Triviality of monomial Higgs potentials

Nuclear Physics B329 (1990) 574-582 North-Holland

TRIVIALITY OF MONOMIAL HIGGS POTENTIALS Hing Tong CHO and Kimball A. MILTON Department of Physics and Astronomy, University of Oklahoma, Norman, OK 73019", USA and Department of Physics, Ohio State University, Columbus, OH 43210, USA

James CLINE and Stephen S. PINSKY Department of Physics, Ohio State University, Columbus, OH 43210, USA

L.M. SIMMONS, Jr. Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA and The Santa Fe Institute, 1120 Canyon Road, Santa Fe, NM 87501, USA

Received 7 June 1989

The 8 expansion method is used to study all scalar field theories with symmetric monomial self-interactions of the form (¢2)p in four dimensions. With a momentum cut-off A it is shown that these theories are all distinct, admit spontaneous symmetry breaking, and have a nonperturbative structure in the coupling constant ~. In the limit that the momentum cut-off is taken to infinity, it is shown that the 8 expansion of all these theories degenerates to the renormalized expansion of a single theory independent of p. That expansion is the renormalized weak-coupling expansion of hq~4.It is argued that all theories of this class are the same in the limit A --, 00. This universal behavior of all monomial interactions with the discrete symmetry ~, ~ - ¢ is interpreted as a strong indication that all such theories are free, as ~,4 is believed to be.

1. Introduction Self-interacting scalar fields play a central role in the Higgs m e c h a n i s m of the s t a n d a r d m o d e l of electroweak interactions. However, because of strong indications of the triviality of the ?~4 theory in four space-time d i m e n s i o n s (for a recent review a n d references to the literature see ref. [1]), the f u n d a m e n t a l n a t u r e of the Higgs m e c h a n i s m remains in doubt. It is therefore i m p o r t a n t to explore a larger class of scalar p o t e n t i a l s which, like ;k~4, admit s p o n t a n e o u s s y m m e t r y breaking. W i t h the a d v e n t of the 8 expansion [2, 3] it becomes possible to study, in four dimensions, * Permanent address. 0550-3213/90/$03.50© Elsevier Science Publishers B.V. (North-Holland)

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self-interacting scalar fields with a lagrangian of the form £g= ½(0~) 2 + ll~Zq~2+ XM2ck2(ep2/M2) 8 ,

(1.1)

where M 2 is an arbitrary mass introduced to keep h dimensionless for all & As with the ~4~4 theory, eq. (1.1) is invariant under q~~ -q}. Clearly, for #2 < 0, this theory exhibits spontaneous symmetry breaking of the type required to give rise to masses in the standard model. Unlike the expansion in powers of ~,, which would lead to a power series with zero radius of convergence, the 8 expansion in some contexts results in a series with a finite radius of convergence [2, 3]. The coefficients of the 6 expansion have nontrivial ~, dependence and the convergence of the 8 expansion can be greatly improved by Pad6 approximants [4, 5]. In this paper, we introduce a momentum cut-off A and calculate the Green functions belonging to eq. (1.1) to second order in 8. Throughout this paper we restrict ourselves to four space-time dimensions. As long as A is finite, each value of 8 corresponds to a distinct theory. Therefore, if new physics arises at scales of order A, effectively cutting off the Higgs sector, the potentials (q}2)1+8 represent distinct physics for each value of 8, and the Green functions exhibit a nonperturbative structure in h. However, as A --* oo the renormalized Green functions for all 8 reduce to those arising from the weak-coupling expansion of hq,4 field theory. The 8 expansion therefore indicates, as A --* oo, that all theories of the class (1.1) have a universal behavior. Extensive examination of one member of that class, ~q~4, indicates that it is free [1]; therefore, all members of the universality class would be expected to be free. This paper is organized as follows: In sect. 2 the 8-expansion Feynman rules are stated. In sect. 3 the cut-off dependence of the two-, four-, and six-point functions are estimated through second order in ~, and the behavior of the renormalization constants is thereby determined. It is then seen in sect. 4 how the renormalized weak-coupling expansion of the hq}4 theory is reproduced as the cut-off tends to infinity.

2. Feynman rules As usual, the theory is renormalized by computing the Feynman graphs from the "renormalized" lagrangian, oo

.£aR = ½(1 + A ) ( Oq,) 2 + -lzm2,2+ AZMZq} z E ( 8 " / n ! ) [ l n ( ' ~ 2 / M Z ) ] ",

(2.1)

n=l

where the renormalization constants are calculated in powers of & m 2 = rag+ 8m~ + 82m• + . . . , Z = Z 0 + 8Z 1 + 82Z2 +

"'"

,

(2.2)

m 2 = p2 + 22~M z,

A = 8A 1 +

82A2 +

---

.

(2.3)

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We are to determine the renormalization constants in (2.2) and (2.3) so that the Green functions calculated from &oR are finite. The rules for generating the 8 expansion of eq. (2.1) may be stated as follows [6]: Construct first the standard Feynman vertex for 2n external lines as though 8 were a large integer. The 28 + 2 - 2n lines that are not external close on themselves, each loop giving an integral I,

I=j(2~)

(2.4)

4 k2+m2.

Thus, the effective vertex for this theory is ~.M2(28 + 2)!(I/M2) ~ 02"(8) = (8 + 1 - n ) ! 2 n + l - " I " - l "

(2.5)

The primitive vertices that enter into the 8 expansion are

0" v2.(8)

vt2~) = 08---~

(2.6)

In the 8 expansion the primitive vertex is associated with 2n lines, a power 8 " of the expansion parameter, and a factor of 1/m !. The term with n = 1, m = 0 has no powers of 8 and is added to the bare mass term ~/2q~2 as seen in (2.2). Comparing the 8-expansion rules with those of weak-coupling ~,q~4, which has one vertex with four lines and one power of the coupling ~, we see that the 8 expansion has a two-fold infinite set of vertices, v~2,~):2n is associated with all even numbers of lines, and m with all powers of the expansion parameter 8. Using these vertices and the propagator ( p 2 + m2)-1 one calculates the one-particle-irreducible (1PI) Green functions in the usual way. Only normal-ordered graphs (no one-vertex loops) are now included. To second order, it is easy to obtain explicit formulas for the primitive vertices. These are

o(21)=2•M2S, where S = 1 + if(3/2) +

v~22)=2XMz[ S2 +I'(3/2) -1] ,

ln(2I/M2),

and, for n > 1,

vO) 2~,MZ(_l)n(n_2)!(2/I) n-l, 2n o~2) 2n =

4XME(_ 1 ) " ( n

-

(2.7)

2)!(2/I)"-1[S + i f ( l )

(2.8a) - i(n

- 1)]

(2.8b)

In this paper, we are interested in the leading cut-off dependence of the Green

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Fig. 1. N-line graph (not including vertex factors) whose cut-off dependence is given in (3.3).

functions. From eqs. (2.7) and (2.8) the corresponding behavior of the vertices is v~2k) - 7tM 2 lnk(I/M2);

n > 1,

v (k)2,, - ( • M 2 / I " - 1 ) l n k - l ( 1 / M 2 ) .

(2.9)

The fundamental loop integral I behaves like I - A 2, where now we have introduced a momentum cut-off A. We are now ready to estimate the various contributions to the Green functions.

3. Order-8 Green functions and renormalization constants

Here we will estimate the leading cut-off behavior of the Green functions of the theory given by £~0r through second order in 8. The ingredients of this estimation are the vertex behavior given in (2.9), the cut-off dependence of the fundamental loop integral I, and the behavior of the N-line graph G(N, p) (not including vertex factors) shown in fig. 1. The argument is most immediate in coordinate space, where the propagator is A(x)--

m~ (2rr)2 ( mox ) - l K l ( mox ) "

(3.1)

Using the asymptotic behavior for small z, Kl(Z ) - l / z , dependence of

we find the leading

G ( N , p) = f d4x An(x)e ip'x "Ixl> 1/A

(3.2)

as A--* m: 1

G(2, p) - ~

G(3, p ) -

[ 1 p2 [ln(A2/mg) - -6- -mg + . . . ],

6--~-~4 A 2 -

N>3:

G(N,p)

ln(A2/m~)+...

"(A2) u-2 (2~-) 2N

N- 2

,

(3.3a)

(3.3b) p2 (A2) N-3 ] 8 N-3 +''" ' (3.3c)

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(a)

(b)

Fig. 2. One-vertex graphs contributing to the 2n-point functions.

where we have kept only the term of order p2, and the leading behavior, in each case, in A 2. Note that the leading powers of A are not multiplied by logarithms of the cut-off, contrary to what one might have guessed based on knowledge of (3.3a), and similarly no logarithms multiply the leading p2 term for N > 3. Because of this fact, the cut-off behavior estimated below for a single term of a class of diagrams is the same as that for the sum (over internal lines) of the entire class. The 0 ( 8 ) contributions to the 2n-point functions are the single vertex graphs shown in fig. 2. Apart from irrelevant constants, the four-point function to this order is (3.4)

8Zov(1) - 8 ~ Z o M 2 / A 2 '

so if we require a finite, nonzero contribution to G (a) to order 8, we have Zo _ A 2 / M 2. The two-point function is

8Zorn2'' + 8m21 - 8XZo M2 ln( A 2 / M 2) + 8m 2 .

(3.5)

This determines the mass counterterm: for simplicity, we can adopt an oversubtraction scheme in which m 2 is chosen to make (3.5) exactly vanish, so G (2) = m 2. The six-point function is

8 Zov(61) - 8k Z o M 2 / A 4 ,

(3.6)

so with the above choice for Z o, G (6) vanishes as A ---, oo. The classes of graphs that contribute to G(4)(0) in 0(8 2) are shown in figs. [2-4] with four external lines. The contributions from fig. 2 are 82Z00(2) + 82Z10(1) - ( 8 2 h Z o M 2 / A 2 ) l n ( A 2 / M

2) + 8 2 • Z t M 2 / A 2,

(3.7a)

1 Fig. 3. Two-vertex graphs contributing to the 2n-point function. Here 2n =p + q, and the number of internal lines is 1.

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Fig. 4. Graph containing the mass counterterm contributing to the 2n-point function.

and the contributions from fig. 3 are*

32Z2(ot41))2G(2)-32)~2Z2M41n(A2/mE)/A4, 3272i,,~1~ ~0 \ t"2n+ 2 )2G(2n ) _ 82k2Z2M4/A4, ~2'72,,(1)

,(1)('2['), ~Ot,2n+ 2V2n+4V l~,, + 1)

n > 1

~2)~2Z2M4/A4,

(3.7b) (3.7c) (3.7d)

and 2 (1) (1) * 2Zdv2nV2n+4G(2n ) - *2)~2Z2M4//A4,

n > 1.

(3.7e)

T h e graph in fig. 4 cancels the graph in fig. 3 with l = 2 and q = 0 exactly b y virtue of our oversubtraction prescription. T h e order 82 contribution to G¢4~(0) contains coefficients of order h and )~2 as well as complicated h behavior through the ln(m g) t e r m in (3.7b). Given the choice for Z0, (3.7) can be m a d e finite with a suitable choice for Z 1, behaving like Z 1 -

( A 2 / M E ) [ l n ( A 2 / M 2) + M n ( A 2 / m g ) ] ,

(3.8)

which again shows the nonperturbative structure of the ~ expansion for finite A. T h e graphs contributing to the two-point function Gt2~(0) in second order are again s h o w n in fig. [2-4] with two external lines. T h e large-A behavior of fig. 2 is 82Zov~2' + 82Zlv~i' + 82m~ - 82AZo M2 ln2( a2/M 2) + 82AZ1 M2 ln( A2/M 2) + ~2rn22, (3.9a) while that of fig. 3 is

#2Z~( v~+ 2)2 G(2n + 1) - #2A2Z~M4/A 2 ,

(3.9b)

2 (1)v2.+EG(En) (1) $ 2Z~)v2,, - 82)~2ZgM4/A2.

(3.9c)

and

* As shown in ref. [5], when the symmetry numbers are included in (3.7), the sum on n is finite for fixed momentum cut-off A. Therefore, we can examine individual terms in the n sum as far as our estimates of cut-off behavior go.

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Again, the graph in fig. 4 cancels the contribution of fig. 3 with l = 2 and q = 0 exactly by virtue of our oversubtraction prescription. Once more we can choose the mass counterterm in fig. 2b to keep G(2)(0) = m2; therefore,

m~-A2[~2+~21n(A2/M2)ln(A2/m~)+Xln2(A2/M2)].

(3.10)

There are many graphs which contribute to the six-point function in second order. A representative one is shown in fig. 3 with p = q = 3. It has the behavior 82ZoZ( v(91),+4)2G(2n+ 1) -

82~,2ZgM4/( A2)3,

(3.11)

which vanishes as A2 -~ 0o. The same conclusion holds for all the other graphs contributing to the six-point function in 0(82). The higher 2n-point function vanish even more strongly. Finally, we must comment on the momentum dependence of these graphs. With a finite cut-off A it is clear that for p 2 A2 each value of 3 will generate a distinct, nontrivial theory with a nonperturbative structure in ~,. However, for finite p2, as A ~ oo, we see from the results quoted in (3.3) that the graphs shown in fig. 3 with l > 2 cannot have any momentum dependence, since the p2 dependence is down by a power of A - 2. (Since there are no logarithmic corrections to the leading power, the infinite sum over internal lines cannot change the estimate.) The only four-point function graph with surviving momentum dependence is the one in fig. 3 with p = q = 2, with two internal lines. This graph, which has precisely the structure of the O(X2) four-point function graph in ~ 4 , will give a finite contribution.

4. Conclusions Let us restate what happens through second order in 3. As a result of our oversubtraction scheme, and appropriate wavefunction renormalization (choice of the constant A in (2.1)) the two-point function is simply Gt2)(p) = p 2 + m 2"

(4.1)

The only graphs surviving are those for the four-point function, in figs. 2 and 3 with p = q = 1 = 2. The other four-point function graphs simply rescale X, in the zerom o m e n t u m graph, by a term of order (1 + X3). It is perfectly clear, and is borne out by detailed calculation, what happens in higher order in 8. In third order, for example, graphs with the rescaled vertices of the form of fig. 3 with p = q = l = 2 occur in the four-point function, and surviving six-point function graphs, such as that shown in fig. 5 appear. [Since the triangle graph G(1,1,1) is finite, that diagram

33X3Zg(v(41))3(1/mg)G(1,1,1)- ?d33/mg,

(4.2)

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Fig. 5. The sole graph contributingto the six-point functionin order 83 survivingin the limit A ~ m.

a finite result.] Graphs with more internal lines all vanish as the cut-off goes to infinity. The conclusion is clear: only graphs which would appear in the weak-coupling expansion of ~44 appear (apart from the rescaling of h), and the renormalized 8 expansion precisely coincides with the renormalized 2, expansion. This means that all theories in the class (1.1) are equivalent, perturbatively in & This in turn suggests that all these theories are equivalent to the free theory. This at first sight rather surprising conclusion of the 8-expansion analysis of the renormalization of ~(42) 1÷~ may be understood as follows: suppose at the outset we had considered an interaction term of the form

-~'i., = XA2°-*>(42)1÷8.

(4.3)

Here the cut-off A is used in place of M in eq. (1.1). It is well-known [7] that for integer values of 8 :~ 0 the weak-coupling expansion of the lagrangian (4.3) yields renormalized Green functions which at an energy scale E are identical to those of a 44 theory, up to corrections of order E / A . What we have shown here is that the same is true even if 8 is not an integer; in particular, when 0 < 8 < 1, where one might have guessed that the theory would be more interesting because then (42) 1+8 has dimension less than four. We have in fact shown more than this, because the 8 expansion is a nontrivial, frequently convergent, resummation of the X expansion. Important physics is obtained by studying the way these theories depend on the cut-off A. In particular, the value of A at which the theory breaks down, A~, presumably will depend upon 8. Therefore, so will the bound obtained for the Higgs boson mass based on A~ [8]. If new physics provides a finite cut-off for these theories, the 8 expansion provides an important method of studying these theories at momentum scales near the cut-off energy. This study makes it even less likely that the Higgs is a fundamental particle, because in the limit of infinite cutoff the entire class of monomial interactions which could generate spontaneous symmetry breaking is equivalent to )~44, which is believed to be free. (For another 8-expansion treatment of this problem see ref. [9].) One of us (L.M.S.) thanks Ohio State University for its hospitality during a portion of this work. We thank Carl M. Bender for many helpful conversations

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during the course of this work. And we thank the US Department of Energy for partial financial support. References [1] K. Huang, Triviality of the Higgs field, Proc. DPF/APS Meeting, Storrs, CT, August 1988 (World Scientific, Singapore, 1989). [2] C.M. Bender, K.A. Milton, M. Moshe, S.S. Pinsky and L.M. Simmons, Jr., Phys, Rev. Lett. 58 (1987) 2615 [3] C.M. Bender, K.A. Milton, M. Moshe, S.S. Pinsky and L.M. Simmons, Jr., Phys. Rev. D37 (1988) 1472 [4] H.F. Jones and M. Monoyios, Int. J. Mod. Phys. A4 (1989) 1735 [5] H.T. Cho, K.A. Milton, S.S. Pinsky and L.M. Simmons, Jr., J. Math. Phys. 30 (1989) 2143 [6] S.S. Pinsky and L.M. Simmons, Jr., Phys. Rev. D38 (1988) 2518 [7] J. Polchinski, Nucl. Phys. B231 (1984) 269 [8] M. Liascher and P. Weisz, Nucl. Phys. B290 (1987) 25; B295 (1988) 65 [9] C.M. Bender and H.F. Jones, Phys. Rev. D38 (1988) 2526