the effective one-loop action in the strongly interacting standard electroweak theory

Volume 197, number 1,2 PHYSICS LETTERS B 22 October 1987

T H E EFFECTIVE O N E - L O O P A C T I O N IN T H E STRONGLY I N T E R A C T I N G STANDARD E L E C T R O W E A K T H E O R Y ~

Oren CHEYETTE and Mary K. G A I L L A R D Lawrence Berkeley Laboratory and Department of Physics, University of California, Berkeley, CA 94720, USA

Received 20 July 1987

One-loop corrections to the effective gauged a-model of the large Higgs mass limit of the standard electroweak model are presented. The one-loop corrected strong scattering amplitude is obtained by a resummation of the derivative expansion. A sig- nificant enhancement is found for longitudinal W and Z scattering cross sections in the TeV center of mass energy region.

In this letter we extend the results of a calculation [ 1 ] of the one-loop effective action in the heavy Higgs limit of the standard electroweak model to obtain the one-loop corrected scattering amplitude for longitudinally polarized W's and Z's, and we discuss the implications for SSC physics. We find that the dominant correction to the low energy theorems [ 2 ] for elastic scattering of longitudinally polarized W's and Z's is independent of the gauge couplings and therefore may be applicable to a broader class o f models. Its effect is to enhance cross sections in the TeV center of mass scattering energy region as compared with results [ 2] based on the tree amplitudes [ 3] in the limit m~ >> s. In particular, there is a large one-loop contribution to ZLZL elastic scattering, which is absent at tree level.

One may question the validity of a loop expansion in the strongly interacting limit of the theory that we are considering. One point of view is as follows. In the absence o f a "light" (rnH~< 1 TeV) Higgs boson, for any electroweak model possessing a custodial SU(2) which guarantees p = 1 at tree level, the WL, ZL sector is necessarily [2,4] described by a chirally symmetric a-model at energies below the new physics (e.g., resonance) thresholds. This effective theory is nonrenormalizable and quantum corrections include divergent contributions that do not appear in the underlying renormalizable theory. The divergent integrals are regulated by a cut-off that reflects the symmetry breaking scale of the new physics which modifies the effective a-model approximation. Thus we expect these apparently divergent contributions to be a reasonable indicator of true physical effects in the energy region below the new physics threshold. On the other hand, the finite contributions at one loop depend on the detailed manner in which the new physics damps the apparent divergences, and are therefore unreliable. In the following discussion we will denote the mass scale o f the unknown symmetry breaking physics by msB. In the standard model with a physical Higgs we may take msB = mH, but the results presented here are applicable to a larger class of models.

It has previously been shown that the two-derivative [ 5 ] and the logarithmically divergent part of the four- derivative [ 6 ] terms in the one-loop effective action for the non-linear a-model can be reproduced by taking the large-mH limit o f the linear a-model (which is just the Higgs sector of the standard model) if the cut-off

~r This work was supported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the US Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY85-15857.

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in the non-linear model is identified with mn. The results [ 1 ] reported below include gauge and scalar loops in the presence of gauge and scalar background fields allowing, in particular, a determination of the one-loop correction to the p-parameter in agreement with the large-ran limit of previous diagrammatic calculations [7,8]. (There is an indication, however, that this procedure may not be valid at the two-loop level [ 7].)

Most of the results reported below are contained explicitly or implicitly in existing literature [ 5-9 ]. The one loop divergences of the non-linear a-model have long been known [4,9], and Longhitano [8] used diagrammatic methods to find the four-point W and Z effective interactions which diverge as In msa. Here we present the logarithmically and quadratically divergent terms in the covariant [ 5,6 ] derivative expansion [ 10 ] of the one-loop effective action for the standard model in a "renormalizable" gauge, but in the nonrenormalizable limit r n s B ~ . This means that we retain the unphysical Higgs scalars as both quantum and background field degrees of freedom. This permits a direct implementation of the "equivalence theorem" [ 2,3,11,12 ] relating S-matrix elements for WE, ZL to those of the respective "eaten" unphysical scalars w, z in the evaluation of scattering amplitudes. It also displays the results in a way which clearly separates effects specific to the strongly interacting [ 13 ] limit of a large symmetry breaking scale from those that occur in the renormalizable, purely gauge sector of the theory.

In the rnsB-*~ limit of the standard model, the Higgs sector reduces to a non-linear a-model [2,4]. Quantum corrections to the ungauged non-linear a-model have been studied [ 5,6 ] using a covariant derivative expansion which explicitly respects the non-linear realization of the chiral symmetry in all steps of the calculation. In ref. [ 1 ] these methods were generalized to include gauge covariance in the gauged non-linear a-model, or more specifically, the large-rosa limit of the standard model. The salient results are described below.

Neglecting fermions, which are irrelevant to our results, the lagrangian for the standard electroweak model well below the mass scale of the symmetry breaking physics is simply

5a= -½ T r F i w - ¼ F 2 + JDuOJ 2, (1)

where the SU (2) and U ( 1 ) gauge field strengths are defined, respectively, by the 2 × 2 matrix-valued field strength

Fwu~ = (i/g) [0 u - igWu , O . - igW.] , Wu = W ~ z J 2 (2)

and by

VBv.=(i/g') [Ou-ig'Bu, 0 . - i g ' B . ] . (3)

The covariant derivative is

D u =0 u - i g W u - i ( g ' / 2 ) B u, (4)

where g= e/sin 0 = cot 0 g' are the usual weak coupling constants. We parameterize the scalar fields by

where o ~ 250 GeV is the electroweak symmetry breaking scale, and the unitary matrix U is an SU(2) group element with field dependent parameters; U - 1 in the unitary gauge. We work in a renormalizable gauge ( U~ 1), so that the lagrangian (1) is explicitly SU (2) × U (1) gauge invariant for a suitable choice of the transformation property of U. The standard form of the non-linear a-model is recovered for g, g' = 0 with the parameterization

U=(llo)(a+in.z), a 2 = 0 2 - 7g 2 , (6)

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22 October 1987 Volume 197, number 1,2 PHYSICS LETTERS B

for which the kinetic term reduces to

1 [ 7~'0tzT~

tDuq~l 2 ~ ~0~ z~0~zc+ -" 7---~ l=~%" (7)

The zc~ are related by a field redefinition to the 09 -+, z of ref. [2], hence their S-matrix elements are equivalent. To lowest order in the fields they are the same, so the four-field coupling that determines WL, ZL scattering amplitudes is just

LPint = (1/2o 2) (w + 0~w- + w - 0j, w + +z0uz) 2, (8)

giving, among others, the scattering amplitudes [2,3]

jgw_-__J/(w÷ w - ~ w + w - ) = -iulo 2, jgz =~g(zz~zz) =0, ~¢{wz =Jg(w+w - ~zz) =is/v 2, (9a, b, c)

where s and u are Mandelstam variables with obvious definitions. The ultraviolet divergent part of the one-loop contribution to the effective lagrangian has been obtained [ 1 ]

by a covariant expansion of the effective action for the theory defined by the lagrangian (1) with background gauge and scalar fields, with a convenient choice of gauge condition. Details of this technique are given in ref. [ 1 ]; the result is

5°1 = - (1/16zc 2) (rlm~n/O 2) [Du~[ e +In (rl'm~BIm2w) (1/16x 2)

× { ~ g2 t r r ~ v - I g,2F~ _ 1 gg'FB~, tr z3 U-IF~U

- ~ itr[gU-IFwu~g+ (g'/2) FB~.r3] [Cr", ~"1 - ] g2(2+tan20) ID~OI 2

- (3g 2 tanZ0/802) (O*Duq~) 2 + ~ (tr d~)2 + 1 (tr G~G.)2}. (10)

In (10) the 2×2 matrix valued field ~u is defined by

G#=½ (G#'z-G°r3),

where

Ga=-iU-~DaU-~½ ( GF,.r +G°). (11)

(Note that ~ = G for sin 0 =0.) In eq.(10) r /and r/' are parameters of order 1 that depend on the details of how new physics damps the divergences. Actually, we may have different values of r/' for each term in (10); however we will ignore this issue, and in the following we take r/' = 1. The parameter msB is to be identified with the cut-off that defines the threshold for new physics.

The terms proportional to I D ~ l e, which contain the only quadratic divergence in eq. (10), can be absorbed by a renormalization of the Higgs vacuum expectation value v. This is simply a reflection of Veltman's "screen- ing theorem" [ 13 ]. The form (qbtDaq~) 2 has been identified [ 14 ] previously as the only possible two-derivative term that is SU(2)L×U(1 ) gauge invariant but breaks the custodial SU(2). It contains corrections to the low energy scattering theorems [2] based on chiral SU(2) in direct proportion to corrections to the p parameter

p=mZw/m 2 cos2 0, (12)

which is 1 at tree level. This term contains a mass shift for the Z, but not the W, as is easily seen in the unitarity gauge where

(q~t Du~)2--, - (g2o4/16 cos20)Z 2. (13)

Inserting (13) into (10) to obtain the shift in the Z mass gives the correction

p= 1 - (392/64zc 2) tan20 In (m2sB/m~v), (14)

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in agreement with earlier calculations [7,8]. As discussed below, consistency of our approximation requires msB < 3 TeV, which means 1 - p < 0.004, well below current limits (after including other radiative effects). In the following we will neglect the correspondingly small corrections to the low energy scattering theorems arising from this term: since they grow approximately linearly with s, they are small compared to the tree approximation at all energies.

The terms proportional to F 2 in eq. (10) reflect a genuine logarithmic divergence; they are simply the gauge and scalar loop contributions to the r-functions at scales/~ intermediate between the gauge boson masses and the Higgs mass: mw <</z << msB.

The remaining terms in (10), which are finite for finite Higgs mass, represent observable new physics. They contain n-point functions for vector boson (V) scattering with up to four transverse bosons and an arbitrary number of longitudinal ones. This follows from the equivalence theorem which states that to all orders in perturbation theory, on-shell S-matrix elements with any number of external WL and ZL may be identified with those for external w's and z's as extracted from the standard model lagrangian in a renormalizable gauge.

In particular, the term of order gg' gives a momentum dependent correction to the VTVT--'VLVL vertex, while the terms linear in g and g' give a correction to the effective VT--'3VL vertex used in ref. [ 15 ] to estimate three-VL production in the heavy Higgs limit. The one-loop correction contains three derivatives and therefore grows with s relative to the tree contribution. The last two terms of eq. (10) give similar contributions to ver- tices with up to four transverse bosons through gauge covariant derivatives. Since the rate for VT emission from a beam of fermions is enhanced relative to VL emission by a factor In (s/m2), the rapidly growing VTVT~VLVL and VTVL~VLVL amplitudes extracted from (10) could be comparable in the TeV region to the VLVL--'VLVL amplitudes that we will discuss in more detail below. Here we note only that in the effective W, Z approximation [ 15,16 ], fusion [ 17 ] of vector bosons of different polarizations add incoherently, so these contributions can only increase the signals.

Neglecting gauge couplings, the last two terms of eq. (10) reduce simply to

1 ol-:oop _ in ~ [ 2(g° O;niO, ns) 2 + (giJ O#TriOulzj)2], scalar - - 967~204 m w (15)

where

gij=(~iJ-~-7~iT~J/(I)2 --7~ 2 ) (16)

is the scalar metric. This is just the logarithmically divergent four-derivative term in the one-loop effective action of the ungauged non-linear a-model defined by the tree lagrangian of eq. (7). It can readily be extracted from the scalar action of ref. [ 5 ] by expanding up to terms with four derivatives, or, equivalently, terms, quadratic in the scalar curvature. This term has also been computed in ref. [ 6 ] where it was shown to agree with the mH~oO limit of the corresponding one-loop contribution in the linear a-model.

In writing eq. (15) we have cut off the infrared logarithmic divergence at the W mass, which is appropriate for the the physical WL scattering amplitudes we are interested in. However to get the correct kinematic argu- ment of the logarithm - and a more realistic estimate of the scattering amplitudes - we must sum the derivative expansion. As emphasized by Aitchison and Fraser [ 10] the successive terms in the derivative expansion for the unbroken chiral a-model are increasingly infrared divergent, although S-matrix elements are necessarily infrared finite. By the same token, the infrared divergence of eq. (15 ) as rnw~ 0 is an artifact (except for special values of external momenta) of retaining only the ultraviolet divergent terms in the derivative expansion.

Using the covariant formalism of ref. [ 5 ], it is straightforward to resum the derivative expansion to obtain the kinematically correct amplitude involving four external scalar fields (which we interpret as longitudinally polarized W's and Z's). Neglecting gauge couplings, the full one-loop effective action of the a-model is given by [5]

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~ f d4p 5gift = ~ T r l n [(pu+G~O/Op~)2-1~],

with the matrix valued functions/~ and G defined by

o o ' n x-, ( - -1) !_. O n ax =,=o2" 7 . . Jam "'" Du.R(x) Opt, ... Opu~ '

(17)

(18)

( n + l ) 0" G~= ,=o ~ ( - i ) " :(n+2)-------5 D~, ...D~.G~u(x) Opt, ... OPu" , (19)

with

Du-=OuTciDi, RJ =Ou~k Ou7~l RJ~li, G{> =Ovnk O~ntR{t~= -G'}i,u, (20, 21, 22)

where Di is the scalar field redefinition covariant scalar derivative with the connection

Fjk=7~i gjk/O 2 (23)

derived from the scalar metric (16), and

R}kt = ( r~}&k-- (~ikgjl)/O2, (24)

the corresponding scalar Riemann tensor. Since (21) and (22) are at least quadratic in the scalar fields, to get the two-body elastic scattering amplitude we need to retain terms at most quadratic in R and Gu~ and their covariant derivatives. Terms linear in Gu, vanish in the momentum integration by antisymmetry. The term linear in R (which has a quadratically divergent coefficient) is proportional to the tree amplitude and, as discussed above, can be absorbed into the renormalization of v. Finally, since the scalar connection is higher order in the scalar fields and since we can drop total derivatives, we need consider only terms of the form

Tr ~ d4pRf(Ou,Pu)R, Tr f d4pGu~g~"P~(O~,p~)Gpa, (25)

where 0v is an ordinary coordinate derivative and f a n d g are dimension p-4 functions of their arguments; f(O, pu) and g(O, pu) give the logarithmically divergent terms as extracted from either (10) or (17), with

Tr J d4p Gv~gU~P~(O, pu)Gp, ocln mZB Tr Gu~G u" (26)

by Lorentz invariance. If follows from dimensional analysis that the terms of the form Gv~O~OpG p~ must be suppressed by a factor 1/mZB (there are no power-law infrared singularities). Therefore the leading terms, after momentum integration are of the form

YrR[ ln (m~B/02)+a ]R , Tr Guv [ln(m~B/02)+a '] Cr "~. (27)

The constants a and a' of order one, can be absorbed into a redefinition of the cut-off msB, which may then be different for the two terms in (27). We emphasize again that the precise evaluation of a and a' depends on the details of the new physics which regulates the divergences, and is therefore meaningless in the present context. This uncertainty cannot, however, affect the coefficients of the terms in (27). We will assume in what follows that a = a' = 1; we expect the results to be insensitive to this assumption.

Since we know the coefficient of In m 2B in the expressions (27), it is straightforward to extract the full amplitudes. Evaluating explicitly the expressions (27) and using eq. (10), we obtain for the logarithmically divergent four-scalar couplings

~l-loop = (1/64yg204) [ ( N - 2 ) (0urci0un*) In (mgB/O 2) (OvT~jOvT~ j) + (OuTtiOuT~j) In (mZB/o 2) (OvgiOuT~ j) s c a l a r

_ 1 (Ov~O.Trj) In (m~8/O 2) (Ov~sO"~ j) + ~ (OvTriO~=j) In (m~B/O 2) (O"~*O~J)], (28)

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1250

1000

750

5oo

250

- I . . . . I . . . . I . . . . I a longitudinal ~ pair -

producUon rate

- - one loop

20 TeV ~

0.5 1 1.5 WW pair mass (TeV)

1000

BOO

600

400

200

I . . . . I . . . . I . . . . I b longitudinal ZZ pair

production ra te

one loop

20 TeV \ x

I I 0.5 1 1.5 E

ZZ pair mass (TeV)

Fig. 1. W[-WC (a) and ZLZL (b) pair production rates in pp collisions at x/s=20 and 40 TeV' with rapidity cut lYl < 1.5, assumed integrated luminosity of 104o cm-2, and cut-off scale msB = 3 TeV. The amplitudes have been unitarized according to the prescription given in the text.

where N is the number of Goldstone bosons (in this case N = 3). One-loop corrections to specific scattering amplitudes can now be read off from eq. (28). We obtain, for example,

• 1 - loop --1Jaw = (1/64re2v 4) {3s 2 In (m~B/--s)+3t 2 In (m~B/--t)+2u 2 In (rn~B/--u)

--~ t 2 In (m~B/--s)--~ s 2 In (m~B/--t)+~ u 2 [In (m~B/--s)+ln (m~B/--t)]}, (29)

which should be a good approximation to the exact amplitude for W + W f elastic scattering for s, - t , - u >> m~v. The absorptive part of this amplitude is extracted by setting In ( - s ) = I n s - i n . This expression agrees with the corresponding results obtained in the context of pion physics [ 9 ].

Taking any three independent two-body scattering amplitudes, we can determine the isospin I = 0, 1, 2 amplitudes for each partial wave. We have checked that the unitarity predictions based on the tree amplitudes agree with our results for the absorptive amplitudes, which are non-vanishing only for J = 0, I = 0, 2 and I = J = 1. At one-loop all partial wave channels contribute, although only the J = 0, 2 amplitudes grow with ms~.

To compare with the tree level results, consider the one-loop contribution (28) to the 90 ° ( u - - - t = - s / 2 ) amplitude for w+w - elastic scattering, for which the real part is

jg,jioop = (4/zce) [S2/(1 Tev)e ] [4 In (rn~/s) +In 2], (30)

as compared with the tree level amplitude (9a)

j~twree = 8 S / ( 1 T e V ) 2 . (31)

The value of (30) exceeds that of ( 31 ) for x/s >~ 2 TeV if msB > 3.4 TeV, supporting our earlier assertion regard- ing the range of validity of the perturbation expansion. Quite generally, since the a-model coupling constant is o -2, the effective loop expansion parameter is (4zoo)-2~ (re TeV)-2, indicating that the expansion cannot converge if either v/s or msR exceeds 3 TeV. We further recall that tree level unitarity breaks down [2,15 ] at x/~= 1.8 TeV, suggesting a cut-off (i.e., scale of new physics) around 2 TeV. Fortunately, since our results depend only logarithmically on the CUt-Off, they are not strongly sensitive to its precise value.

For purposes of illustration, we display in fig. 1 the tree level and one-loop corrected production rates for W 3 W ~ and ZLZL pairs in pp collisions at 20 and 40 TeV, using the effective W, Z approximation [ 15,16]. We have unitarized the s-wave components of the amplitudes, substituting for the one-loop J = 0 partial wave

a=ao/(1 -al/ao), (32)

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and for the tree level J = 0 partial wave

a=ao/[ 1 - (i/16n)ao], (33)

where ao and al are the calculated tree level and one-loop s-wave amplitudes. The higher partial waves are small in the plotted energy region, so we have not bothered to unitarize them. Fig. 1 illustrates our main results, namely that one-loop corrections to the effective a-model for strongly interacting W's and Z's give production rates that are considerably enhanced relative to estimates based on the tree level amplitudes. We note also that, since all partial waves are present in the one-loop amplitude - with the correct singularity, chiral symmetry and crossing properties - our results represent a plausible model for strong WL, ZL scattering in the TeV energy region in the absence of resonance structure.

A comment is in order on the comparison of our results with previous attempts [ 17 ] to improve upon the tree approximation to the effective strongly coupled a-model. These authors solve, to leading order in 1/n, the O(n) a-model ( n = 4 for the chiral SU(2) of the standard model) defined by the lagrangian of eq. (7), to obtain a fully unitarized scattering amplitude. Setting N = n - 1 for the number of Goldstone bosons, the one- loop effective action in (28) in the leading n approximation reduces to

( n/64n2o a) ( OltTgiOlt[c i) In ( m~B/O 2) ( OvTtjOv TgJ), (34)

which contributes, for example, to w+w - ~ z z only in the I=J= 0 channel, in conformity with the results of ref. [ 18 ]. For n = 4 this contribution is given by

1/1-1ooo = (1/8n2) (s2/o4) [ln (miu/S) +in] , (35) - - . a~ ,~ w z , l a r g e n

which can be recovered by expanding the same amplitude [2] extracted from the results of ref. [ 18] in powers ofs/n2o 2. This may be compared with the I=J= 0 contribution to the same channel as obtained from the lagrangian (28):

"l-loop 2 2 4 ln(mZB/s) l~ 4in]. --l~wz, l=J=o=(S /96n o ) [ ~ + ~ + (36)

Thus in this order the l/n approximation overestimates the I = J = 0 amplitude by a factor of about 2, while missing the other partial waves, in particular the logarithmically divergent contributions to the I = 2 and J = 2 partial waves.

Finally, we comment on the validity of our results in the context of models more general than the standard electroweak model. In any model for electroweak symmetry breaking with a custodial SU(2) symmetry, the strong coupling of the longitudinal bosons at sufficiently low energy is governed by an effective tree lagrangian of the form (7), so one would expect one-loop effects in the same energy region to generate the terms in (10) which give chiral SU(2) symmetric corrections quartic in momenta to the amplitudes (9) that follow from low energy theorems for this class of models. For any model in which the effective tree lagrangian (7) is obtained as a particular limit of a renormalizable tree lagrangian, the divergent one-loop corrections of the effective theory below new physics thresholds should, as in the standard model, correspond to the ms•--, 09 limit of the one- loop corrections to the renormalizable theory. The corrections calculated above should be universally applicable to this class of models.

To conclude, we have found substantial one-loop corrections to the effective a-model that characterizes strong WL, ZL scattering amplitudes in the large Higgs-mass limit of the standard electroweak model. We wish to emphasize that we regard our results as only indicative of corrections to the low energy scattering theorems that apply to any phenomenologically acceptable model (p~ 1 ) if there is no light (mH << 1 TeV) Higgs boson or other resonance. Our results are encouraging, as they suggest an enhancement of scattering cross sections below the energy scale at which new physics should appear if this scale considerably exceeds 1 TeV. Specifically, the approach to new physics is characterized by more rapidly rising cross sections than those obtained [2] at

211


tree level. These results emphasize the need for high effective vector boson luminosity which requires both high primary proton beam energy and luminosity.

We thank Bob Cahn and Howard Georgi for useful remarks, and Mitch Golden for help in producing the graphs.

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the effective one-loop action in the strongly interacting standard electroweak theory

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