered by the lack of ability in calculating hadronic matrix elements of the relevant effective operators. For example, in order to be able to understand the implications of the observed neutral mixing of the B system, we require the calculation of the AB= 2 matrix element and the B decay constant. Recently, there have been a number of sophisticated computations of relevant quantities for heavy systems using lattice simulations of QCD [ 1 ] and sum rules [2,3]. In this paper we present an alternative approach to the calculation of neutral mixing matrix elements in heavy meson systems using a static cavity model which includes O(g 2) corrections to the usual valence result.

The model is based on the construction of O(g) Fock state wavefunctions, corrected for centre-of- mass effects, using j= ~ quark and 1= 1 gluon modes in the static cavity [4,5 ]. These wavefunctions have previously been applied to the calculation of QCD corrections to the K-rt matrix elements of the bare effective AS= 1 operator [6], and to the kaon B parameter [7]. Although the model parameters and masses of the strange and heavier quarks are determined by fitting to the ground-state meson mass spectrum and charge radii, the light quark mass, mu.d, is not determined by this procedure. However, it was found in ref. [6] that the A /= rule was uniquely

reproduced for some value ofmu.d~O(140) MeV for each basis size considered. The corresponding value of the kaon B parameter at the largest basis size studied, N~ = 6 (where NB is the maximum quark or gluon mode number in the wavefunction), was Bn = 0.58.

In this paper we calculate the matrix elements of the AC= 2 and AB= 2 operators to give the quantity fM ~ (M=D and B), for NB= 1, 2, 3 and 6, using the model parameters of ref. [ 6 ] which reproduce the K-n A/= rule.

2. The model

The basic starling point of the model is the construction of large basis wavefunctions which may be written schematically as [ 5 ]

N~

I~'> = Iqct> + ~ ~ a({n} ), a)IqdlG>/~).. In} c~

NB

+ ~ ~. ~ b({m},fl)lqClq~lG>r~,.~.p, (1) {n,I} f ,B

where I qdl > is the usual valence state from which the higher components evolve. For the bremsstrahlung ( I q~lG > ) and vacuum fluctuation states ( I q1q~lG > ) respectively the sets, {n} and {m}, contain the mode numbers of the quarks and gluons in terms of the basis of states in the static cavity, whilst the remaining quantum numbers (orbital, spin and colour) are de-

265


noted by ot and ft. For the vacuum fluctuation states the flavour of the quark "sea" is summed overf=u, d, s, c, b. The amplitudes a({n}, a ) and b({n},fl) are calculated from QCD in the static cavity approximation for which the quark and gluon fields are known. The basis size, Nr3, serves as the cut-off parameter; i.e. a given calculation at basis size Na includes all modes n, mjo+ ~, la"12(p2)") ' (3)

I(P)I 2 2 ~A 2. (p2) = (27t)3 f d 3 p ~ p = 4 (6)

The model is characterised by just four parameters: ct (effective quark-gluon coupling), B (con- finement pressure), Zo (zero-point parameter) and q (CM parameter). These parameters are fitted in the following manner. At each basis size, Nr3, and a given value of mu.d the n and P masses are used to determine a and B whilst the parameters Zo and r/are determined from the n and K charge radii. The quark masses ms, mc and mb are fitted to the meson masses K(496), D(1867) and B(5273) respectively. Fi- nally, the values of mu.d for each basis size are obtained from the K-n AI= t rule [6].

3. Results

For the details of the calculation we refer the reader to the computation of the kaon B parameters in ref. [7] from which all the necessary expressions for the matrix elements required here can be obtained. The results for the neutral mixing matrix elements are shown in fig. 1 (the normalization used here corresponds to.f,,= 93 MeV ) whilst the ratio is given in fig. 2. At NB=6 the results forf~g v/~-" are

.fDx/BD=89.8MeV, fB V /~ =62.0 MeV. (7)

where N is the normalization of the wavefunction ( 1 ), and the parameter q, governing the centre-of-mass prescription, is determined in the fitting procedure.

Plane wave states are constructed using a wave- packet projection

f 3 O(p) I v )= d P2-E~ IV (P ) ) , (4 )

w i th a gaussian paramcter i za t ion o f the d i s t r ibut ion

amplitude

( 2 ,3/4 /2E(p) [__p2"~ O(P)=k~2j V ~ exp~,~-- J 5)

The parameter, A, is fixed by the consistency condition that ~(p) should give the same value of (p2) as determined by (3), i.e.,

>

100

95

90

85

80

75

70

65

60

55:

t} D

o o B o o

50 ~ 2 3 4 5 " g " ~ " Baxis Stzc. N~

Fig. I. Basis size dependence o f the quantities fD ~,fBD and / . f . v'BB.

266


1.80 - , - , - , - , " , " ,

1 .70

160

1 .50

14 I

130

1 .20 '

0

0

0

NRQM

I 2 3 4 5 6 Bas is S i ze . N ,

Fig. 2. The ratio fD V/ffDD/f8 ~ as a function of the basis size. The dashed line indicates the non-relativistic quark model (NRQM) prediction.

A property of heavy meson systems, dlQ, where q is light, is that in the non-relativistic quark model (NRQM) it is expected that the decay constant scales as 1 /n /~ as rno-~oo (up to logarithmic corrections [ 1,8 ] of the order of 10%) whereas the B parameter is expected to be roughly constant. Hence, the D to B ratio of the quantity fro x /~ should scale as

fo,/To ~ (~)1 /2 1.68 (8)

In fig. 2 we see that this ratio is close to the NRQM result; the basis size dependence will tend to improve this agreement. On the other hand, the ratio 0CD/ fB) = 1.09+0.23 for the sum rule calculations [2,3] does not reflect the NRQM limit to the same degree. The extrapolated lattice value [ I ] offa was obtained using the NRQM limit.

That these values are smaller than those obtained from sum rule techniques, for which [2,3] fo~ 141 + 26 MeV and fB ~ 129 + 13 MeV (we have com- bined the results of ref. [ 2 ] and ref. [ 3 ] ), can be seen by assuming that the B parameters for B and D are close to unity. From lattice QCD calculations one has [ I ] fD = 137 + I I MeV with an extrapolated value of the B decay constant offB = 85 MeV. However, as in- dicated in fig. 1 the results are still sensitive to the basis size indicating that a more realistic model prediction, i.e. at larger NB, will be in better agreement. Unfortunately, limitations on computer time pre- vented computation for NB> 6.

References

[ 1 ] M.B. Gavela, L. Maiani, S. Petrarca, G. Martinelli and O. Pene, Phys. Left. B 206 (1988) 113.

[2] S. Narison, Phys. Lett. B 198 (1987) 104. [ 3 ] C.A. Dominguez and N. Paver, Phys. Lett. B 197 ( 1987 ) 423. [4] L.C.L. Hollenberg and B.H.J. McKellar, Phys. Rev. D, to

appear. [ 5 ] L.C.L. Hollenberg and B.H.J. McKellar, J. Phys. G, in press. [6] L.C.L. Hollenberg and B.H.J. McKellar, University of

Melbourne preprint UM-P-89/51. [7]L.C.L. Hollenberg and B.H.J. McKellar, University of

Melbourne preprint UM-P-89/52. [ 8 ] I. Shifman and M. Voloshin, ITEP preprint 86-54 ( 1986 ).

267


COLLAPSE OF THE WAVE FUNCTION, ANOMALOUS D IMENSIONS AND CONTINUUM L IMITS IN MODEL SCALAR FIELD THEORIES

Elbio DAGOTTO Institute for Theoretical Physics, University of California, Santa Barbara. CA 93106. USA

Aleksandar KOCI(~ ~.2 Institut J~r Theoretische Physik. Universitiit Regensburg, D-8400 Regensburg, FRG

and

John B. KOGUT 3 Department of Physics, University of lllinots at Urbana-Champaign, 11 I0 West Green Street, Urbana, IL 61801, USA

Received 27 December 1989

We study a model scalar field theory with vector couplings in the ladder approximation. A massless bound state appears in the spectrum when the coupling exceeds a critical value a ~ 1. The physical mechanism behind this critical point is "collapse of the wave function", the same as for quenched spinor QED. Composite operators acquire large, negative anomalous dimensions and 08 interactions become renormalizable above the critical coupling.

What happens when the strength of attraction in a bound-statc problem becomes large? Normally, a stable bound state is formed due to a balance between zero-point motion and attraction. However, if the attraction is sufficiently strong, it can overwhelm the zero-point repulsion and collapse occurs. We want to understand the physics of this situation within the framework of relativistic quantum field theory. In this Letter we study the behavior of a model scalar field theory with vector coupling. Our main motivation is to dctermine the possibility of the existence of a non- asymptotically free (NAF) scalar field theory, and, since our basic question is spin-independent, confront our results with those of the spinor QED. We hope this will enable us to understand the main ingre- dients responsible for the existence of a nontrivial continuum limit and devclop a consistent physical

On leave from Department of Physics, University of Arizona, Tucson, AZ 85721, USA.

2 Alexander von Humboldt Fellow. Guggenheim Fellow.

picture of coupling constant renormalization beyond perturbation theory.

The physics of collapse was studied recently in strongly coupled spinor QED [1-3]. There, it was found that if the charge exceeds a critical value, vacuum rearrangement takes place and the theory undergoes a transition to a phase where chiral symmetry is spontaneously broken. However, chiral symmetry breaking is just one of a host of interesting phe- nomena that accompany this phase transition [3]. Here, we focus our attention on one such effect, namely large anomalous dimensions of composite operators [4]. We motivate our interest by the following observation. The increase of attraction reduces the average radius of a bound state and increases the probability to find the bound particles near one another. A useful example that illustrates these effects is the problem of a Dirac particle in a Cou- lomb center. At short distances (r


function develops a 1/r singularity and the probability to find the particle near the origin (P(r)~ r 21 ~(r) 12 ) becomes O ( 1 ). In this regime the particle is squeezed beyond its Compton wavelength and this, we believe, is the origin of new physics.

With fermions, this physical effect causes dynamical mass generation for a very' simple reason. Mass can be best understood as a frequency ofhelicity flip. For example the mass term in the lagrangian is

* + * ). On the other hand, in a t r /~= -- m(~YL~ R I/J R ~J L two-body approximation there is a relation between binding and helicity. A bound state is a standing wave which is essentially a supe~osition of an incoming and a reflected wave. If the interaction is spin independent, then the bound state has a net chiral charge of two units because, at the boundaries of the potential well, momentum, but not the spin, is reversed. Therefore, when a panicle is bound to a region of size smaller than its Compton wavelength (which equals the life-time of a state with definite handedness), helicity is flipped at a rate higher than the intrinsic rate (mass) [ 5 ]. Fermions adjust to this situation by increasing their mass through chiral symmetry breaking. This is one reason why, for example, the pion radius equals the Compton wavelength of the constituent quark [6 ].

In our previous work we have argued that it was the peculiar behavior ( l /r singularity in the wave function ) of the short-distance sector of the theory at strong couplings that makes such theories nontrivial [2,7]. Reference was not made to chiral properties, etc., but rather to the anomalous dimensions of the composite operators as the manifestation of collapse within the operator product expansion (OPE) language in field theory. To illustrate the connection between anomalous dimensions and triviality, consider a well-known example in 2~ 4 scalar field theory [ 8 ]. Let x approach y in a euclidean formulation of the theory and consider the operator ~4,

04(X) ,~ 02(X)~2 (y )

Co + C2 2 (x_yV,~,~ (x_~)~.~ ~(x)

G 4 X + (x_y)2a,~_,~,,R()+ .... (1)

where d~2 and d~, denote the relevant scale dimensions, and 0~ are suitably renormalized operators. Obviously, if d,~,>2d~2, the series truncates at 2R(X), i.e. the renormalized lagrangian is quadratic and the underlying theory is free. Conversely, if d~, ~< 2d~2 there is an interacting continuum limit, and the series does not necessarily terminate with the quartie term. Since the canonical dimension of the 0 4 operator is 4, one defines d~, =4+r/~, where ~/,,, is the anomalous dimension. In that case, the condition for nontriviality (d~, ~


tions. Our aim here is to obtain the bound-state wave function and show that, for sufficiently strong coupling, it behaves in the same way as its analog in spi- nor QED. From there, we will compute the higher- point functions and deduce, from their asymptotic behavior, the anomalous dimensions of composite operators. We shall sketch our analysis here and point out analogies to QED. We hope that this work will motivate more detailed studies of scalar theories.

We begin with the analysis of the bound states in a model scalar field theory in order to see whether our simple picture, based on relativistic quantum me- chanics, generalizes properly to a field theory. We treat the bound-state problem through the Bethe- Salpeter (B-S) equation in the ladder approximation. In fermionic QED the application of the ladder approximation was justified because chiral symmetry breaking occurred at high momenta where vertex corrections were constrained to be soft by the vector Ward identity. It is not clear whether similar arguments hold for scalar QED. Here we consider a scalar-vector theory with an interaction given by LI = e.~,A., where ... = i ( 0 + %,0- 0,,0 + ). The application of the ladder approximation in this model is more reasonable because of the absence of seagull graphs. In terms of the B-S amplitude ZP (k) and the scalar propagator J (p ) , the bound-state equation reads

d- t (k - ~P)d-'(k+ ~P)zp(k)

=4ie 2 f g~,~(k-k' )2_ (k -k ' )u (k -k ' )~

k" (k -k ' ) 4

x (k+ P)~(k- P)~z~(k' ). (5)

Here P is the total four-momentum of the bound state (p2=M2). After continuing to euclidean momenta and doing the angular integration, eq. (5) becomes, for the M= 0 state,

(x+m2) 2 z(x)

X

x 2 A2 3a[{ (~) f ] -- 4re d), Z(y)+ dyz(y) , x

(6)

where we denoted x=k 2. This integral equation can be converted into a boundary value problem,

d [- 3 d f (x+m2) 2 . "~] 3a x ;

A2Z' (A2)+ 3x(A2)=O, (7)

where we have assumed that the renormalized scalar mass m = X(p 2= 0) was much smaller than the cutoff A. It is easy to solve eq. (7) in the asymptotic regime. For a ac,

1 Z(X) ~ -x5 sin (x /a - /~ - 1 In (x/122 ) ), (9)

where12 is an infra-red scale. In the supercriticai case, the boundary condition (for A-~oo) is satisfied if,

~-aT-a--5 ln(A2/122) =zr. (10)

This gives a familiar equation for the coupling constant flow [ 1 ]

n2 ot/c~c = 1 + ln2(A2/122 ) , ( 11 )

or, equivalently, an expression that fixes 12,

122=,4 2 exp( - -n /~- 1 ). (12)

Therefore, at a= ac the relativistic bound-state wave function exhibits the same kind of behavior as its quantum mechanical analog. Collapse is manifested through the oscillations in the wave function and the system is stabilized if the coupling constant is renormalized according to eq. ( I 1 ).

How does the (renormalized) scalar mass (mR) scale in the critical region? In fermionic theories one finds that the dynamical mass scales the same way as 12 (eq. (12) ). This follows from the relationship between the gap and pseudoscalar bound-state equations that is insured by the Goldstone theorem. How- ever, in scalar theories no such correspondence exists, and the solution for the scalar mass does not (in principle) exhibit special dependence on the coupling constant near the critical point. It is easy to convince oneself of the correctness of this statement within the approximation that we employed: summing the ladder diagrams in the gap equation produces a nonvanishing self-energy for any non-zero coupling. This occurs because, unlike fermionic theories, a massless

270


scalar theory does not have more symmetry than the massive one [9]. Therefore, radiative corrections produce non-vanishing mass corrections even if the bare theory is massless and, to keep the renormalized mass fixed, the theory requires fine tuning. In our calculations we must, therefore, control mR through appropriate counter terms in the lagrangian. Imple- mentation of this requirement might present a nontrivial problem in computer simulation studies of the physics discussed here. It is also a serious matter of principle.

Next we sketch the calculation of anomalous dimensions of composite operators in the ladder approximation. It is easy to solve the equation for the four-point function G4(Pl P2P3P4). After some rear- rangements of the variables we find that

G4(pl, P2, P] P'2) =&(Pl -P2 -P ' l -l-P2 )

(74(~(p, +p2), t (pq +p~); p, -p2), (13)

with the momentum assignment as shown in fig. 1. The equation for (74(q, q'; P) for large q exhibits the

Pl P2 o) Pl P~

b)

P7- - P6 Ps

P l~ P,. Pz P3

c)

Fig. 1. (a) Four-, (b) six- and (c) eight-point functions in the ladder approximation.

same anomalous scaling as that of the ,rP(q) in eq. (9) with the anomalous dimension r /=- l+ x/ l -a /a t for a~< O~c, and r/= - 1 for a>_-ac.

From this point, our task is simple if we note that in the ladder approximation, the higher-point functions can be built from G/s. After some kinematics, G6 and G8 satisfy the same factorization of the total momentum as it was demonstrated for (74 in eq. ( 13 ). Then, the corresponding (76 and (78 can be expressed as convolutions of (74's with respect to the second variable (q ' ) . For example, when all the external momenta correspond to the kinematics of the short-distance expansion, i.e. pt=p2, P3--P4, Ps=P6, fig. lb gives

(76({P}) ~ f (74(P,, q' )3 - ' (q ' )(74 (P,, q' )A - ' (q ' ) q"

(74 (P3, q' )3- ' (q' ). (14)

In this case the anomalous dimension of the 06 operator can be read offeq. (14). Since there are three powers of (74, each contributing q to the large-momentum limit, we find that qo~ = 3r/and, at strong couplings do6 = 6 + qo~ = 6 + 3q = 3.

Analogously, for (Ts the large momentum behavior is determined by four powers of (74 and t/o, =4q and do8 =8+4q=4 at a=a. Clearly, for a>~ac all three operators 04, 06, 08 have dimension ~< 4.

We see that, as a result of the non-asymptotically free nature of the vector coupling, higher dimensional operators (06 and ~) enter the renormalized lagrangian through operator mixing. In fermionic theories we interpreted this effect as the non-decou- piing of heavy modes [ 3 ]. An important application of this result would be to the abelian-Higgs model. There are three issues that need to be discussed in that context. They are: triviality, symmetry breaking and fine tuning. We briefly discuss why we believe that at strong coupling all three appear in a new light relative to perturbation theory.

Triviality. Recent results suggest that the ,;t04 theory has a trivial continuum limit [ 10]. Unless gauge couplings can remedy it, that would mean that the Higgs mechanism, fermion masses etc. would not ex- ist except in a theory with a finite cutoff. Relevance of the 4, 06 and 08 operators could very well alter this situation in the sense that the renormalized theory would produce a more sophisticated potential,

271


V(O)=m202+ 24 2606 + 28 08 . (15)

In fact, i f28> 0, the coupling 2 4 can take any value. At strong couplings do, ~< 4, and thus the renormalized lagrangian is already interesting without any reference to 26 and 28. The fact that ot is large, is crucial here. We believe that an asymptotically free gauge theory could not affect the short-distance structure of the Higgs sector and thus, influence the question of triviality. For example, SU(2) gauge theory with Higgs fields exhibits triviality of the Higgs coupling as if the gauge fields were absent [ 1 I ].

Symmetry breaking. The emergence of 06 and 08 as relevant operators in the effective potential in eq. ( 15 ) might lead to a symmetry breaking pattern that is quite different from that in perturbation theory [ 12 ]. If this is the case, radiative corrections at strong couplings could induce an entirely new phase diagram that cannot be obtained perturbatively through the loop expansion (Coleman-Weinberg effect [ 12 ] ). To emphasize this difference, we recall that in ref. [ 12 ], radiative corrections to the effective potential were obtained from the summation of photon loops whose origin was the e21012 2 A u piece in the scalar QED lagrangian. This led to the one-loop improved potential

204 ( 522 3e 2"] (02 I /=~ +\ l142n2+64n2/O' ln

(16)

which has a minimum away from the origin. The ladder approximation discussed here neglects this interaction completely and the effective potential in eq. (15) comes from an entirely different class of diagrams.

l, ine tuning. In four dimensional scalar theories at weak coupling there are only lwo relevant interactions: ejj/l~, and ,,~404. Because of that, it is not possible to eliminate the quadratic divergences in the scalar selfenergy to all orders, without requiring (unnaturally) fine tuning of the coupling constants. One of the remedies for this problem is supersym- metry [13]. One introduces new (fermionic) de- grees of freedom that, if coupled supersymmetrically, lead to complete cancellation of the quadratic divergences. Namely, the extra symmetry requires the de-

generacy of the matter fields so the scalars acquire the same radiative corrections as fermions and their mass is free of quadratic divergences because of approximate chiral symmetry. There are two ingredi- ents that are relevant for naturalness. The condition for the cancellation of the quadratic divergences defines a surface in coupling constant space (this surface exists because the parameter space is enlarged by the additional couplings). The fact that the couplings are supersymmetric guarantees that this surface is stable against radiative corrections, namely cancel- lations occurs at all orders. This example suggests how the relevance of 06 and 08 in strongly coupled scalar theories might cure the problem of fine tuning. The parameter space is now enlarged to four couplings (0~, 24, 26, 28) and there are only two equations they have to satisfy. They are the fixed point conditions (an analog of eq. ( I 1 ) ) and the condition for vanishing of the scalar mass. These two equations would define a critical surface in coupling constant space on which a continuum limit of the theory exists. It remains to be seen under what conditions this surface remains stable.

It is worthwhile to pursue these problems in more detail by studying scalar QED with V(0) given by eq. ( 15 ). This study is underway.

And finally, one would like to know if any of this physics survives the inclusion of vacuum polarization effects due to the charged scalar fields. Collapse of the wave function provides a new source of coupling constant rcnormalization, eq. (11), in the quenched theory and new operators mix with the theory's lagrangian. Therefore, the usual perturbative arguments for triviality (the Moscow zero charge effect) do not apply. However, new methods of analysis or very powerful lattice gauge theory computer simulations may be necessary to decide this crucial issue. Even if vacuum polarization renders all such four-dimensional field theories trivial, the novel physical effects described here might occur in other models, in other dimensions. For example, the Schwingcr-Dyson equations for quenched QED4 occur in the large-N expansion of multi-flavor QED in three dimensions as well as in various grand unified technicolor models in four dimensions [ 7 ]. It would be quite interesting to find analogous examples of the scalar-vector ladder graph equations discussed here

272


as well as scalar f ields with large, negat ive anomalous

d imens ions .

We are grateful to E rhard Seiler and Peter Weisz

for s t imulat ing discussions. The work of A.K. is suppor ted by A lexander yon Humboldt Sti ftung. The

work of J .B.K. is part ia l ly suppor ted by the NSF under grant NSF PHY87-01775. E.D. is suppor ted by

the NSF grants PHY82-017853 and DMR-88-128852

and in part by funds f rom NASA.

References

[ 1 ] P.I. Fomin, V.P. Gusynin and V.A. Miransky, Riv. Nuovo Cimento 6 (1983) I; V.A. Miransky, Nuovo Cimento 90 A ( 1985 ) 149.

[2]J. Kogut, E. Dagotto and A. Koci~, Phys. Rev. Len. 60 (1988) 772;61 (1988) 2416.

[3] J. Kogut, E. l)agotto and A. Koci6, Phys. Rev. Lett. 62 (1989) 1001; A. Koci~, E. Dagotto and J. Kogut, Phys. Lett. B 213 ( 1988 ) 56.

[4] C.N. Leung, S. Love and W. Bardeen, Nucl. Phys. B 273 ( 1986 ) 649.

[5] A. Casher, Phys. Lett. B 83 (1979) 395. [61A. Koci~, Phys. Lett. B 207 (1988) 489. [ 7 ] J. Kogut, E. Dagotto and A. Koci~, Nucl. Phys. B 317 ( 1989 )

253; B 317 (1989) 271. [8] K. Wilson, Phys. Rev. 179 (1969) 1499. [ 9 ] G. 't Hooft, in: Recent developments in gauge theories, eds.

G. 't Hooft et al. ( Plenum, New York, 1980). [ 10 ] M. LiJscher and P. Weisz, Nucl. Phys. B 290 [ FS20 ] ( 1987 )

25; for a recent review see, for example, K. Huang, J. Mod. Phys. A4 (1989) 1037.

[11 ] A. Hasenfratz and P. Hasenfratz, Phys. Rev. D 34 (1986) 3160.

[ 12] S. Coleman and E. Weinberg, Phys. Rev. D 7 (1973) 2887. [ 13 ] M. Veltman, Acta Phys. Pol. B 12 ( 1981 ) 437;

R. Kaul and P. Majundar, Nucl. Phys. B 199 (1982) 36.

273


A RECALCULATION OF THE GRAVITATIONAL MASS D IFFERENCE BETWEEN THE K AND ~o MESONS

1.R. KENYON School of Physics and Space Research, Birmingham University, PO Box 363, Birmingham B I 5 277] UK

Received 27 October 1989; revised manuscript received 7 December 1989

The method first applied by Good to determine the gravitational mass difference A,'/s between the K and I

Volume 237, number 2 PHYSICS LETTERS B 15 March | 990

will lead to a phase difference between the K and I~ components of K~_ (or K ): their de Broglie frequen- cies will then differ by AMg~0g. Such a phase differ-

0 0 ence, if it exists, would lead to Ks-KL mixing and would contribute to the observed 2n decay of KL. It is straightforward to express this contribution in terms of the standard mass matrix for the neutral K meson system. Referred to the (K , I~ ) basis this mass matrix has the form

M,, M,2) Mt2 M22 '

where the quantity

6=Re(Mr1 -M22)

is the CPTviolating mass difference between K and I~ , and

~'=Im(Mtl -M22)

is half the difference in their widths. When 6 is interpreted as arising from a difference in gravitational energy we have

6= AMs~0 8 . (1)

Barmin et al. [ 11 ] have made a detailed study of the consequences of CPT and non-CPT violating transi- tions for the neutral K meson system. They introduce the parameter

1 M l l -M22 zJ= 2 AM-~iAF '

whose real and imaginary parts are

l AM6- AFT I AMy+ AF6 Re( J ) = Im(A) = 2 ~,~2 [_ 14/~'2 ' 2 ~r2. .{_ lap2-

Here AM is the mass difference between the KL and K mesons and AT'is the difference between their decay widths. From the Particle Data Group tables [ 12 ]

AM+ iAF= (3 .52- 3.94i ) 10- 12 MeV.

Then we have, since AM and AF are nearly equal,

),+6 ~-6 Re(,cl)~ 4AM' Im(Lt)~. 4A~-~"

Barmin et al. [l l] reach the conclusion that IAI ~< l0 -4 with the main uncertainty coming from the poor determination of Poo. In subsequent anal-

yses Tanner and Dalitz [ 13 ], and Briere and Orr [ i 4 ] put upper limits of around 10 -4 on both Re(A) and Ira(A). These results indicate that both 6 and y are small. For the purposes of the discussion here, it will be assumed that whatever CPTviolation is present is due entirely to 6, as would be the case if its origin lies in a difference in gravitational energy between K and I~ . Then (dropping the sign )

d Im(A) ~ 4M" (2)

Combining eqs. ( 1 ) and (2) gives

AMg~0g =4AM Im (A), or

A,~ G =4A~4 Im(d) . (3)

The experimental values of AM, Im(zl) and cp s are therefore required in order to set a limit on AMg. The experimental value of AM is 3.52 10 -~2 MeV. The NA31 Collaboration have reported the most precise determination of neutral K meson decay CP viola- tions decay phases to date [ 15 ]. They use the Bell- Steinberger relation

ira(A) -~o , - ~0+ _ - -~ ~0oo

Re(~)

to calculate Im(zl) from their measurements, and obtain

Im(A)~


mass attributed by Lynden-Bell et al. to the cluster, over and above the mass normally occupying such a volume, is 2.7 1016 h~ "t solar masses or 5.4 10 46 hff ~ kg. Then if t08 is determined mainly by the "great attractor"

(08 = G M / rc 2 ,

where G is the universal gravitational constant, M is the excess mass of the supergalactic cluster and r is its distance from the earth. This gives

tp 8 =2.99 10 -5 ,

in which the dependence on ho has cancelled out. Substituting these current measurements of AM,

Im(LJ) and q~8 into eq. (3) yields

4X2X 10-4X 3.52X 10 -12 2.99X 10 -5

=9.5X 10 - I t MeV,

and

zXM8 ~


References

[ I 1L.I. Schiff, Phys. Rev. Lett. 1 (1958) 254. [2] P. Morrison, Am. J. Phys. 26 (1958) 358. [3] M.L. Good, Phys. Rev. 121 (1961) 311. [41 R.V. E/Stv6s, V. Pekar and E. Fekete, Ann. Phys. (Leipzig)

68 (1922) 11. [51P.G. Roll, R. Krotkov and R.H. Dicke, Ann. Phys. (NY)

26 (1964) 442. [6] L.I. Schiffand M.V. Barnhill, Phys. Rev. 151 (1966) 1067. [7] F.C. Witteborn and W.M. Fairbank, Phys. Rev. Lett. 19

(1967) 1049; Rev. Sci. Instrum. 48 (1977) 1. [8 ] W.M. Fairbank, Experiments to measure the force of gravity

on positrons, in: Proc. XXIII Rencontre de Moriond (Les Arcs, Savoie, France, 1988), eds. O. Fackler and J. Tran Thanh Van (Editions Fronti~res, Gif-sur-Yvette, France).

[ 9 ] S. van der Meer, Stochastic damping of betatron oscillations in the ISR, CERN report CERN/ISR-PO/172-13 ( 1972 ).

[ 10] N. Beverini et al., A measurement of the gravitational acceleration of the antiproton, CERN report CERN/PSCC/ 86-2, PSCC/P94 (1986).

[ 11 ] V.V. Barmin et al., Nucl. Phys. B 247 (1984) 293. [12] Panicle Data Group, G.P. Yost et al., Review of particle

properties, Phys. Lett. B 204 (1988) I. [ 13 ] N.W. Tanner and R.H. Dalitz, Ann. Phys. ( NY ) 171 ( 1986 )

463. [ 14] R.A. Briere and L.H. Orr, Phys. Rev. D 40 (1989) 2269. [ 15 ] NA31 Collab., D. Fournier, XIV Intern. Syrup. on Lepton

and photon physics (Berkeley, CA, August 1989); K. Peach, private communication.

[ 16] D. Lynden-Bell et al., Astrophys. J. 326 (1988) 19. [ 17 ] V.B. Braginskii and V.I. Panov, Sov. Phys. JETP 34 ( 1972 )

463. [18] K.I. Macrae and R.J. Riegert, Nucl. Phys. B 244 (1984)

513. [ 19 ] J. Badier et al., Phys. Left. B 93 (1980) 354.

277


COHERENT PRODUCTION OF L IGHT SCALAR OR PSEUDOSCALAR PARTICLES IN BRAGG SCATI 'ERING

W. BUCHMOLLER ..t, and F. HOOGEVEEN a

a Institutfur Theoretische Physik, UniversitiR tlannover, D-3000 Hannover, FRG b Deutsches Elektronen-Synchrotron DESE D-2000 Hamburg, FRG

Received 6 October 1989; revised manuscript received 29 December 1989

X-rays penetrating crystals can produce light scalar or pseudoscalar particles via the Primakoffeffect. We compute the intensity of scalar and pseudoscalar particles in the reflected beam of a Bragg reflection. We then estimate the sensitivity of a realistic experiment, using synchrotron radiation and a double crystal spectrometer, which can test for the existence of light scalar and pseudoscalar particles.

New interactions with a characteristic mass scale M can manifest themselves at energies much smaller than M through pseudo-Goldstone bosons, whose masses and couplings to "l ight" particles scale as 1 / M. Two particularly interesting examples, which arise in minimal extensions of the standard model, are the axion and dilation. The axion [1,2], the pseudo- Goldstone boson of a spontaneously broken chiral Peccei-Quinn symmetry invented to solve the strong CP problem, acquires its mass through the vacuum expectation value of the chiral anomaly; the dilaton, a Brans-Dicke type scalar [3 ], which arises in theories with spontaneously broken scale invariance [4], obtains its mass from the vacuum expectation value of the conformal anomaly [5,6]. Both scalar particles, axion and dilaton, appear together if the standard model is the low energy limit of a theory with spontaneously broken superconformal invariance [7].

Light scalar particles can be produced and detected by means of the Pr imakoff process (cf. fig. I ), i.e.,

Y- i . . . . . (3,0-

E, 8

the mixing with photons in an external electromagnetic field. Following Sikivie [8], various sugges- tions have been made to search for light scalars by means of external magnetic fields [9,10 ]. Here we will explore the feasibility to make use of the strong electric fields seen by X-rays penetrating crystals for photon-scalar conversion. The basic idea is illus- trated in fig. 2: The incident X-ray is reflected from a crystal under a Bragg angle OB; the reflected beam contains scalar particles which are produced in the crystal via the Pr imakoff effect; only the scalar particles penetrate the absorber and produce in a second Bragg reflection the outgoing electromagnetic wave which is detected.

Let us now calculate the intensity of the final photon beam. To be specific we concentrate on the pseu-

CRYSTAL

X-RAY I : : .i=, I y\ eB,'~.OB

007/ SORBER ;/

Xx~A[ :: :1 DETECTOR - , , , . y ,,J'a',;-

CRYSTAL

Fig. 1. Primakoff process: photon-axion (dilaton) conversion in an external electromagnetic field.

Fig. 2. Experimental setup to search for light scalar particles in Bragg scattering (see text).


Volume 237. number 2 PHYSICS LETTERS B 15 March 1990

doscalar axions whose electromagnetic interactions are described by the lagrangian density

if,= t 4F~,vl,'j,v,b (Oua)~ 1,~2~2 1 __ _ ~ . . . . . - "~ Fu~PU"a .

(1)

The corresponding field equations read

1 (D+m~)a=- -~E.B , (2a)

0 ) VB- ~E=- B~a-EXVa , (2b)

V-E= IB .Va . (2c)

We note that in models where the two-photon coupling of axions is generated through the triangle anomaly, M is about two orders of magnitude larger than the mass scale fo f the spontaneous symmetry breaking.

In the electrostatic field of a screened Coulomb potential,

Ze E=-VO, O=-~nreXp( - r / ro ) , (3)

where Z is the nuclear charge and ro the screening length, an incoming plane electromagnetic wave,

B(t, x) =B()exp[ i (ogt -k .x) ] , (4)

generates an outgoing spherical axion wave

0 a ( t ,x ) - = ~a(t ,x )

- F" (20) e~'B( ) lexp[ i (cot -k r ) l , (5a) 4riM r

where

k~(20) =k 2 J" d3x O(x) exp(iq.x) ,

q=k' -k , k'=ke~, k=lk l , e ,=x/ r ,

20= ,~ (k, k' ) . (5b)

Here we have neglected the axion mass. In the case of non-vanishing mass k' = ( k2 - m ] ) ~ /2. For a screened Coulomb potential one easily verifies:

Zek 2 Fa(20) = ( 1/ro)Z+2kZ( i - cos 20) " (6)

It is convenient to define an average electric field by

1 E(k, 20)= ~-3 Fa(20) , (7)

where d is the lattice spacing of a cubic lattice. Eq. (5a) yields for the differential cross section of

unpolarized photon-axion conversion:

do" a 1 d,Q - 32n2M "2 F~(2O) sin220. (8)

The corresponding Thomson cross section for electromagnetic scattering by an atom reads [ 11 ]

- i

do'~_d_~_ (a ) - F~(2o, 1 + c O s 2 2 O m 2 ' (9a,

with

1 f d3xp(x) exp(iq-x) , F , (20) = e

Fv(0) =Z. (9b)

Here rn is the electron mass, Fv is the atomic structure factor and p is the electron charge density of the atom. The formfactors Fa and k'~ satisfy the relation

ek2 [z -Fy(20) ] (10) F, (20) = ~-

Inserting this equation into eq. (8) yields an expression for the differential photon-axion cross section which has previously been derived by Raffelt [ 12 ]. We note that for a neutral atom, contrary to the Thomson cross section, the photon-axion cross section reaches its maximum at 20~ n/2 and vanishes in the forward direction.

Given the scattering amplitudes for elastic photon scattering and photon-axion conversion by a single atom the coherent scattering by a crystal can be calculated using "Darwin's dynamical theory" [ 11 ]. Here the first step is to compute the wave scattered from a single layer of pointlike scattering centers. In- terference of the outgoing spherical waves leads to reflected and transmitted plane waves. In the case of Thomson scattering one finds for the two polariza- tions parallel and perpendicular to the scattering plane (cf. ref. [11 ], fig. 3) of reflected (R) and transmitted (T) waves:

EcR) )(20)El~ exp[i(oot--t ' -x)] ,( )(t, x) =ip, ( , ) , ( l l a )

279


Fig. 3. Scattering plane and Bragg angle.

E~r: ~ )(t, x )= [! + ip~.t , (0) ]

X ~-~o~ exp[i(oJt-k'x)] "'i1( ) ( l lb )

where

P l (20) = aFr(20)N~2 Icos2OI (12a) rosin O

aFv(20)N'2 -p (20) . (12b) p~ (28)= ms inO

In eq. ( 1 I b) we have neglected the attenuation of the transmitted wave. In eq. ( 12 ) N~ denotes the number of scattering centers per unit area.

The scattering amplitude for photon-axion conversion can be read off from eq. (5). For the inverse process one obtains from eqs. (2):

,%(20) e,. X (e,. X/~) B( t ,x ) - 47tM

xa I exp[iUot-k'x) ] , r

k=k/ Ik l . (13)

In eqs. (5) and (13) only the polarization contrib- utes where the magnetic field is parallel to the scattering plane, i.e., B=B~. After integration over one layer of atoms one obtains from eqs. (5) and (13) for the reflected waves

a~(t,x)=i(20)BlO)exp[i(o~t-k' .x)], (14a)

b~R)(t,x)=i(20)atO~exp[i(~ot-k'.x)], (14b)

where

F~(20)N,). . (28) - ~ ~ s ln2e . (15)

The transmitted waves are not modified since the forward scattering amplitude vanishes.

From eqs. ( 11 ) and (14) we can now calculate the intensities of the outgoing axion (photon) waves

produced by the photon-axion (anion-photon) conversion in a crystal. Let the layers of atoms be located at z = 0, z = - d, .... z = - nd. The amplitudes of transmitted and reflected waves at the nth layer are B~r)-- 4,+l, BIR~=Bn, dtT~=--Cn+l, d~R)=-Dn (cf. fig. 4). From ( 11 ) and (14) we then obtain the set of coupled equations (p=p(20), po=pFv(O)/ Fr( 20), =~( 20) ):

B~ =ip exp( -2inO)A, + ( 1 +ipo)B,+ ~

+i~exp( -2 in)C , , (16a)

A,+ ~ = ( 1 +ipo)A~ +ipexp(2inO)B,+l

+i~exp(2 inO)D,+, , (16b)

Cn+~ =i~exp(2in)B~+ ~ +C~. (16c)

Dn =ieexp( - 2in )A, + D~+ ~ , (16d)

where 0=kds in 8. Clearly, for 0= the Bragg condition is fulfilled and one has constructive interfer- ence. In the case ~=0 eqs. (16) reduce to the equations well known from ordinary Thomson scattering (cf. ref. [ 11 ] ).

In order to obtain the intensity of the outgoing electromagnetic wave after the double scattering shown in fig. 2 one has to compute from eqs. (16) first the ratio Do/Ao with the boundary condition go = 0, and second the ratio Bo/Co with the boundary condition Ao=0. The calculation can be carried out as for Thomson scattering (ref. [ 11 ] ) and one obtains up to terms oforderp~, C':

.%% '~ B.-.Uo

- "~ '~ z=O

B- ?b. I A'~'C" z=-d

z=-2d

Bn..."D~.~ C nl, :1.~

z=-nd

Fig. 4. Beams of photons and scalar particles in the crystal: A(B)=Iransmitted (reflected) photon beam, C(D)=trans- milted (reflected) axion/dilaton beam.

280


Do Bo i~ Ao - Co - i - rexp( -2 i0 ) ' (17a)

where

r=x+[x2-exp(2 i0 ) ] 1/2, I r l< l ,

1 ~= [(l+ipo)2+p2+exp(2iO)]. (17b) 2( 1 +ipo)

It is instructive to express the ratio (17a) in terms of the deviation of the scattering angle O from the Bragg angle OB defined by

2dsin OB (1 l -n ) s[n-~OB =m2, m=l ,2 ..... (18)

where

po), n=l - ~ndsin O, (19)

is the index of refraction. After some algebra one obtains (cf. ref. [ 11 ] )

Do ~ id Ao pi(Ao+d)+-(A2--62) t/2'

d=O-Oa, (20)

where

1). 1 A-

n / sin 2On '

Ao=d Fv(0) (21) Fv (2On) '

l - '= a__ N2Fv(20a). (22) m

I is the penetration depth of the X-ray into the crystal, and N denotes the number of scattering centers per unit volume. From eq. (20) it is obvious that the ratio Do/Ao is strongly peaked for scattering angles O in the range [On-A, OB+A], which is the width of an ordinary Bragg peak.

Within the width of the Bragg peak the transition probability for photon-axion and axion-photon conversion takes a simple form. From eqs. (7), (12), ( 15 ), ( 17 ), (20) and (22) one obtains

p= D~o 2 = ~B 2= ~_~)2

- 2

(23)

This means that the photon-axion transition probability is essentially determined by two parameters: the average electric field E seen by photons in the crystal and the penetration depth / into the crystal. Eq. (23) is analogous to the expression obtained for the photon-axion transition probability in an external magnetic field [ 10 ].

What sensitivity with respect to the mass scale M can be reached in a realistic experiment? Typical values for penetration depth and Bragg angle are 1~ 1 lam and O~ 10; for the average electric field E (cf. eq. (7)) one finds for Z= 10, d=2 A and ro= (0.1, 0.5, 1.0) ,g, [13] the values E= (0.07, i.0, 1.8) keV 2. Note that the average electric field E strongly depends on the screening length ro which is smaller than the lattice spacing d. We emphasize that this micro- scopic electric field is much stronger than the mac- roscopic magnetic field attainable with dipole mag- nets ( 1 Tesla ~ 200 eV 2 ). For a photon energy to ~ 10 keV the width of the Bragg peak is d~ 10 -4. An in- tense source of X-rays will be provided by the Euro- pean Synchrotron Radiation Facility. By means of undulators one expects to achieve a brightness of q~~ 10tS/s (0.1% BW); the divergence of the beam is given by 6~'/-j =m~/E~ 10 -4 for 5 GeV electrons [ 13 ]. From d2/,;t = cot O dO and dO~ 10- 4 one finds that photons within a bandwidth d2/2 ~ 10-3 contribute to the Bragg reflection. Hence the number of photons N "bs in the final state is given by the product of the brightness tD, the probability p2 of the photon-axion-photon transition and the running time T. Realistic requirements are N~= 10 and T= 100 d. From eq. (23) we then obtain for the mass scale M:

/~ l sin 20"] M>I103GeV l~-eV zlJam b-~]

( 7" lo '" X 10~S/s0.1%BW lOOdNObS ] (24)

This lower bound will be slightly decreased if finite detection efficiency and temperature effects, i.e., the

281


Debye-Waller factor, are taken into account. In the case of nonvanishing axion mass the axions are emit- ted under an angle Oa < OB. In principle the experiment is sensitive to masses ma < o sin On, where oJ is the photon energy.

In addition to the Primakoffeffect light scalar particles can also be produced in a Compton-type process via their direct coupling to electrons. For a Yukawa coupling strength g~ rn/fwe expect the production cross section for scalars ,t to be of the same order of magnitude or even larger than the Primakoff cross section, whereas for pseudoscalars the cross section will be suppressed by (v/c)'-, where v is the electron velocity. If the two-photon coupling of the scalar particles is radiatively generated through the triangle anomaly. The strongest bound on the mass scalefofspontaneous symmetry breaking could come from this Compton-type process. However, further investigations arc necessary in order to clarify under what conditions this scattering can take place coherently.

Instead of Bragg scattering one can of course also consider Laue scattering, where the penetration depth is much larger. For 100 keV photons and scattering angle 8~ 1 one can achieve /~ l cm [13]. This would improve the lower bound (24) on the mass scale Mby thrce orders of magnitude up to ~ 106 GeV. Such an experiment would clearly be very interesting. It remains to be seen, however, whether the cf- fective electric field f which appears in eq. (23) is the same as for Bragg scattering. A detailed calculation will be published elsewhere.

We conclude that the proposed Bragg scattering experiment can test for the existence of light scalar par- ticlcs with masses up to 10 keV and a mass scale for the two-photon coupling up to 103 GeV; in Laue scattering it may be possible to reach even l0 ~ GeV of the interaction mass scale. This range of parameters has not yet been explored by other laboratory experiments [ 14,10] and is not excluded by astrophysical bounds from the Sun which apply to scalars with masses below ~ 1 keV [ 12,15 ]. An interesting laser cxperiment for photon-axion conversion is an external magnetic field, which has recently been proposed

~'~ Here "'scalars" denotes particles with Jt'=O+. Otherwise in this paper "scalar" refers to particles with JP=0 + and/or Je=0- .

[10], is sensitive to interaction mass scales M> 5 108 GeV, possibly even M> 1 10 t~ GeV, but only to masses below ~ 1 cV. Both experiments, as well as the recent proposal based on the M68baucr effect [16], cannot cxceed a range of parameters which appears to be almost excluded by astrophysical bounds inferred from helium burning stars [ 17 ], which apply to scalars with masses below ~ 10 keV. However, since laboratory cxperiments are independent of models of stellar evolution, they can never- theless significantly contribute to our present knowl- edge about masses and interaction strengths of very light scalar panicles, and complement astrophysical considerations.

We would like to thank H.U. Everts, M. Henzler, P. Kienle, G. Raffelt and H. Schulz for helpful comments. We are indebted to G. Materlik for his help in estimating the sensitivity of realistic experiments.

References

[ I ] R.D. Peccei and H. Quinn, Phys. Rev. Lcn. 38 (1977) 1440; S. Weinberg, Phys. Rev. Lett. 40 (1978) 223; F. Wilczek, Phys. Rcv. Left. 40 (1978) 279.

[2] J.E. Kim, Phys. Rev. Lett. 43 (1979) 103; M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B 166 (1980) 493: M. Dine, W. Fischler and M. Srednicki, Phys. Len. B 104 ( 1981 ) 199.

[3] P. Jordan, Z. Phys. 157 (1959) 112; C. Brans and R.H. Dicke, Phys. Rev. 124 ( 1961 ) 925.

[4] G. Mack, Nucl. Phys. B 5 (1968) 499; P.G,O. Freund and Y. Nambu, Phys. Rev. 174 (1968) 1741.

[5] R.D. Peccei, J. Sola and ('. Wetterich, Phys. Left. B 195 (1987) 183.

[6] W. Buchmfiller and N. Dragon, Phys. Lett. B 195 (1987) 417.

[ 7 ] W. Buchmiiller, Erice Workshop on H iggs particles ( 1989 ). [8] P. Sikivie, Phys. Rev. Len. 51 (1983) 1415. [9] A.A. Ansel'm, Sov. J. Nucl. Phys. 42 (1985) 936;

L. Maiani, R. Petronzio and G. Zavattini, Phys. Left. B 175 (1986) 359; M. Gasperini, Phys. Rev. Len. 59 (1987) 396; G. Raffelt and L. Stodolsky, Phys. Rcv. D 37 (1988) 1237.

[10] K. van Bibberet al., Phys. Rev. Len. 59 (1987) 759. [ 11 ] See, for instance, B.E. Warren, X-ray diffraction (Addison-

Wesley, Reading, MA, 1969 ). [ 12] G.G. Raffelt, Phys. Rev. D 33 (1986) 897. [ 13 ] G. Materlik, private communication;

see also ESRF report ( 1987 ).

282


[14] M. Davier, in: Proc. XXIII Intern. Conf. on High energy physics (Berkeley), ed. S.C. Loken (1986) p. 25.

[ 15 ] M. Yoshimura, in: Proc. XXIII Intern. Conf. on High energy physics (Berkeley), ed. S.C. Loken (1986) p. 189; G.G. Raffelt, in: Proc. XXIV Intern. Conf. on High energy physics (Munich), eds. R. Kotthaus and J.H. Kiihn (1988) p. 1519.

[ 16 ] A. de R6jula and K. Zioutas, Phys. Lett. B 217 ( 1989 ) 354. [ 17 ] G.G. Raffelt, private communicalion;

G.G. Raffelt and D.S.P. Dearborn, Phys. Rev. D 36 ( 1987 ) 761.

283


NEGATIVE B INOMIAL MULTIPL IC ITY D ISTRIBUTIONS AND THE ENERGY DEPENDENCE OF THE PARAMETERS

C.P. SINGH, Saeed UDDIN Department of Physics, Banaras ttindu University, Varanasi 221 005, India

and

A.N. KAMAL Theoretical Ph),sics Institute and Department of Physics. University of Alberta, Edmonton, Canada T6G 2Jl

Received 27 November 1989

A parametrization relating the parameters k and r~ of the negative binomial distribution (NBD) is proposed; the energy dependence of the parameter k is derived and compared with the available experimental data.

The study of multiplicity distributions in high energy hadronic [ 1 ], semileptonic, ieptonic [2,3], as well as hadron-nucleus [ 4 ] and nucleus-nucleus collisions [5] has revealed many striking systematics. For example, charged particle multiplicity distributions have a negative binomial (NB) shape over a very wide energy range,

P~(h ,k )=(n+~- l ) (~/k )~( l+t i /k ) -~-k , (1)

where r~ and k are two free parameters varying with energy. The parameter r~ has the interpretation of the average multiplicity. These two parameters are re- lated to the dispersion D= (n -~- r~ 2 ) 1/2 as follows:

D=a+a2/k , (2)

and k decreases with increasing energy; i.e., the distribution becomes broader and broader than the Poisson distribution. One of the challenging problems facing us today is the interpretation of the em- pirical relation ( 1 ) in terms of a general mechanism common to hadronic, leptonic and semileptonic pro- cesses. Some attempts have already been made to derive the negative binomial distribution (NBD) from general principles using a stochastic model [6], a string model [ 7 ], a cluster [ 8 ], a stationary branch-

ing process [9] and a two-step model of binomial cluster production and decay [ 10 ]. However, it is still difficult to understand why the same distribution fits such different reactions. Moreover, there is still no understanding of the behaviour of the parameter k which, according to the fits, decreases with energy. For example, in the model of Giovannini and Van Hove [ 8 ], k is the ratio of cascading and partial stim- ulated emission and it is rapidity dependent. How- ever, the dynamics is missing and k appears to be just a parameter required to fit the data. In particular we are not able to derive the correct energy dependence of k from either of the models available so far in the literature [ 1 1 ]. The parameters r~ and k are found to vary. smoothly with energy indicating the breakdown of KNO scaling [ 12 ]. However, it appears that the energy dependence of r~ and k cancel out so that an approximate scaling is observed over a very wide range of energy. In this paper, we attempt to param- etrize the relation between r~ and k and thus obtain the relation illustrating the correct energy dependence of the parameter k.

Recently, UA5 data [ 13 ] have demonstrated that the energy dependence of 1/k is linear in In x/s where v/~ is the total centre-of-mass energy. When corn-


Vo ume 237 number 2 PHYSICS LETTERS B

bined with ISR, FNAL and Serpukhov data [ 13 ] the linearity in In x/~ is extended to the energy range x/S= 10-900 GeV. The fitted form of l /k is

l/k=a'+b' Inx/~ , (3)

with a ' - - -0 .104+0.004 and b'=0.058+0.001. However, Szwed et al. have noticed that this linear parametrization fails at energies below 10 GeV where Ilk not only turns out to be negative but increases more rapidly as the energy increases [ 14]. Recently, Chliapnikov and Tchikilev have derived [ 9 ] a relation between the parameters k and m (=a/k) using the concept of the stationary branching process [ 15 ]:

l /k=a+blnm, (4)

where the parameters a and b are determined from a fit to the experimental data and their values are a=0.119_+ 0.003, b=0.068 + 0.004. Data on pp and 1013 inelastic and non-diffractive samples do not quite satisfy (4) very well and hence the presence of a quadratic term In2m in (4) is strongly suggested by the data. Chliapnikov and Tchikilev have already recog- nised that (3) and (4) are incompatible.

We suggest a parametrization relating Ilk with as follows:

l/k=aj +bl In a . (5)

We now confront relation (5) with the presently available experimental data for the non-diffractive sample of charged panicles in pp and plb collisions. The values of 1/k and a for pp and P0 data are taken from the compilation in ref. [ 14]. We notice (fig. 1 ) that the experimental data can be described reasona- bly well (x2/NDF=3.40) by the relation (5) with fitted values of the parameters a~ =-0.253_+0.005 and b~ = 0.154 + 0.003. The relation (5) thus reduces the two-parameter NBD to a one-parameter distribution. In this connection we want to mention that we have analysed the data published up to 1986. Re- cent results from the UA5 group indicate that the NBD does not describe their 900 GeV data. More- over, the old data have been updated [ 16]. The inclusion of this recent data into the form (5) in fact improves the fit and yields the parameters a t=-0 .245+0.004, bt=0.149-+0.002 with X2/ NDF=3.96.

We can now obtain the energy dependence of 1/k from (5) using the experimentally known energy de-

15March 1990

0.6

0.4

~' 0.7 [

-0.2

-0.41 i , i i , , , I i i i i , , , , I 10 100

Fig. 1. Parametrization of eq. (5); solid line. Data taken from the compilation ofSzwed et al. [ 14J.

pendence of a. As pointed out by the UA5 group, the energy dependence of the average multiplicity can be described by the following relation [ 13 ]:

a=A'+B' In x /s+C' ln2v/S, (6)

where A'=2.7_+0.7, B'=-0.03_.+0.21 and C'= 0.167 + 0.016 as obtained from the fit to the experimental data. Alternatively, we find another useful relation [ 13]:

a=a+b's~, (7)

where o t=-7 .0+ 1.3, f l=7.2+ 1.0 and 7=0.127+ 0.009. Both the above relations (6) and (7) reveal a very close agreement with the experimental data in the entire energy region up to x/~=900 GeV. Using eqs. (6) and (7) in (5), we obtain the energydepen- dence of the parameter k as follows:

1/k=a, +b~ ln(A' +B' In v/s+ C' ln2x/~) (8)

and

l /k=A+Bln x/~+Cln( l +D/s y) , (9)

respectively. HereA=at +b~ in ,8, B=2b~y, C=b~ and D= o~/,8. The result of our calculation is shown in fig. 2 and compared with the experimental data. We find that the same energy dependence of 1/k is obtained from eqs. (8) and (9) at high energies. However, at low energies where 1/k is negative, the experimental data show close agreement with the energy dependence derived from eq. (9). At this point we can also

285

qcd corrections to cp violation in higgs exchange

Documents