



SLAC-PUB-95-6904 W Anomalous Moments and the Polarization Asymmetry Zero

in fle ! W *

\Lambda

Stanley J. Brodsky and Thomas G. Rizzo

Stanford Linear Accelerator Center Stanford University, Stanford, California 94309

and Ivan Schmidt Universidad Federico Santa Mar'ia

Casilla 110-V, Valpara'iso, Chile

ABSTRACT We show from general principles that there must be a center of mass energy, ps0, where the polarization asymmetry A = \Delta oe(fle ! W *)=oe(fle ! W *) for circularly-polarized photon and electron beams vanishes. In the case of the Standard Model, the crossing point where the asymmetry changes sign occurs in Born approximation at psfle = 3:1583 : : : MW ' 254

GeV. We demonstrate the sensitivity of the position of the polarization asymmetry zero tomodifications of the SM trilinear flW W coupling. Given reasonable assumptions for the

luminosity and energy range for the Next Linear Collider(NLC) with a backscattered laser beam, we show that the zero point, ps0, of the polarization asymmetry may be determined with sufficient precision to constrain the anomalous couplings of the W to better than the 1% level at 95% CL. In addition to the fact that only a limited range of energy is required,the polarization asymmetry measurements have the important advantage that many of

the systematic errors cancel in taking cross section ratios. The position of the zero thusprovides an additional weapon in the arsenal used to probe anomalous trilinear gauge couplings.

Submitted to Physical Review D. \Lambda Work supported in part by Department of Energy contracts DE-AC03-76SF00515 and DE-AC02- 76ER03069, and by Fondo Nacional de Investigaci'on Cient'ifica y Tecnol'ogica, Chile, .

I. INTRODUCTION Precision measurements of Z-pole observables at LEP and SLC [1] combined with the new W mass determinations [2] and the discovery of the top quark at the Tevatron [3, 4] have demonstrated that the Standard Model (SM) provides an excellent description of physics below the electroweak scale. There are many reasons to believe, however, that new physics beyond the SM must exist, but it remains unclear just how or where it will first be directly observed. The scale of such new physics may not be far away, perhaps ' 1 TeV, in which case it will surely manifest itself at existing or planned colliders.

One of the most sensitive measures of new physics beyond the SM are the values of the electroweak moments of the various leptons, quarks, and gauge bosons. In the SM, the anomalous magnetic moments \Delta _ = (g \Gamma 2)eS=2M of spin S = 12 and S = 1 fundamental fields (with mass M )and the anomalous electric quadrupole moments \Delta Q = Q + e=M 2 of the vector bosons vanish at the tree level due to the requirements of gauge invariance and renormalizability, thus ensuring a quantum field theory which has maximally convergent high energy behavior. Deviations from these canonical values of the magnetic dipole and electric quadrupole moments beyond the usual SM radiative corrections [5] may reflect new physics or new interactions at high energies such as supersymmetry [5], technicolor, or compositeness [6].

Precision measurements at the Z-pole and elsewhere have already placed rather stringent restrictions on anomalous (V = fl; Z)f _f couplings [7]. However, direct experimental probes of the trilinear gauge boson couplings, V W +W \Gamma , are still at a rather early stage [8, 9, 10]. If the energy scale of the new physics is indeed of order 1 TeV, it is anticipated on rather general grounds that these anomalous trilinear couplings can be no larger than O(10\Gamma 2) [11]. Experiments have yet to achieve sensitivity at this level. However, it is ex2

pected that the vector boson couplings will eventually be probed at the precision of 1% or better at high energy hadron and e+e\Gamma colliders through processes such as W -pair production from fermion pair annihilation, e+e

\Gamma ! W +W \Gamma , q _q ! W +W \Gamma , and associated production,

q _q0 ! W Z0; fl.

The advent of backscattered laser beams at e+e\Gamma colliders will allow tests of the anomalous couplings of the gauge bosons through measurements of the high energy photon

collision processes flfl ! W +W

\Gamma ; flfl ! Z0Z0; fle ! Z0fl; and fle ! W *: A distinctive

feature of the fle ! W * process is that it isolates the on-shell photon flW W vertex in a model-independent manner. In this paper we shall show that measurements of the fle ! W * cross section with polarized photon and polarized electron beams can be used to test novel features of the canonical couplings of the W and provide high precision measurements of its magnetic and quadrupole moments at a precision below 1%.

A remarkable consequence of the canonical couplings of fermions and gauge bosons in the SM is that the integral that appears in the Drell-Hearn Gerasimov sum rule(DHG) [12, 13] vanishes. This interesting observation was first made for quantum electrodynamics and also for the more general case of the SM by Altarelli, Cabibbo and Maiani [14]. Even more generally, one can use a quantum loop expansion to show [15] that the logarithmic integral of the spin-dependent part of the photoabsorption cross section, i.e.,Z

1 *th

d*

* \Delta oeBorn(*) = 0 (1)

for any 2 ! 2 SM process fla ! bc at the Born level. The particles a; b and c are arbitrary (so long as a carries non-zero spin!) and can be identified as leptons, photons, gluons, quarks, elementary Higgs, vector bosons, supersymmetric particles, etc. Here * is the photon laboratory energy and \Delta oe(*) = oeP (*) \Gamma oeA(*) is the difference between the photoabsorption cross section for parallel and antiparallel photon and target helicities. Similar arguments also

3

imply that the DHG integral vanishes at tree level for virtual photoabsorption processes such as `fl ! `Q _Q (with Q being a heavy fermion) and `g ! `Q _Q; the lowest order sea-quark contribution to polarized deep inelastic photon and hadron structure functions. Of course the sum rule does receives individual nonzero contributions in higher order perturbation theory in the SM from quantum loop corrections and the production of higher particle number final states. However, The DHG sum rule predicts that the final result is very small, of order ff times the square of the target's anomalous magnetic moment. The DHG sum rule thus also provides a highly non-trivial consistency check on calculations of the polarized cross sections.

In principle, one could use measurements of the logarithmic integral of the polarized photoabsorption cross section in Eq. (1) as a way to isolate the higher order radiative corrections and bound the deviations from the canonical SM couplings. Some of the most interesting applications and tests of the DHG sum rule in the SM would be to apply Eq. (1) to the reactions flfl ! W +W \Gamma , flfl ! Z0Z0; fle ! Ze, and fle ! W *: The delicate cancellation of the positive and negative contributions [16] of \Delta oe(fle ! W *) to the DHG integral calculated in Born approximation is evident in Fig. 1. On the other hand, if the W were to have non-zero anomalous magnetic and electric quadrupole moments, i.e., \Delta _W ; \Delta QW 6= 0, then the DHG integral for fle ! W * is not zero since the cancellations no longer take place [13]. In fact if the W had a point-like anomalous magnetic moment, then the DHG integral for the 2 ! 2 process diverges logarithmically at high energies.

In this paper we shall exploit the fact that the vanishing of the logarithmic integral of \Delta oe in the Born approximation also implies that there must be a center of mass energy, ps0, where the polarization asymmetry A = \Delta oe=oe possesses a zero, i.e., where \Delta oe(fle ! W *) reverses sign. We shall demonstrate the sensitivity of the position of this zero or `crossing point' (which occurs at psfle = 3:1583 : : : MW ' 254 GeV in the SM) to modifications of the

4

0.08 \Delta s( seg

)/s

0.06 0.04

-0.04

7883A12-95 1 10

pa2fl MW sin2qw

seg /MW

s = = 49.8 pb

100 0.02 -0.02

0

2~ ~

~

Figure 1: The Born cross section difference \Delta oe for the Standard Model process fle ! W * for parallel minus antiparallel electron/photon helicities as a function of y. The logarithmic integral of \Delta oe vanishes in the classical limit.

SM trilinear flW W coupling. As we will see, given reasonable assumptions for the luminosity and energy range for the Next Linear Collider(NLC), the zero point, ps0, of the polarization asymmetry may be determined with sufficient precision to constrain the anomalous couplings of the W to better than the 1% level at 95% CL. Since the zero occurs at rather modest energies where the unpolarized cross section is near its maximum, we will also see that an

electron-positron collider with ps = 320 \Gamma 400 GeV is sufficient for our analysis, whereas other techniques [17, 18] aimed at probing the anomalous couplings through the fle ! W * process require significantly larger energies. In addition to the fact that only a limited range of energy is required, the polarization asymmetry measurements have the obvious advantage that many of the systematic errors cancel in taking cross section ratios. The position of the zero thus provides an additional weapon in the arsenal used to probe anomalous trilinear gauge couplings.

5

II. CALCULATION OF THE POLARIZATION ASYMMETRY The total polarization-dependent cross section for fle ! W *, in the case where only the C- and P -conserving anomalous W W fl couplings are non-zero, can be written in the form

oe = (1 \Gamma P )(oeun + ,oepol) : (2)

Here, \Gamma 1 ^ P ^ 1 denotes the initial e

\Gamma beam polarization, oepol = \Delta oe, and the Stoke's

parameter, \Gamma 1 ^ , ^ 1, describes the circular polarization of the back-scattered laser photon.p

sefl is the center of mass energy of the e \Gamma fl collisions. We shall consider deviations from

the SM where the W has point-like (momentum-independent) anomalous magnetic and quadrupole couplings. The polarization-dependent part of the cross section, oepol, is then given by

oepol = eoe32x3 hT1 + (\Delta ^ + *)2T2 \Gamma \Delta ^T3 \Gamma *T4i ; (3)

where x = y2 = sefl =M 2W , eoe = p2GF ff ' 49:8 pb, and

T1 = \Gamma 24 \Gamma 80x + 104x2 \Gamma 32x(3 + x) log(x) ; T2 = x + 2x2 \Gamma 3x3 + 4x2 log(x) ; T3 = 48x(1 \Gamma x) + 16x(2 + x) log(x) ; (4) T4 = 64x(1 \Gamma x) + 32x(1 + x) log(x) :

We have used the standard notation of Hagiwara et al.[19] for the anomalous static moments of the W : \Delta _W = e2M

W (\Delta ^ + *) and \Delta QW =

\Gamma e M2W (\Delta ^ \Gamma *). The corresponding polarizationindependent term, oeun, is given by

oeun = eoe hT5 + \Delta ^T6 \Gamma *(* + \Delta ^)T7 + *2T8 + (\Delta ^ + *)2T9i ; (5)

6

where

T5 = (1 \Gamma 1x)(1 + 54x + 74x2 ) \Gamma (2 + 1x + 1x2 ) log(x)x ; T6 = \Gamma 1 + x + 2x

2

2x2 \Gamma

2 + 3 log(x)

2x ;

T7 = 12 ^\Gamma 1 + 1x + log(x)* ; (6)

T8 = (\Gamma 1 + x)

2

8x ;

T9 = 1 + 2x + x

2

32x2 \Gamma

1 + (1 \Gamma x) log(x)

8x :

These results were obtained through the use of MATHEMATICA and REDUCE and differ somewhat in their non-SM terms from other explicit results in the literature. However, the integration over angles of the helicity amplitudes obtained by Raidal [18] nicely reproduces the above expressions. Note that the effective values of \Delta ^ and * that are probed in the fle ! W * reaction are for on-shell photons and may in principle differ from those probed in e+e

\Gamma ! W +W \Gamma where the photon is time-like with q2 ? 4M 2

W .

In its fle ! W * manifestation, the DHG sum rule implies thatZ

1 1

oepol(x)

x dx = 0 ; (7)

for the tree graph SM cross section where the couplings of all the particles involved in the process are canonical. (The electron mass is also neglected here.) In Ref. [15] it was pointed out that the vanishing of the above integral is due to a rather delicate cancellation between the regions where oepol is positive (y = psefl=MW ! y0) and regions where it is negative (y ? y0). Here we denote the cross-over point where the integrand vanishes (i.e., the zero position) by y0 ' 3:1583. in the SM case. When anomalous W W fl couplings are present,

7

several things happen. First, since the couplings are no longer canonical the DHG sum rule will be violated. Indeed, since the fle ! W * cross section is not well-behaved in the y ! 1 limit when these point-like anomalous couplings are non-zero, we might also expect that the DHG integral does not even converge! This expectation is indeed realized by performing an explicit calculation employing a cut-off parameter, xm AE 1; to leading order in x\Gamma 1m we obtain (i.e., dropping all terms of order x

\Gamma 2

m or higher)Z

xm 1

oepol(x)

x dx = e

oe 64 "(\Delta ^ + *)

2 13 \Gamma 6 log(xm) \Gamma 8 log(xm)

xm !

\Gamma 16* + 64 log(xm)x

m `1 + * +

1 2 \Delta ^' # ; (8)

where we then take xm ! 1. Here we see that non-zero values of the sum \Delta ^ + * result in the DHG integral becoming logarithmically divergent. (Of course, as x gets large new physics effects, such as form factors and new particle production, arise to prevent the integral from truly diverging. This apparent divergence is simply the result of the break down of the point-like approximation for the anomalous couplings.) If \Delta _W = e2M

W (\Delta ^ + *) = 0 then

the integral converges and yields a finite result proportional to *. Note that the well-known

radiation amplitude zero [20] (which takes place at cos ` = 1 in both the doepol and doeun angular distributions) also occurs for this process whenever \Delta ^ + * = 0. Thus, if we could determine the value of the DHG integral directly from experimental data it would provide us a unique handle on possible non-zero values of \Delta ^ and *.

In practice, the collider energy as well as the maximum energy fraction carried by the backscattered laser light are restricted. For a 500 GeV (1 TeV) e+e\Gamma collider, only the range 1 ^ y ^ 5:4(10:4) is kinematically accessible. This range is far too small to allow a direct confrontation with the sum rule since we are still very far from the asymptotic region. Thus

8

4 y0

y0

5 2.6 2.8 3.0 3.2

3

-0.4 -0.2 0.2 0.40

-0.4 -0.2 0.2 0.40

l 2

(b) \Delta k

(a)

5-95 7964A2 Figure 2: Separate \Delta ^ and * dependence of the value of y0.

9

we must turn to more subtle methods.

Since the null result of the DHG sum rule arises from a delicate cancellation not present when anomalous couplings exist it is clear that these coupling must modify both the

shape of oepol(x) as well as the location of the place where the integrand vanishes, y = y0. Past analyses [17] have focussed on the overall shape of the polarization asymmetry, whereas here we will focus mainly on the zero's position. Figures 2a-b show the separate \Delta ^ and * dependence of the value of y0. Several features are immediately apparent from these plots: (i) If * = 0; then the deviation of \Delta ^ from zero perturbs the value of y0 to smaller values; (ii) if \Delta ^ = 0, the variation of * from zero can push y0 in either direction depending on the sign of *; (iii) the value of y0 shows a significantly greater sensitivity to non-zero values of * than \Delta ^. Thus if measurements determine that the energy where the polarization asymmetry changes sign is higher than that predicted by the SM, then * must be non-zero. It is also apparent that probing the location of the asymmetry zero will lead to a stronger bound on * than on \Delta ^.

0 0.02AD HG

-0.02

0.04 0.06 0.08

2.4 2.6 2.8 3.0 3.2 3.4 3.6

y5-95 7964A3

Figure 3: ADHG in the region near the SM value of y0. The solid curve is the SM predictionwhile the others, from top to bottom on the left, are for * = \Gamma 0:1; \Delta ^ = \Gamma 0:2; \Delta ^ = 0:2; * = 0:1, respectively.

10

In order to ascertain how much quantitative information we can obtain on the values of \Delta ^ and * from measuring the crossing point y0, we must perform a Monte Carlo study. Specifically, we will first want to know the constraints we can place on these anomalous coupling parameters if the SM situation is realized. We take the y region surrounding the SM value of y0 and divide it into 11 bins each of width \Delta y = 0:2. Note we have not yet tried to optimize either bin size or the distribution of integrated luminosity. Instead of considering oepol, we form an asymmetry using the ratio of both cross sections

ADHG(y) = oepol(y)oe

un(y) ; (9)

thus removing a number of systematic errors from the analysis. Figure 3 shows that not only does the value of y0 change when anomalous couplings are present, but so too does the shape of ADHG in the region near the zero. We assume as input into our Monte Carlo study that each \Delta y bin receives an equal integrated luminosity of 5f b\Gamma 1 and that the e\Gamma beam is 90% left-handed polarized, i.e., P = \Gamma 0:90. Next we generate Monte Carlo "data" (assuming the SM is correct) and try to fit the resulting distribution to the \Delta ^- and *- dependent functional form of ADHG. If *(\Delta ^) is zero, this procedure yields the fit shown in the first line of Table 1(with 95% CL errors). If we assume that both \Delta ^ and * non-zero, we obtain the 95% CL allowed region shown in Fig. 4. As we have anticipated, we obtain a far more restricted range of * than we do \Delta ^: We expect that somewhat better limits may be obtainable by optimization of our parameters. Notice that we have only performed our fit by covering the y region 2.0-4.2, which could just as well have been done by an e+e\Gamma collider with ps ' 420 GeV with the same integrated luminosity.

Do the constraints improve if we fit ADHG over the entire y range accessible at a 500 GeV collider with the same total integrated luminosity? To address this question, we now take 22 bins of width \Delta y = 0:2 covering the range 1 ^ y ^ 5:4 with an integrated luminosity

11

l 0

-0.01

0.01 0.02

-0.02

-0.10 0.10-0.05 0.050

s

\Delta k5-95 7964A4 Figure 4: 95% CL in the \Delta ^-* plane for the 11 bin fit described in the text. `s' labels theSM prediction.

of 2:5f b

\Gamma 1 per bin and re-do our fit. (We again note that we have not tried to optimize either

the bin size or the distribution of integrated luminosity; we are simply seeing the sensitivity of our results to different fit assumptions.) In the case that either \Delta ^ or * is zero we find the fit values displayed in Table 1. The result in the case where both anomalous coupling parameters are non-zero is shown as the solid curve in Fig. 5. In either case we see no substantial improvement in the bounds we can obtain on * but fitting the entire accessible y range significantly reduces the allowed range of \Delta ^ at 95% CL.

If we keep the collider energy fixed and double the integrated luminosity per bin, how do our results change for the SM example because of the improved statistics? In the case where either \Delta ^ or * is non-zero we find the values shown in Table 1. If both parameters are non-zero, we obtain the allowed region shown as the dashed curve in Fig. 5. The doubling of the statistics results in a significantly smaller allowed range for both of the anomalous coupling parameters.

How does the size of the bin affect these results? If we double the number of bins

12

0 l

0

-0.01

0.01 0.02

-0.02 -0.01-0.02-0.03 0.01 0.02 0.03

\Delta k5-95 7964A5

s

Figure 5: The solid(dashed, dash-dotted, dotted) curve corresponds to the second(third, fourth, fifth) case described in Table 1. `s' labels the SM prediction.

(i.e., change to \Delta y = 0:1) and keep the total integrated luminosity and center of mass energy fixed for the same total y range, do we improve our sensitivity? In the case where either \Delta ^ or * is zero we find the values shown in Table 1. The result in the case where both anomalous coupling parameters are non-zero is shown as the dash-dotted curve in Fig. 5.

In this case we see a very slight degradation of the limits obtained previously but no truly significant changes.

What happens at a 1 TeV collider with higher integrated luminosity? Since fitting the entire distribution gave the best results in the 500 GeV case, we will consider only this situation. We keep the bin size and integrated luminosity per bin fixed, but extend the y range up to 10.4, and repeat the above procedure. In the case that either \Delta ^ or * vanishes we obtain (with 95% CL errors) the values in Table 1. When both anomalous couplings are present, we obtain the 95% CL allowed region inside the dotted curve in Fig. 5. The size of the allowed region in this case is somewhat smaller than the corresponding one obtained for the 500 GeV collider but not as significantly improved as that obtained by doubling the luminosity in the 500 GeV case. A short analysis shows that essentially all

13

of the improvement in the * determination comes from increasing the integrated luminosity whereas the improved \Delta ^ determination derives both from better statistics as well as the expanded energy range covered by the fit.

Thus given fixed integrated luminosity, the value of * is well constrained from fits to the data in the region near y0 whereas, the optimal limits on \Delta ^ requires data fit over a large energy range.

0.03 0.02 0.01

0 -0.01

0.10-0.10 -0.02

-0.05 0.050

\Delta /c

l s

x

x

5-95 7964A6 Figure 6: 95% CL regions in the \Delta ^-* plane for the SM case as well as for the \Delta ^ and *values marked by an `x'. Note that these cases do not have overlapping allowed regions.

Let us now suppose that one or both \Delta ^ or *, are non-zero at the percent level; can such anomalies be distinguished from the SM? For purposes of comparison, we again consider a 500 GeV machine with L = 55f b\Gamma 1 equally distributed over 22 bins with \Delta y = 0:2 which covers the range 1 ^ y ^ 5:4. Figure 6 shows two simple cases with their corresponding 95% CL ellipses: (i) \Delta ^ = \Gamma 0:02 with * = \Gamma 0:01, and (ii) \Delta ^ = 0:02 with * = 0:01. In either case we see that the SM is excluded at 95% CL, i.e., the ellipses do not overlap the SM result. The non-SM and SM cases are thus seen to be cleanly separated. The polarization asymmetry zero can be used to constrain \Delta ^ and * if the SM is realized, but it can also

14

discover anomalous couplings if they are indeed present.

0 0.0001 0.0002

-1.0 1.0-0.5 0.50

cosq

ds

pol/

dco

sq (ar

b. u

nits )

5-95 7964A7 Figure 7: Angular distribution of the polarization-dependent part of the fle ! W * crosssection in the SM for y = y

0. Note the radiation zero at cos ` = 1 and that the integral overcos ` vanishes.

Thus far we have only considered an equal distribution of integrated luminosity in a fairly wide region surrounding y0. What happens in the reverse scenario, i.e., when we concentrate all of our luminosity in a single bin surrounding the SM value of y0 ' 3:158? In this case an e+e\Gamma collider with an energy of only ps ' 320 GeV is needed. From the previous discussion we expect to obtain a relatively very poor constraint on \Delta ^ since the energy range is so limited. As a benchmark, let us again take L = 55f b\Gamma 1 and consider bin widths of \Delta y = 0:1; 0:2, and 0.3. If \Delta ^ = 0, we find in all three cases that * = \Gamma 0:0009 \Sigma 0:0086, whereas if * = 0 we obtain instead \Delta ^ = \Gamma 0:028+0:107\Gamma 0:095. As anticipated, \Delta ^ is relatively poorly determined, but * is as well-constrained as in the case of the wide range 11 or 22 bin fits. Thus an excellent bound on * can be achieved even with an e+e

\Gamma collider of rather

modest center of mass energy (' 320 GeV); achieving strong limits on \Delta ^ will requires a higher energy machine.

As a last case, we take the same integrated luminosity as above and reintroduce 11,

15

\Delta y = 0:2 bins. However, instead of giving all 11 the same luminosity, we preferentially weight the bins closest to the central bin where the SM value predicts the change in sign of the polarized asymmetry. As an example, we take 1/2 of the total luminosity in the y = 3.05- 3.25 central bin, 1/8 in the two adjacent bins, 1/16 in the next pair and so on until the y range 2.05-4.25 is covered. (The central bin thus receives a luminosity of L = 55 \Delta 12 \Delta 6463 f b

\Gamma 1.)

In this case we find that \Delta ^ = 0:0165+0:0657\Gamma 0:1313 when * = 0 (a very poor determination, as expected) and * = \Gamma 0:0018+0:0080\Gamma 0:0085, when \Delta ^ = 0. Note that this * range is quite comparable to that found in all the other cases above with the same total luminosity. If we changed the initial weighting fraction of 1=2 for the central bin to 1/f , with f ? 1, then we obtain the results displayed in Table 2. In all cases the \Delta ^ determination is quite poor while * is well determined.

At the crossing point of the polarization asymmetry, the integral over angles of the of the polarization-dependent part of the fle ! W * cross section vanishes. By the mean value theorem there must be an angle `0cm where the polarization-dependent part of the differential cross section vanishes. This is illustrated in Fig. 7 for the case of the SM. In principle, measurements of this angular zero can also be used to limit possible new physics and anomalous couplings of the W; but the limited angular acceptance will cause a corresponding detriment to the statistical significance.

III. DISCUSSION AND CONCLUSIONS There has been a remarkable progression of improvement in the absolute precision of the measurements for the lepton magnetic moments - to parts in 10\Gamma 8 for the electron and parts per 10\Gamma 6 for the muon [21]. The lepton g \Gamma 2 measurements exploit the fact that the canonical Dirac coupling implies the equality of the Larmor and spin-precession frequencies of charged particles of any non-zero spin in a constant magnetic field. Unfortunately, it is

16

unlikely that this precise technique could be directly exploited to determine the moments of particles as short lived as the vector bosons of the standard model.

In this paper we have exploited two other unique features implied by canonical magnetic and electric quadrupole couplings: (1) the fact that the logarithmic integral of the polarized photoabsorption cross section vanishes identically, and the corollary (2) that there exists a specific energy where the polarization asymmetry must reverse sign.

In particular, we have shown that measurements of the DHG integral for the process fle

\Gamma ! W \Gamma * can bound the anomalous magnetic moment of the W: The polarization

asymmetry for this process is particularly sensitive to the anomalous coupling \Delta ^ at high energies. The SM also predicts that polarization asymmetry for fle

\Gamma ! W \Gamma * vanishes at a

precise energy psfle = 3:1583 : : : MW ' 254 GeV in Born approximation. We have shown that measurements of any deviation from this value can provide 95% confidence level limits on the W coupling * at a precision significantly below 1%, assuming the projected NLC luminosity. The precision of this polarization asymmetry measurement will benefit from the relatively low linear collider energy and the smaller systematic errors associated with asymmetries.

IV. ACKNOWLEDGMENTS We would like to thank J.L. Hewett for discussions related to this work. One of us (TGR) also thanks the members of both the Phenomenology Institute at the University of Wisconsin-Madison and the Argonne National Laboratory High Energy Theory Group for the use of their computing facilities as well as their hospitality.

17

References

[1] See the talk given by M. Calvi at the XXXth Recontres de Moriond, Electroweak Interactions and Unified Theories, Meribel, France, March 1995.

[2] F. Abe et al., CDF Collaboration, Fermilab reports PUB-95/033-E and PUB-95/035-E,

1995.

[3] F. Abe et al., CDF Collaboration, Fermilab report PUB-95/022-E, 1995. [4] S. Abachi et al., D0 Collaboration, Fermilab report PUB-95/028-E, 1995. [5] G. Couture and J. Ng, Z. Phys. C35, 65 (1987); G. Couture, J. Ng, J. L. Hewett and

T. G. Rizzo, Phys. Rev. D36, 895 (1987) and Phys. Rev. D38, 860 (1988).

[6] An analysis of the relationship of anomalous couplings to internal structure for spin-1

bound states is discussed in F. Schlumpf and S. J. Brodsky, SLAC-PUB-95-6860.

[7] T. G. Rizzo, Phys. Rev. D51, 3811 (1995); A. Grifols and A. Mendez, Phys. Lett. B255,

611 (1991) and erratum Phys. Lett. B259, 512 (1991); B. Ananthanarayan and S. D. Rindani Phys. Rev. Lett. 73, 1215 (1994); G. K"opp et al., Z. Phys. C65, 545 (1995); F. del Aguila and M. Sher, Phys. Lett. B252, 116 (1990); R. Escribano and E. Masso, Phys. Lett. B301, 419 (1993)and Nucl. Phys. 429, 19 (1994); W. Bernreuther, O. Nachtmann and P. Overmann, Phys. Rev. D48, 78 (1993); G. Couture, Phys. Lett. B305, 306 (1993)and Phys. Lett. B272, 404 (1991); G. Domokos et al., Phys. Rev. D32, 247 (1985); J. Reid, M. Samuel, K. A. Milton and T. G. Rizzo, Phys. Rev. D30, 245 (1984.)See also, P.D. Acton et al., OPAL Collaboration, Phys. Lett. B281, 305 (1992); D. Buskulic et al., ALEPH Collaboration, Phys. Lett. B297, 459 (1992).

[8] M. S. Alam et al., CLEO Collaboration, Phys. Rev. Lett. 74, 2885 (1995).

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[9] F. Abe et al., CDF Collaboration, Phys. Rev. Lett. 74, 1936 (1995) and CDF report

CDF-ANAL-ELECTROWEAK-CDFR-2951, 1995.

[10] S. Abachi et al., D0 Collaboration, Fermilab report PUB-95-044-E, 1995. [11] H. Aihara et al., Report of the Subgroup on Anomalous Gauge Boson Interactions of

the DPF Long-Range Planning Study, Fermilab report PUB-95/031, 1995. For a recent review of gauge boson self-interactions, see I. Hinchliffe, talk presented at the International Symposium on Vector Boson Self-Interactions, UCLA, February 1-3, 1995 and F. Boudjema, talk given at Beyond the Standard Model IV, Lake Tahoe, CA, December 12-16, 1994.

[12] S. D. Drell and A. C. Hearn, Phys. Rev. Lett. 16, 908 (1966); S. Gerasimov, Yad.

Fiz. 2, 598 (1965) [Sov. J. Nucl. Phys. 2, 430 (1966)]; L. I. Lapidus and Chou KuangChao, J. Exptl. Theoretical Physics 41, 1545 (1961) [Sov. Phys. JETP 14, 1102 (1962)]; M. Hosada and K. Yamamoto, Prog. Theor. Phys. 36, 426 (1966). For a recent review of the empirical tests of the DHG sum rule see B. L. Ioffe, preprint ITEP-61 (1994); D. Drechsel, University of Mainz preprint, 1994.

[13] In principle, if all elastic and inelastic contributions to the DHG integral from the new

physics were included,then the evaluation of DHG sum rule for \Delta oe(fle ! X) would yield the corresponding new physics correction to the electron anomalous moment.

[14] G. Altarelli, N. Cabibbo and L. Maiani, Phys. Lett. 40, 415 (1972). [15] S. J. Brodsky and I. Schmidt, SLAC-PUB 95-6761 (1995). [16] I. F. Ginzburg, G. L. Kotkin, S. L. Panfil and V. G. Serbo, Nucl. Phys. 228, 285 (1983).

See also S. Y. Choi and F. Schrempp, Phys. Lett. 272, 149 (1991).

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[17] The possibility of using the fle ! W * process to probe for the anomalous couplings

of the W has been investigated by a number of authors. See, for example, E. Yehudai, Phys. Rev. D41, 33 (1990); S. Y. Choi and F. Schrempp, Phys. Lett. B272, 149 (1991); G. Couture, S. Godfrey and P. Kalyniak, Phys. Rev. D42, 1841 (1990); K. A. Peterson and S. Godfrey, in The Fermilab Meeting DPF92, Batavia, IL, 10-14 November 1992, (World Scientific, Singapore), C. H. Albright et al.eds., pp.370; K. Cheung et al., Phys. Rev. D51, 4 (1995).

[18] M. Raidal, Helsinki preprint HU-SEFT-R-1994-16, 1994. [19] K. Hagiwara et al., Nucl. Phys. B282, 253 (1987). [20] See, for example, R.W. Brown et al., Phys. Rev. D20, 1164 (1979); K.O. Mikaelian et

al., Phys. Rev. Lett. 43, 746 (1979).

[21] See the articles by T. Kinoshita and D. R. Yennie and by F.J.M. Farley and E. Picasso

in Quantum Electrodynamics, edited by T. Kinoshita, T., World Scientific Press (1990).

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ps(GeV ) L=bin f b\Gamma 1 N

bins \Delta ^ *

500 5 11 \Gamma 0:008+0:046\Gamma 0:054 0:0004+0:0085\Gamma 0:0086 500 2.5 22 0:002+0:011\Gamma 0:012 0:0029 \Sigma 0:0086 500 5 22 0:0014+0:0087\Gamma 0:0086 0:0022+0:0062\Gamma 0:0063 500 1.25 44 0:0037+0:0139\Gamma 0:0140 0:0005 \Sigma 0:0092 1000 2.5 47 0:002 \Sigma 0:011 0:0012+0:0078\Gamma 0:0080

Table 1: 95% CL constraints on \Delta ^ and * for the different scenarios described in the text.

f * 1.1 \Gamma 0:0008+0:0087\Gamma 0:0085 1.5 \Gamma 0:0018+0:0087\Gamma 0:0085 2.0 \Gamma 0:0018+0:0080\Gamma 0:0085 3.0 \Gamma 0:0015+0:0087\Gamma 0:0085 4.0 \Gamma 0:0012+0:0086\Gamma 0:0086

Table 2: 95% CL constraints on * for the different values of the parameter f described inthe text.

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