Volume 240, number 3,4 PHYSICS LETTERS B 26 April 1990

T H E p P A R A M E T E R IN AN Sp(2N)× U(I) M O D E L

Peter CHO t Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA

Received 31 October 1989; revised manuscript received 30 January 1990

We present a model based upon the gauge group Sp (2N) × U ( I ) which contains the SU (2) × U ( 1 ) electroweak model as an exact subsector. The p parameter in this symplectic model is calculated to leading order in l IN. We illustrate the general depen- dence ofp upon the Higgs coupling constant k with particular interest in the large k limit. Finally, the l/N result is compared with conventional one- and two-loop expressions for p in the standard model.

1. Introduction

Could a very massive Higgs be indirectly detected via its radiative corrections to physical observables? Efforts in the past to address this question have re- lied upon conventional perturbative analysis [ 1,2]. However, as perturbative unitarity arguments indi- cate, ordinary perturbation theory breaks down if the Higgs is very heavy and the scalar self interaction is strong [3,4]. Instead, one really needs to use a non- perturbative calculational scheme to answer the question that unfortunately does not exist for the S U ( 2 ) × U ( l ) electroweak theory. Some authors have previously studied strongly interacting S U ( N ) × U ( I ) models and have attempted to ex- tract qualitative information about the radiative cor- rections to vector boson masses when the Higgs cou- pling constant ;t is large [ 5,6]. Such models can be solved to leading order in l /N. We have considered a different large N model based upon the gauge group Sp (2N) × U ( 1 ). Unlike the SU (N) × U ( l ) models, the Sp ( 2 N ) × U ( l ) theory contains the electroweak model as an exact subsector. The effects of a heavy Higgs upon radiative corrections to various electro- weak observables associated with this subsector may be investigated. In particular, we have calculated to leading order in l / N the radiative corrections to the p parameter of the symplectic theory. This quantity is of interest since the standard p parameter has been

J Research supported by NSF contract PHY-87-14654.

previously proposed as one observable that might re- veal traces of a heavy Higgs at low energies [ l ] . Moreover, the l / N result calculated to all orders in 2 can be compared with one and two-loop perturbative expressions forp in the standard model. The contrast between their large ,~ behaviors suggests that detec- tion of a heavy Higgs via its radiative effects may be more unlikely than previously believed.

2. The symplectie model

Since the classical group Sp(2N) does not appear frequently in particle physics, we will briefly review some of its relevant properties. Sp(2N) is the set o f all linear transformations under which a nondegener- ate skew-symmetric bilinear form is left invariant. Equivalently, for any ge Sp (2N),

g T j g = j , (2.1)

where J is a skew symmetric metric. We will work in a basis in which J = I u× u®itr2. Setting g = exp (i0Ta) with T~ an arbitrary 2N dimensional hermitian gen- erator of Sp(2N) and expanding (2.1) to lowest or- der in 0 yields

Ta = l ~ , x N ® i O ' 2 " - - T * ' l u x N ® - - i o ' 2 • (2.2)

As T~ is equivalent to - T~* and the equivalence ma- trix lux~,®ia2 is antisymmetric, the fundamental representation 2N is pseudoreal. Its generators can be written as tensor products of N × N and 2 × 2 hermi-

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Volume 240, number 3,4 PHYSICS LETTERS B 26 April 1990

tian matrices of the form S ® a and A® 12×2 where S and A are symmetric and antisymmetric respectively. One basis for the N× N hermitian matrices is

(Hm)q= ~d~,,~., l<~m<~N,

1 (S~h)'J= ~ (6',~Jb +6~6;) , l <~b<a<~N,

i i 1 t (A,b)'J= ~ (6~6sb--6~6b), I <~b<a<~N.

The resulting 2N 2 + N hermitian Sp (2N) generators

H,,,®a, Sat, Go , A,o®12x2 (2.3)

are normalized so that Tr(T,,Tb) = ½~b in the funda- mental representation.

We now construct a model based upon the gauge group Sp (2N)× U (1)v which spontaneously breaks to Sp(2 ( N - 1 ) ) X U ( 1 )EM. Our symplectic model is a straightforward albeit unrealistic generalization of the electroweak model without fermions. We use standard notation wherever possible to underscore the similarities between the two theories. The coupling, generators, gauge fields, and structure constants as- sociated with Sp(2N) are denoted by g, T~, A, u and f,h,-. Similarly, the U( I )r coupling, generator and gauge field are written as g' , Y and B u. The gauge couplings g and g' in the Sp(2N) × U ( 1 ) model are assigned the same values as the corresponding cou- plings in the SU (2) X U ( 1 ) theory.

The desired pattern of symmetry breaking in the symplectic model is achieved via the Higgs mecha- nism. We first introduce a set of complex scalar fields

o:(:i/ that transform as 2N under Sp(2N) and whose weak hypercharge assignment is Y ~ = 1 ~. The form of the gauge invariant, classical lagrangian is then essen- tially the same as in the standard model:

F u~F - :,Gu~G

+ (Du~) tDU~ - U(~*q~) . (2.4)

a a a b e b c Fu~=OuA" , -O~Au+g) c A . A . .

the covariant derivative is

O u ~ - I g A u 1 ~ - ~lg B u Y ~ , Duq~ = - ~ - i. ,

and the classical scalar potential is

U(q~t~) = _ ~ zq~tq~+ 2 (q~tq~) 2.

As usual spontaneous symmetry breakdown occurs for positive values of/z 2 and 2, and the scalar fields develop a VEV. Without loss of generality, the VEV may be rotated into the form

with v=x /#2 l),. A shifted scalar field

O,

0N ( 1 / d S ) ( a+ iX)

can subsequently be defined in which the Higgs field (1 and all the Goldstone fields have zero vacuum ex-

pectation value. The hermitian Sp (2N) generators listed in (2.3)

are not always the most convenient ones to use in constructing the gauge theory. We will often take lin- ear combinations of these generators which corre- spond to gauge fields of definite electric charge:

I I ~ = H,,, ® ( (1, -+ ia2)/x/~,

3 H,,, = It,,, ® (I 3 ,

Rdb = (Sah® 0"3 g iAat,® 12x2)/w/~'

l",~b=S,b®(al +-ia2)/X/~. (2.5)

We will focus our attention upon the three fields as- sociated with the generators H~ and H 3 which we name W + and W 3. These W's are of special interest because they are the analogues of the vector bosons with the same names in the standard model.

The gauge field strengths in (2.4) are

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3. Custodial SU(2) symmetry

An important, approximate SU(2) symmetry ex- ists in this symplectic theory that becomes exact as the U ( 1 ) r gauge coupling goes to zero. We will gen- eralize the discussion of this custodial SU(2) pre- sented in ref. [ 7 ] for the SU (2) X U ( 1 ) theory to our Sp(2N) × U ( I ) model. To uncover the presence of this custodial symmetry, we first define a scalar 2N- tuplet ~=lux,~®ia2 'q~* which transforms like under Sp (2N) but has opposite weak hypercharge. We then combine • and • into a 2 N × 2 matrix O = ( ~ ) and rewrite the lagrangian in (2.4) in terms of this matrix:

L#~ = - 4-u~-lt:~ r u ~ _ ¼Gu~GU~+½Tr(DuO*DUO)

- U(~Tr(O*O) ) ,

where

DuO= OuO- igA~u T, O+ ½ ig' BuOtr3 . (3.1)

In the absence of the hypercharge coupling, c~¢~ pos- sesses an Sp (2N) L × SU (2) R invariance given by

O--,L(x)OR t ,

i Au -+L (x ) A uL ) ( x ) - g [ 0uL(x ) ]L ) (x ) ,

B u --. B u .

Obviously, the classical lagrangian remains invariant under Sp(2N)L gauge transformations parameter- ized by the 2N×2N matrix L(x) . However, the ap- pearance of the 2 × 2 matrix R in the transformation law reveals an accidental global SU (2) R symmetry of ~ . The presence of a3 in the hypercharge term of (3.1) explicitly breaks this extra SU (2)R symmetry i fg ' ¢ 0.

When the U ( I ) r coupling constant is set to zero, the symmetry breaks down to S p [ 2 ( N - I ) ] L × SU(2)L+R where the custodial SU(2)L+R is gener- ated by

(Hu ®~r,)L ® ( 12×2)R + ( 12N× 2U)L ® ( ~ai)R,

i = 1 , 2 , 3 .

The fields in our theory transform under the original symmetry group and its unbroken subgroup as

Sp (2N)L × SU (2)R --,Sp (2 ( N - 1 ) )L X SU (2)L+R

O~(2N, 2 ) - - , ( 2 N - 2 , 2 ) @ ( l , I ) @ ( 1 , 3 ) ,

A u~ (2N2+N, 1)

- , ( 2 ( N - 1 ) 2 + ( N - 1 ), 1)

@ ( 2 N - 2 , 2)@(1, 3 ) ,

B u ~ ( l , 1 ) ~ ( 1 , 1) .

The W' s are the bosons in A u that transform accord- ing to the adjoint representation 3 of SU (2) L+ a. Be- cause radiative corrections cannot distinguish among the members of this irreducible triplet, the custodial symmetry forces p--, 1 as g' -,0. As we shall see, this important constraint considerably simplifies the p parameter computation.

4. The p parameter

It is useful to recall the significance o f thep param- eter in the standard model: p is a measure of the rel- ative strength of neutral to charged weak interac- tions. This phenomenoiogical parameter is conventionally defined as the ratio of the coefficients of the neutral and charged current-current interac- tions in the effective lagrangian for neutrino-hadron interactions at low momentum transfer [ 8 ]:

G

+p(j~ _sin20w jEM) (j3U_sin20w J~M) ] •

In tree approximation, p = 1 since ~weak must be a singlet under the custodial SU(2) when sin 0w=0. Radiative corrections to the vacuum polarizations of the W -+ and W 3 propagators induce deviations o fp away from unity. Such radiative corrections are of course energy dependent. However, the polarization tensors/7u+~ - (qZ) and H ~ (q2) are essentially con- stant for q2 << M2w and may be approximated by their values at q2=0:

//'u~- (q2) ~ +- + ~F/~ (0)=H -(0)gu~, 33 2 ~ 33 33 Hu~(q )~Hu~(O)=H (O)gu~.

The radiative corrections top are thus given to lowest order by

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Volume 240, number 3,4 PHYSICS LETTERS B 26 April 1990

H+-(O)-H33(O) A p = p - 1 = M2 w (4.1)

We adopt (4.1) as the definition o f t h e p parame- ter in the symplectic model. We will calculate Ap to lowest order in the gauge couplings and to leading or- der in 1/N. Since the number o f graphs contributing to a particular process at a fixed order in perturba- tion theory grows with N, radiative corrections di- verge as N--,oo. Therefore, to obtain meaningful re- suits wc must rescale the coupling constants and VEV of the theory:

g+g/x/N, g'+g'lx/~, 2-,2/N, b ' 2 - - + N l ; 2 .

Now as N grows large, the increase in the number of graphs with closed loops around which N virtual par- ticles can propagate is offset by the decrease in the interaction strengths. Moreover, the VEV redefini- tion ensures that all I/N dependence explicitly ap- pears in subsequent expressions. Thus quantities such

,IAt2 1 ~ ,2 , , 2 as tree level masses , , *w=a6 ~ and M~=21~z= 22v 2 are independent of N.

The diagrams contributing t o / 7 + - and/73s may be classified according to their dependence upon the gauge couplings and 1/N. After extracting a common factor of Mw from the vacuum polarization tensors, we find that the H ÷ - and H 33 diagrams of lowest or- der in the gauge couplings are O(g2M2w ) and contain only one loop if scalar radiative corrections to scalar propagators are temporarily ignored. There arc a large number of these one-loop graphs. However, thc cus- todial SU(2) symmetry guarantees that all of the H ÷ - diagrams precisely cancel the H 33 diagrams in expression (4.1) when sin 0w = 0. Much of this can- cellation remains intact even when sin 0w ¢- 0. In fact, the only uncancelled one-loop graphs contributing to Ap contain internal vector bosons, ghosts or scalars that belong to the standard model subsector. These diagrams are listed in fig. 1. Although each of the graphs is UV divergent, we have organized the dia- grams in fig. 1 into combinations in which the ultra- violet infinities cancel. So no renormalizations for any of the parameters appearing in the gauge sector of the theory are needed in the p parameter calculation.

W 3 W 4"

W- W-

W- W+ W:]:

2) .~_,. "~.~v + ~ ' ~ - ~,-,~{ ,;~'-~ W- "'..,-." W" W- " ' - o - " W- W s "'++" W 3

W 3 W 3 W- +

3)

W a W ± W ±

W- W- W" W" W a W s

4)

5)

4,. ~,

~ w - - w ~ w 3

W ~ " -- W ~ W 3

B W- Z

Fig. 1. Lowest order graphs containing gauge bosons (waw lines), ghosts (dashed lines), or scalars (solid lines) which contribute to &). The Higgs propagator is marked by a circular blob.

5. T h e H i g g s p r o p a g a t o r

We have momentari ly ignored scalar radiative cor- rections to scalar propagators. Such radiative correc- tions are especially important in the limit o f a large scalar coupling constant. In this section we calculate these corrections to leading order in 1/N.

We begin by returning to the unbroken lagrangian in ( 2.4 ), and we decompose the complex scalar fields in the unshifted 2~:tuplet q, into their real and ima- ginary components:

~.~ *c2 + in2~.+ 2 ( ~ = : ~ : .

(~N N/L 2/'2N- J + i7c4,~,'- I

~ 0 ~ ' ] 7t2N + iTt4N

When rewritten in terms of the real n fields, the un-

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broken scalar sector of the classical lagrangian ap- pears as

~- ' " ' = ,~ , ½0.,,~,0"~,+½~ o~'~'- ~(~,~,)~ .

(The zero subscripts affixed to 20 and /-to are re- minders that these parameters really represent bare quantities.) We earlier analyzed the spontaneous symmetry breaking in this O (4N)-symmetr ic scalar sector in the semiclassical approximation. We now reanalyze the symmetry breaking including quantum corrections following the approach of Coleman, Jackiw and Politzer [ 9,10 ].

A term involving a new auxiliary field X is added to the scalar lagrangian:

,~ca,.r--" ~c.~., + - - X - ~;n, +/1o 2 42o N

The added term has no effect upon the dynamics of the theory, for the functional integral over Z is a triv- ial gaussian which multiplies the generating func- tional of the theory by an irrelevant constant. How- ever, the added term does cancel the scalar mass and quartic self interaction terms, and the new lagrangian is quadratic in the n, fields. After integrating out these 7t degrees of freedom in the connected generating functional W[J, Z] for the scalar sector of the theory, we obtain the effective action via the Legendre transformation

F[nc,X] = W[J,x]- f d4xJ'n~

N 2 NII~ X) = f d4x( - ~ltc['-Ilt c -4- 4~oX --½X. 2 +

+2Ni Tr log( [] +Z) •

When the fields in the effective action are held con- stant, F reduces to minus the integral of the effective potential V:

N v = - x 2 + ~xn~

42o

Nlzo2 z + 2 N i ~ __d4k _ _ _ ~ l o g ( - k 2 + x ) . 22o J 2rt 4

Spontaneous symmetry breaking takes place in the scalar sector if the classical fields to not vanish at the

stationary point of the effective potential. From the derivative conditions specifying the location of the min imum of V, we find the following classical field values at thc minimum point:

)~=0 ,

(5.1)

The ultraviolet cutoffA appearing in ( 5. l ) can be ab- sorbed into definitions of renormalized mass and scalar coupling parameters

2 _ _ /,12 /12 /~2 A 1 1 1 log 2 = 2o 4n 2' 2 - 2o ~- 41t 2 M--/'

where M denotes an arbitrary renormalization scale. After substituting these renormalized quantities back into (5. I ) we reconfirm that the symmetry breaks as ~t develops the VEV (Tt)2=Ni22/2-Nv 2. Because this VEV appears to equal the one found semiclassi- cally, the tree masses of vector bosons and ghosts in the gauged theory are unaltered in the new analysis so long as this new definition o fv 2 is used.

Having determined the ground slate of the scalar sector, we introduce the shifted field a=Tt 2 x - (Tt) and rewrite F in terms of this Higgs field:

4N--If [ N X 2 F = ,=,T~ d4x - ~ < D < - ~ a D a +

- ½1~<- - ~Za2-Za< n>

. {-mUo . Z ~ 2~-o - ½ (n>2 + 2Ni Tr log(F-1 + Z ) .

This expression for the effective action is the formal solution of the O(4N) theory to leading order in l / N. Any I PI Green's function is obtainable from F b y functional differentiation. We are particularly inter- ested in the l PI two-point functions because they are the inverse scalar propagators.

By inspection, there are no O ( N °) radiative cor- rections to the Goldstone propagators in the un- gauged scalar theory. However, there are radiative corrections to the Higgs propagator. We find the leading order expression for the Higgs propagator in thc aa entry o f the matrix propagator for the a-X system:

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i[1 + 2 / 4 n 2 log( - e M 2 / k 2 ) ] D"'( k2) = k2[l + )t/ 4n 2 log( - e M 2 / k2 ) ] - 2 I t z"

( 5 . 2 )

The physical Higgs state corresponds to the pole in the Doo propagator. For small scalar coupling, the pole so hides on the second Riemann sheet just below the logarithmic cut along the positive real axis. As 2--,0, so approaches the perturbative position k2= 2/~ 2, and the propagator Doo reduces to its standard tree form. For strong coupling, the pole location is determined by

_ e M 2 so ~ 8nZv z log - -

So

Neglecting the right-hand side's mild logarithmic de- pendence upon so, we observe that the scale of the Higgs pole is set by v z. Consequently, although the perturbative mass M~ =2/~2= 22v 2 grows large in a strongly interacting Higgs sector, the physical mass remains around the weak interaction scale. Such an upper bound for the Higgs mass in a strongly inter- acting O(N) model has been previously discussed in the literature [ 5, I 1 ].

6. N u m e r i c a l r e s u l t s

Having calculated the Higgs propagator, we now evaluate the diagrams in fig. 1 which contribute to the p parameter. We let A, 1 ~< i ~< 5, denote the values of the five finite combinations of graphs listed in the figure. Computing the diagrams in the first three sets is straightforward provided that one remembers var- ious symmetry factors and ghost-loop minus signs. The last two groups of graphs are more problematic since they involve the complicated Higgs propagator. We can slightly simplify their evaluation by exercis- ing our freedom to set the renormalization scale to M 2 = v2/e. However, the logarithms appearing in the Higgs propagator render impossible closed-form, an- alytic expressions for the fourth and fifth sets of graphs. Therefore, ~4 and ,,15 must be evaluated numerically.

Adding together all of the diagrams, we at last ob- tain the radiative corrections to the p parameter in the symplectic model:

g2 ! Ap= 64--~N [ ( 1 6 + ~ ) ( 1 l°g c°s20w']

+ sinZ0w ,/

- 12 log cos 20w + tan20w { 3 [log ]g2_ R (2) ]

log cosZ0w - 1 }] , (6.1) ~ 4

where R(2) is the numerically determined function. As promised, Ap does indeed vanish when sin 0w=0 and the custodial SU(2) is restored. Numerically evaluating eq. (6.1), we find

NAp= [ - 1 .457-0.565R(2) ] × 10 -3 . (6.2)

This result is based upon the value sin 20w = 0.229 de- rived from a global fit to all experimental data [8]. Amusingly, NAp falls within the error bars of the ex- perimental value Ape~p = ( - 2 + 8.6 ) × 10- 3 for all 2.

We can compare our value for the symplectic the- ory p parameter with conventional perturbation the- ory predictions for p in the electroweak model with- out fermions. To obtain the standard model p at one- loop order, we set N= 1 and evaluate the graphs in fig. 1 in Feynman gauge with the usual tree Higgs propagator substituted in place of the O (N o) expres- sion given in (5.2). The resulting equation for Ap~,i ~°°p~ is identical to (6.1) except that R(2) is re- placed by log 2. Consequently,

Ap~oop) = ( _ 1.457-0.565 log; t )× 10 -3 . (6.3)

As illustrated in fig. 2, NAp and Ap~oop) agree for small ;t as indeed they must. The 1IN method is a valid approximation scheme for large ). and large N provided that the expansion parameter) t /N is small. Of course the method is also valid for small 2. For sufficiently small 2, N may be set equal to one and ordinary perturbation theory is recovered.

It is interesting to examine the large ). behavior of the symplectic model p and to compare it with the large 2 limit of the standard modelp calculated to two- loop order in ref. [ 1 ]. When the Higgs is very heavy, ref. [ ! ] gives the two-loop radiative correction to p a s

. ~ 4, ~,,4~. M~. Ap~oop~= - 5 . 6 6 × Iu - log-~w~2.85 × 10 -7

M2w "

Adding to this the contributions from those graphs in fig. I which do not contain an internal Higgs propa-

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Volume 240, number 3,4 PHYSICS LETTERS B 26 April 1990

-t.~ ' ' ' 1 . . . . I . . . . I . . . . I . . . . I . . . .

- 2 . 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- 2 . 5 ~. L

x - 3 . 0 \ "

- 3 . 5 \ x --.

• 0 , . . . . , =

0 25 50 75 I00 125 150 X

a constant . This f ini te l imi t ing behav io r o f NAp in

the large N theory suggests that the presence of t e rms that formal ly diverge as 2 ~ oo in per tu rba t ive expres- sions for observables such as the p pa rame te r do not necessari ly indica te a potent ia l sensi t iv i ty to a very massive Higgs. On the contrary , ou r result re inforces the view that even a s ignif icant ly improved experi- menta l m e a s u r e m e n t of Ape~p would be unl ikely to provide a useful cons t ra in t on an ext remely massive Higgs. Thus we conclude with a specula t ive bu t plau- sible negat ive answer to our open ing quest ion.

Acknowledgement

Fig. 2. NAp in the symplectic model (solid curve) and Ap in the standard model at one-loop order (dashed curve) and at two- loop order (dot-dashed curve). The experimental value Ape,p= ( - 2 + 8.6 ) × I 0- 3 is also indicated without its error bars (dotted line ).

The au thor thanks Howard Georgi for helpful dis-

cuss ions and Mar t in E inhorn for useful referee com- ments . This work was suppor ted by N S F contrac t PHY-87-14654.

gator and reexpressing the result in te rms of our in- d e p e n d e n t var iable ;t, we f ind

Ap~i'°°p~ = ( - 1.457 - 0 . 5 6 6 log ;t +0 .00569) , ) (6 .4) 10 -3 .

(6 .4)

This equa t ion is plot ted a long with eqs. (6 .2) and (6 .3) in fig. 2. The agreement be tween the two-loop and al l-orders curves at large ;t is clearly bet ter than that be tween the single-loop and al l-orders curves. Whereas the con t r i bu t ions from single-loop graphs ini t ia l ly d ip p below its tree value an d con t inue loga- r i thmically downward , correct ions from two-loop and al l-orders d iagrams both tu rn p back upwards , albeit at qui te different t u r n - a r o u n d poin ts a n d varying rates. However, unl ike A p ~ i t°°p~ which eventua l ly di-

verges l inearly, NAp in the symplect ic theory tends to

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