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Dual models and spontaneous symmetry breaking II.
Nuclear Physics B70 (1974) 397-413.
DUAL MODELS
North-Holland Publishing Company
AND SPONTANEOUS BREAKING
SYMMETRY
II. *
K. BARDAKCI Department
of Physics, University of California, Berkeley,
California 94 720
Received 19 November 1973 Abstract: The stability of the vacuum of the orbital dual model with zero intercept with respect to spurion emission is investigated. Although the model is not fully satisfactory because of the presence of negative norm states, it serves as a simplified laboratory for the more realistic (and more complicated) models. Feynman rules for the calculation of the effective potential are developed, and the problem of finding the Goldstone minimum is reduced to a non-linear integral equation. The propagator of the model is also determined, and trajectories are shown to remain linear.
1. Introduction In a previous paper [l] referred to as (I) in what follows, a method for generating spontaneous symmetry breaking in certain dual models was proposed. This was achieved by introducing a unit amount of fifth momentum or “charge”into the orbital model with unit intercept, and using the resulting zero mass charged particles as supurions to be emitted to destabilize the vacuum. The whole problem then boils down to the determination of the minimum of the effective potential, which governs the self interaction of the zero momentum spurions. In (I), a method for the systematic construction of the effective potential was outlined. This method is feasible for the calculation of the first few lower order terms of the effective potential; however; for the higher terms it becomes very tedious and cumbersome and the need for an alternative approach becomes clear. It is the purpose of this note to develop a more efficient technique of calculation based on Feynman graph expansion. The application of this new technique to the model presented in (I) is unfortunately still beset by algebraic complications. In view of this, we present here the treatment of the orbital dual model with zero intercept, thereby avoiding some of the algebraic complications and hopefully illustrating the main ideas of the method. Of course, the problem we are tackling is somewhat academic; since the zero intercept * Research supported by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under Contract number F44620-70-C-0028. This document has been approved for public release and sale: its distribution is unlimited.
K. BardakEi, Dual models
398
orbital model is known to have negative norm states, they are expected to survive the spontaneous symmetry breaking. The particular problem treated here then serves as a training ground, and we hope to apply the techniques developed to the more interesting and important case of the model with unit intercept described in (I). In the following sections, we shall briefly review the zero intercept orbital model, the idea of the lowering of the energy of the original vacuum by spurion emission and finally the construction of the effective potential for the spurions. Although some of this is a repetition of the material covered in (I), it is included here to make the present paper self contained. The new ingredient of this paper is the development of Feynman graph techniques for the calculation of the effective potential. In terms of appropriately defined fields, the interaction turns out to consist of only a trilinear term. The problem of finding a non-trivial minimum of the effective potential is then shown to reduce to a non linear integral equation. We solve this equation in a very crude approximation and obtain a non-trivial minimum. Finally, we derive the propagator for the model in the usual operator formalism. The propagator depends on the solution to the integral equation described above, and apart from involving this one unknown function, it has simple explicit form. The trajectories stay straight, but in general the intercept shifts.
2. The effective potential
for the zero-intercept
model
The zero intercept orbital model is defined by the following formula for the Npoint amplitude in the tree approximation: 1 BN(Pl’
t perm. p2> .‘., pN) = s du X J n u?ii-' 11
(2.1)
ij
0
The symbols have the same meaning as in (I), and we briefly repeat the definitions for the sake of completeness. Fig. 1 shows the indices 1 to N arranged in planar
N-1 N 1
*
3
Fig. 1. The N-point
2 amplitude.
K. Bardakgi, Dual models
Fig. 2. A typical
399
tree graph
cyclic order, and the Mandelstam variables Sii are given by the formula sji = &i + pi+l + .. . t pi)2, where the indices are again ordered in the cyclic configuration of fig. 1. The Koba-Nielsen variables uji are defined in (I) or in any standard reference on dual models [2]. The integration in eq. (2.1) is over any set ofN-3 variables corresponding to a complete set of non-overlapping channels. The volume element J then depends on the choice of the variables of integration. To determine it, one has first to draw a planar tree graph as shown in fig. 2. The various internal lines are then labelled by the corresponding U’Saccording to the channel they represent. By the rules of the game, only trilinear vertices are allowed. Each propagator is then taken to be unity, and a typical vertex where three lines carrying variables ~1, ~2 and u3 meet is given by l/V, where, W1’
U2’ u3> =
[4 UI#2U3(2-U1-U2-U3)
+(I-utu2-u2u3-~~u3)2]~.
(2.2)
The overall volume element J is then the product of the trilinear couplings defined by eq. (2.2), so that we can write the following symbolic equation: /=+.
(2.3)
If one of the lines of the vertex is external, the corresponding set equal to zero, and eq. (2.2) simplifies to the following form V(0,
U2’
u3) = l-U2U3.
variable, say ul, is
(2.4)
The factorizability of the volume element, explained above, will be important in what follows. Finally, we have to take care of Bose symmetry since the dual amplitude, as it stands, has only cyclic symmetry. This means that (N-l)! terms, corresponding to all the permutations of the external lines axcept the cyclic ones, have to be added by hand. The terms represented by “perm” in eq. 1 are supposed to take care of the Bose symmetry. The lowest mass state in the model is a zero mass scalar, and it is an ideal candidate to serve as a spurion, since it can be emitted at zero four momentum, staying
K. BardakEi, Dual models
400
on the mass shell and without violation Lorentz invariance. We remind the reader that since there exists no completely satisfactory off mass shell continuation in dual theories, staying on the mass shell seems to be a technical requirement. We now very briefly review the construction of a new set of amplitudes from the old ones by emitting spurious into vacuum, and refer the reader to (1) for more details. A spurion is taken to be the zero mass scalar at zero four momentum, and a “mixed” amplitude BN, is defined through the following sum: B,(k,,
k,, .. . . kN; c) = c
cmS BN+m(kl,
k,, .. .. kN; 0, 0, .. . . 0).
m=O
(2.5)
zzz
The m zero four momenta stand for the m spurions emitted into vacuum, and S is the appropriate Bose symmetrization symbol, and c is the spurion to vacuum transition factor. If a Goldstone solution exists, BN is a multivalued function of c, and one is supposed to take the limit c + 0 after going around the cut in the complex c-plane to a different sheet *. Rather than tackle this problem directly, it is covenient to approach it through the effective potential or the efJfective Lagrangian [4]. Given the amplitude BN, one defines a truncated amplitude BN from which all the zero mass one particle poles have been subtracted. One then sets all the external momenta equal to zero to define&,(O). The effective potential is then given by the following expression.
where
L-z= ga$Lq2 - V(@).
(2.6)
The field 4 in the above equation represents the spurion, and the coefficients c, are determined by the requirement that in the tree approximation the above Lagrangian reproduces all the BN(O). Setting 6 c 3 = g, where g is the coupling constant of the model, we have, (QNEN(0) =N! cN .
(2.7)
The Goldstone solution corresponds to a non-trivial minimum of V, if such exists. To investigate the model more completely, and to determine the mass spectrum etc., one has to extend the effective Lagrangian to states other than the spurion, and this will be dealt with shortly. We first adress ourselIes, however, to the problem of determining I’. As explained in (l), the amplitudes BN can be determined by a subtraction procedure, and the cases N = 4 and 5 are dealt with in the appendix, using this procedure. However, this method is not practical for large N, and in sect. 3 we shall develop an alternative method.
* For a treatment
of a field theory
model along these lines, see Lee
[ 31.
K. Bardakci, Dual models
401
3. The graph expansion Consider eq. (2.1) in the limit of all Sii + 0. We are prevented from directly setting sjj = 0 since the integral then becomes divergent. This is, of course, a reminder that the single particle poles at zero mass have to be eliminated from all channels before letting sii + 0. The problem is further complicated by the appearance of various overlapping channels in eq. (2.1), whereas we know that they cannot simultaneously develop poles. To sort this problem out, we invoke the fundamental identity that the Uii’s satisfy [2] : uii +
l-I
Uk,l =
I,
(3.1)
k,l
where the pair of indices (k, 2) run over all the channels that overlap with the channel (i, j). The pole in the variable Sii comes from the region of integration when uii is near zero, and the above identity then requires all the overlapping Uk,J’s to be near one, and prevents poles from developing in the overlapping channels. We wish now to use the above identity to write eq. (2.1) as a sum over its non-overlapping channel poles, rather than the apparent product from of (2.1). Ordering the momenta as in fig. 1, we have the following set of identities as special cases of (3.1), k=i I=N-I
I=N-1 5,2
+
r-I I=3
=l, u2,1
Uli+
n
+J
= l,
(3 < i
k=2, I=i+l
I=N-2 Ulfl
+
n I=2
qJ=
1.
(3.2)
If we multiply the first term of each identity by the left-hand side of the following identity and sum, we obtain the equation below, I=N- 1
N- 2 m=i-2
n 1=3
i=3
in=2
k=i, I=N-1 Ul,m
n k=2, I=i+l
I=N-2 Uk,[ +
n
ul,J = 1.
(3.3)
1=2
This is the final form the fundamental identity needed in what follows. We make use of the above identity as follows. Let us multiply eq. (2.1) by unity, and reexpress unity as a sum over products of u’s through eq. (3.3). This is equivalent to multiplying (2.1) by the left-hand side of (3.3). The expression so obtained can be simplified further as follows. Since we are only interested in the limit sii + 0, we can cancel various factors of u against the singular terms in the integrand; for example, we can write, lim
(uuu~sij_‘)= 1.
(3.4)
We are permitted to do this because the resulting expression is non-singular in the relevant variable and the corrections vanish in the limit sii + 0. On the other hand, all the factors that are not cancelled in this manner must be kept as they are;
K. Bardakfi, Dual models
402
i.e., the dependence
on Sii should not be dropped. We then have the following result:
N-l
lim BN = c
u~!,‘-~, (3.5) ’ where the symbol Gi stands for a set of pair of indices (k,l) and is defined as follows. For i = 2, both k and 1 range from 3 to N. For i = N-l, they range from 2 to N-l. For any i in between these values, k and 1 both range either simultaneously from 2 to i, or from i + 1 to N. We now pause to explain this result and to provide some delayed motivation for the identity of (3.3). Consider the external lines labeled from 1 to N. The line 1 is treated on a different footing from the rest. (Of course, the choice of line 1 for this privileged status is arbitrary, and any other line would do as well.) The rest of the external lines, labeled from 2 to N, are divided into two bunches, one running from 2 to i, and the other from it1 to N, where i is the index of summation on the right hand side of eq. (3.5). The main point is that the ith term then involves only those channels which lie completely either in one bunch or the other, (see fig. 3). For an alternative but equivalent description consider the following three collections of external lines: the first is the line 1 only, the second includes lines 2 to i, and the third it1 to N. In the ith term on the right hand side of (3.3), factors involving momentum transfers connecting any two of the three sets of lines are suppressed. We shall call this result a partial tree decomposition; this nomenclature is justified in fig. 3. The solid line 1, which stands for an external leg, goes into two wavy lines, representing internal propagators. The wavy lines then branch into various external legs, and one then sums over all possible partitions of the external legs keeping the cyclic order intact. The important point is that the wavy internal line is a scalar propagator; it carries no momentum transfer. This then partially solves the problem of overlapping channels, and is the beginning of a Feynman graph expansion. To get a full graph expansion, it is clear that we have iterate (3.5) till a complete decomposition into tree graphs is achieved. This involves, however, first generalizing S-+0
i=2
Jdu
J
n
k,ltGi
Fig. 3. Partial
tree decomposition.
K. Bardakgi, Dual models
403
m
F
4
II
Fig. 4. Partial
tree decomposition
with one internal
”
line.
the identity of (3.5) to amplitudes which contain one internal wavy line in addition to the external legs. The amplitude in question, depicted in fig. 4, has the external legs numbered from m to n in cyclic order, and the wavy line is inserted between the legs numbered by m and n respectively as shown. We denote the invariant energy of the wavy line by s,,, E s, and the corresponding internal variable by u,,,~ e U, and the amplitude itself by B n_m+l(S;P,v Pm+13 '..3P,>.
The new amplitude
satisfies an integral representation
B n_m+l(s; p,. ... . p,) = ]duXJm
n F
identical to eq. (2.1)
uii-sii-l,
(3.6)
11
0
where i and j range from m to n. The only difference between (3.6) and (2.1) is in the range of indices i and j; the variables s and u that appear in eq. (3.6) are a subset of the variables that appear in (2.1). Similarly, Jm,n is that part of the volume factor which applies to the subset in question. For the sake of completeness, explicit expressions for the U’Sand the volume factor are given below in terms of the independent variables u and Ui,n, where 1 +m
=(1-u
‘l,k
=(I-
m+l,n
...Uk,n)(l--uUm+l,n”‘Uk,n
‘k+l,n)
Um+l,n...Uk,.)(l-Um+l,n”‘Uk,nUk+l,n)
’
Cl-q+1,n-.+ ,n)(l-Ul,n-Uk+l,n) where mtl lows:
J
Ul+l,n...Uk+l,n)(l-Uln..,Uk,n)
< k < n-l,
l
m.n = l-14 urn+r n
(3.7)
’
m+l < 1< n- 1 and I < k. The expression
x
1 I-%+l,n
x %+2,n
1
x ...
for .I, n is as fol-
l-T-23
%-l,n
.
(3.8)
K. Bardakfi, Dual models
404
Now we are ready to derive identities similar to (3.1) and (3.2) for the above set of u’s The first identity is the following: (3.9) where, as before, u, n G u, and u, n = 0. This identity is easily verified using eq. (3.7). Proceeding as before, we can derive the analogue of eq. (3.5)
lim Bn_m+1(S;Pm9 .... P,)
(3.10)
s-0 =
i=E1 jdu
i=m
,
J,,,,
0
where the Gimn of pairs of indices (k,Z) is defined by the requirement that either m~kkiandm~l~i,oritl~k~nandit1~I~nholds.Thisisthepartial tree decomposition for the amplitude with the wavy line, and it is pictorially represented in fig. 4. We note that the only difference between eq. (3.5) and (3.9) is the extra factor on the left of eq. (3.9), which disappears as u + 0. This is as it should be, since at u = 0, the wavy line becomes an external line of mass zero, and the subset of u’s that appear in (3.9) become a complete set of Koba-Nielsen variables in their own right for the subamplitude. One slightly unpleasant feature is the left-right asymmetry of the extra factor in (3.9). Clearly, this must disappear in the final answer after the integrations are carried out. However, it is convenient, although not essential, to eliminate this asymmetry by writing the mirror reflection of identity (3.9) and averaging, with the following final result,
lim Bn_m+l(S;Pm,.... p,j=‘=fj’
i=m
s-+0
l-;.,
r-f /=m+1
r-7 ’ k,lEGi.m,
Uk [-Sk,1 -1 ’
j=n-1
j=i+l
x
]duJmn 0
U&U
17 j=i
Umi , 4
(3.11)
where the symbols have the same meaning as in (3.10). Eq. (3.11) can be iterated in an obvious way to generate a complete graph expansion at the tree level. As mentioned earlier, one of the external lines has to be taken off mass shell (the wavy line), and has to be treated on a different footing then the other ones. Of course, in the end, one has to set the internal variable u of this line equal to zero; the wavy line becomes an ordinary on the mass shell external line and the symmetry is restored. There is one further problem with Bose symmetry, however. The original dual amplitude had only cyclic symmetry, and the complete Bose symmetry was achieved by explicit symmetrization of the external legs. When the external momenta all vanish, the order of the external legs is immaterial, and the net effect is multiplication by iV!/IV = (N- l)! The correction factor 1/IV takes care of the overcounting due
405
K. Bardakri, Dual models
,’
/“2
,’
Kh,,
us-1
u2,uJ
=
=
I’
u, -’
‘.\
++..&.+X______U
Fig. 5. The vertex and the propagator. to cyclic symmetry.
On the other hand, in the graph expansion, only the N-l external lines should be symmetrized, since we treat the wavy line distinct from the rest. This procedure generates non-planar tree graphs, and again introduces a factor of (N-l)!, in agreement with the previous counting. After these preliminaries, the Feynman rules are easy to read off from eqs (3.6) and (3.11). The vertex is always trilinear, and one leg of the vertex, the wavy line, is distinct from the other two legs, the dotted lines, as shown in fig. 5. We assign a field $(x, u) to the wavy line, and a field 4(x, U) to the dotted lines, where x is the usual space-time coordinate and u is an internal variable that ranges from 0 to 1. The propagator is always non diagonal. a wavy line always goes into a dotted line and vice versa, and the factor associated with the propagator is u-r-‘. The vertex, on the other hand, consists of two factors. The first factor comes from the projective volume element J and is given by eq. (2.2). The second factor is the term in parentheses on the right-hand side of eq. (3.1 I), which has to be expressed in terms of the vertex variables U, U,,i and Ui+l,n. Starting from the definitions given in eq. (3.7), and after some straightforward algebra, we have the following:
where T/is given by eq. (2.2). The discussion of the previous two paragraphs is then summerized by the following effective Lagrangian density: 1
._ie.{$(x;,,,+
1
1
dul du, dU3 K(ul; u2’ u3>
(3.12b)
K. Bardakfi, Dual models
406
where
The Lagrangian of eq. (3.12) is supposed to reproduce, in the tree approximation, the dual amplitude when all the external momenta are close to zero. However, before we can actually set them equal to zero, we must first truncate the amplitude by subtracting all zero mass one particle poles. This can easily be achieved by means of the following artifice. First, we define two new fields by scaling the old ones as follows: (3.13)
Gt = u G, $ = u@.
In addition, we introduce two new fields, Y and Z, which couple the same way as \llt and $+, except at u = 0. Their couplings are arranged so as to cancel the singularity at p2 = 0 in the propagator of the original fields. We note that the external fields are rCltand & as before, and Y and Z appear only internally. Since the one particle singularities are now eliminated, we can set all the momenta equal to zero, and drop the space-time dependence of the fields. The resulting truncated Lagrangian is given below:
x @,(u,)
#,(u,)
+ K(y
; 0, u3) G,(u,)
+ K(O; U2’ 0) YGQ) G&u,) + K(O;
0, 0) Y(y)
=w,)
-%43) -w,)l.
+ K(y;
-w,)
G,(u,)
030) qq
m4,> Z(u3) (3.14)
It is now straightforward to compute the coefficients c, of eq. (2.13) from Feynman rules. A sample calculation along these lines is carried out in the appendix. It is, however, not necessary to compute these coefficients; we can directly determine the minima (stationary points) of the Lagrangian by varying it with respect to the fields tit, @r, Y and Z. The resulting “equations of motion” are given below:
407
K. Bardakgi, Dual models
’ Z(u) = -4,(O),
Y(u) = 4,(O),
4J,(4 +;g
[
‘duz du, [-, q M(rV~,
Uj)
x G,(u,) @,(u,) ~ K(u; 0, UJ G,(O) @,(u,) -- K(% 5’ 0) @,(u,) @r(O) + K(u; 0, 0) 4,(O) 4,(0)1 = 0, G,(U) +g 11% 0 0 x @,(u,) - Wl
- GO;
5
5
CK(U1;U2, u) $&)
u2
u2’ u) G,(O) @,(u,)
; 0, u> !qq
@,(O)+ WC 0, u> YQ,K9(#@)I = 0,
(3.15)
where the kernel K is defined in eq. (3.12). The above set of non-linear integral equations are the fundamental result of this paper. In form, they are similar to Hartree-Fock equations or to the gap equations of many body physics. The remaining problem is then to ascertain whether these equations possess non-trivial solutions, and if so, determine the solution. As will be shown shortly, once tit and #+ are known, the new model can be constructed in terms of these functions. We do not know how to solve eq. (3.15) in closed form, so we must resort to approximations. Actually, with the help of a conputer, this should not be so difficult, since $r and @r are expected to be smooth non-singular functions of u in the interval 0 to 1. The kernel K can also be easily shown to have only integrable singularities in the same interval, which makes various numerical approaches feasible. Here, we will not attempt any sophisticated analysis of the above equations, since the problem under consideration is not a physically interesting one. The purpose of the whole exercise was to develop the necessary analytic machinery that would hopefully prove useful in more realistic models. For the sake of completeness, however, we shall attempt to find a solution in the following crude approximation. We expand the kernel in a power series in its arguments and retain only the lowest terms. The following approximate expression then follows: K(upT
u3)z
1 +Iu2
In this approximation, $,(u)
= a + b u, &(u)
++,r+
(3.16)
+u2u3.
rllr and @r become linear functions,
so that we can set,
= c + du, and obtain the following set of equations:
ct;g[~d2tc2+cd]=o, a+g{~bdtfactfadt~bc}=O,
dt$g~[ct$d]=o, btg[$zdtbct$zd]
=O.
(3.17)
This set of equations have the non-trivial solution gc = -2, b = d = 0, a = anything. We note that the scale of Jlt is always arbitrary, since it satisfies a homogeneous equation. In fact, it is remarkable that we were able to get a non-trivial solution at all, since the equation $, satisfies is an eigenfunction equation with a given eigenvalue! (g drops out of the problem). We consider these as unpleasent features of
K. Bardak& Dual models
408
the model, hopefully absent from more realistic models. In addition, ation scheme may be misleading.
our approxim-
4. The propagator In this section, we shall determine the propagator of the new model in terms of the functions r+Q and 4 of sect. 3, assuming, of course, that a non-trivial solution to the equations (3.15) exists. It is then possible, at least formally, to determine the vertex and hence the whole amplitude; however, this shall not be attempted here. Now consider a spray of spurions coming out of an internal line that joins a left cluster of lines to a right cluster. The two clusters may themselves contain spurions. Let us number the lines in the left cluster from 1 to 1, the lines in the right cluster from m t 1 toN, and the spurions in between from I + 1 to m. Fig. 6 shows these spurions being emitted in the multiperipheral configuration. As we let the spurion momenta go to zero, various subenergies have the following limits: si it + 0, Sij+Sim> (4.1) ‘j,k +%+l,k’ , 1 I aspj+O,where 1 Gill, 1+ 1
‘ir”(Ui,.JW + (jE uii)-si~m-l, j=I+E
j=m n
-Sm+l,k-1
(t.$k)-si’k-l
+
j=l+l
(:)::> n
uj,k
,
(4.2)
where the ranges of the indices are restricted as in (4.1). It is clear from eq. (4.2) that, in the final answer, the variables ui i appear only in the form of products that appear on the right hand side of that equation. We now express those products in terms of the independent set of variables Us,_ where 2 < r < N-2. j=m
n
j=l+l
u..=
I”
(1-xi,/ ‘1 (‘-‘l,i_l
xi,[ Z”l,m)
tlpxi,l Z”l,m)tl--ul,i_l xi,[ ‘1 ’
Fig. 6. Factorization
of the propagator.
K. Bardakgi, Dual models
‘f?f“j
k =
tl-Ym,k Z)(l-U1,&k (l-%,lym,k
j=l+l
’
‘)(l-Ym,k
’
409
Ul,k) ‘l,k) ’
(4.3)
where r=l Xi/znu,r> r=i
r=k-1 ymkE
r=m-1
n r=m
ul,,
Z=
n
Ulr.
r=l+l
’
The main lesson to be learned from eq. (4.3) and the discussions preceding it is the following. We can derive the subenergies into two classes: the “hard” subenergies that stay finite in the limit of vanishing spurion momenta, and the “soft” subenergies, or the spurion subenergies, that go to zero in the same limit. Similarly, in an obvious analogy, we can call the internal variables that have a hard subenergy as an exponent a hard variable, and those with a soft exponent, a soft variable. The important point is that the dependence of the hard variables on the soft ones is through the product z defined in eq. (4.3), and not through the individual factors. In view of this result, we can detach the middle internal line along with its spurion spray from the left and the right clusters. The propagator defined in this fashion depends only on the variables x E ul,/, y E u~,~ and z as defined in eq. (4.3). In fig. 7, the propagator is shown with the spurions emitted in the multiperipheral and tree configurations respectively, and the above comments become clearer in the tree configuration. We note that the solid lines in this figure are associated with the variables u~,~ and u~,~, and the wavy line with the variable w G u~+~ m, which are related to x,y and z as follows: x = QJ
(1 -z)( 1-xyz) YzUl,m)
WEUl+l,m
=(l_zy)(l-xy)’
(4.4)
Now let us list the various factors that make up the propagator: where r ranges from I to m. These fac(a) Subpropagators of the form u R-pZ-1, Lr tors multiply to the expression xR-p ml zR--p2-l yR-p2-1, where,
p2=Sl,l
= S1,J+1 = . . . = Sl,m,
R
=
c rz(as]+a,,, . n=l
Fig. 7. Spurion
emissions
from the propagator.
410
K. Bardakgi, Dual models
For the definition of the harmonic oscillator operators and the details of factorization in the dual models, we refer the reader to any standard review article [2]. (b) A factor of the form (1 -xzy)-l, connected with zero intercept, which again results from factorization. (c) A projective volume factor Vt(x, w, v), where Vis defined by eq. (2.2). (d) Finally the summation of the spurion lines have to be carried out. This summation is identical to the one carried out in sect. 3, and the result is again represented by the Lagrangian of eq. (3.12). This shows that, consistent with our earlier notation, the wavy line of fig. 7 stands for the field +, which has to satisfy eqs. (3.15). Putting everything together, we have the following result: 1
1
1
sj-j-dx
Ab2)=D(p2> +g
du dw I’-l&y,
0 0 x #-p2-1
$?-p2-1
w)
0
zR-p2 -1
(1 -XJJZ)_1 l//(w),
where A is the final propagator, D is the bare propagator
D(p2)=
]du
(1-u)-l
defined by
u~-+-~,
0
and $ is the solution to equations (3.15). The expression given above is somewhat formal as it stands. For example, it suffers from spurious infrared problems which require a more careful treatment. Also, there is a problem of twist invariance, which again can be taken care of fairly easily. We shall not dwell on these problems here, since our purpose was solely to develop the techniques needed in arriving at eq. (4.5), and hopefully thereby pave the way to a more realistic calculation in a better model. We end this section with the observation that the spectrum of the propagator we have just calculated consists of straight line trajectories. This follows from the fact that it only depends on the operator R. We believe this to be a general result applicable to any model constructed along the lines given in (l), and we have the following simple argument to support it. The spurions, since they are emitted at zero four momentum, cannot carry any angular momentum. Since any curvature in the trajectories must come from an angular momentum dependent interaction, the preceding argument seems to exclude it.
5. Conclusions and new directions In the preceding sactions, we have treated a somewhat artificial model, with a view to developing tools for the treatment of more interesting models. The main question is then, for example, in the case of the model defined in ref. [ 11, can we sum the spurion graphs by means of a compact Lagrangian similar to the one given by eq. (3.12)? The crucial difference between the model treated here and the one
411
K. BardakGi, Dual models
defined in (1) is the existence of a “fifth” momentum in the latter case, which is not set equal to zero at the end. This has the consequence that, in a realistic model, both the fields and the interaction in the effective Lagrangian will involve an infinite set of harmonic oscillator operators. This complicates the situation quite a bit, and converts a quantum mechanical problem into a field-theoretic one. Fortunately, there seems to be a compensating simplification available. Since the model has unit intercept, it is well known that an infinite set of Ward identities can be derived [5]. These identities seem to be sufficient in number to eliminate all the oscillator states in the effective Lagrangian and thereby reconvert the problem into a quantum mechanical one. Work is under progress on this exciting possibility.
Appendix Here we wish to compute the contact terms in the effective Lagrangian corresponding to four and five point interactions. Our aim is to compare the results of the subtraction procedure, developed in ref. [ 11, with the new technique developed in this paper. We remind the reader that in the case of a four point function, only two poles, zf the ,form 1/s12 + 1/sZ3, need be subtracted to arrive at the truncated amplitude B. Setting g = 1 for simplicity, have the following: g4
=
]&
{x-slZ-1
(1_x)-s23-1
_
x-w-l
_
(1 _x)-s23
-l}.
(A.11
0
As explained in I, we can now let s12 + 0, ~23 + 0, in the integrand obtain the following result: E4(0) = 0.
directly and
(A4
Similarly, for a five point function, we have five double poles to subtract, of which a representative one is, for example, ST; s$. The truncated amplitude is given by the following expression:
x
q3-1
q4-1
u-s45-l
u-s51-1
45
51
_
u
-slz-l -s45-1 15 u12 u45
-
. ..}.
(A.3)
where l-u45
l-U12 ‘23
= 1-u12u45
t
‘34
= 1-u12
U45’
U15 =
1&U12 U45.
The dots in the above formula stand for the other four poles which the reader can fill in easily, using cyclic symmetry. We can again set all the subenergies equal to zero, and after some algebra, obtain the following: (A.41
K. Bardakgi, Dual models
412
2,..,
+
.\ I’ z
,’ I’ 3
I ’ ___m(
2 \\ <’3 ‘\ , \. ,’
,/4
Fig. 8. Feynman
graphs
$I
’ ___-4 ‘\
,f’
+
‘%5
3..
-. / .. ,’
+ 2____
1 I 5
5
for four and five point
/4
amplitudes.
Let us now rederive (A.2) and (A.4), using the interaction of eq. (3.12). The relevant Feynman graphs for the four and five point functions are given in fig. 8. We note that, as remarked earlier, the interaction vertex always involves a wavy line and two dotted lines, and the propagator converts a wavy line into a dotted one and vice versa. The external legs are on the mass shell, with u = 0. For the calculation at hand, we only need the following special values of the kernel K,
K(z;O,y)= +$
K(O;v)=&
For the four point function, B, ‘v ]&X-S12-’ 0
K(O;O,y)=1.
(A.5)
we have the following:
+ ]dyy-%3-l.
(A-6)
0
The poles can now be subtracted tained.
trivially, and again the result of eq. (A.2) ob-
K. Bardakci, Dual models
The calculation B, u /
+ 1-i
/dx
is similar for the five point function, dy
&x-s23-1
XY x-s’2-l
Y_s4s_l
413
and the result is given below:
y-s45-1 + 1;Ax;yx-s15-1
+
1-i
xy
p15-l.y-s34-1
+ 1-i
The poles can easily be isolated and subtracted The result is.
0
0
(A.4).
References [l]
y-Q4-1
by expanding in powers of xy.
~5(0)=3]~dxdq+,, again confirming
xyp12-l
(A.7)
1-xy
1-xy
1-xy
y-%-l
K. Bardakgi,
Nucl. Phys. B68 (1974) 331. Proc. of Brandels Summer School; J.H. Schwarz, Phys. Reports 8 (1973) 270. [3] B.W. Lee, Nucl. Phys. B9 (1969) 649. [4] J. Goldstone, Nuovo Cimento 19 (1961) 154; J. Goldstone, A. Salam, and S. Weinberg, Phys. Rev. 127 (1963) [5] M.A. Virasoro, Phys. Rev. Dl (1970) 2933.
[ 21 S. Mandelstam,
965.
.
DUAL MODELS
North-Holland Publishing Company
AND SPONTANEOUS BREAKING
SYMMETRY
II. *
K. BARDAKCI Department
of Physics, University of California, Berkeley,
California 94 720
Received 19 November 1973 Abstract: The stability of the vacuum of the orbital dual model with zero intercept with respect to spurion emission is investigated. Although the model is not fully satisfactory because of the presence of negative norm states, it serves as a simplified laboratory for the more realistic (and more complicated) models. Feynman rules for the calculation of the effective potential are developed, and the problem of finding the Goldstone minimum is reduced to a non-linear integral equation. The propagator of the model is also determined, and trajectories are shown to remain linear.
1. Introduction In a previous paper [l] referred to as (I) in what follows, a method for generating spontaneous symmetry breaking in certain dual models was proposed. This was achieved by introducing a unit amount of fifth momentum or “charge”into the orbital model with unit intercept, and using the resulting zero mass charged particles as supurions to be emitted to destabilize the vacuum. The whole problem then boils down to the determination of the minimum of the effective potential, which governs the self interaction of the zero momentum spurions. In (I), a method for the systematic construction of the effective potential was outlined. This method is feasible for the calculation of the first few lower order terms of the effective potential; however; for the higher terms it becomes very tedious and cumbersome and the need for an alternative approach becomes clear. It is the purpose of this note to develop a more efficient technique of calculation based on Feynman graph expansion. The application of this new technique to the model presented in (I) is unfortunately still beset by algebraic complications. In view of this, we present here the treatment of the orbital dual model with zero intercept, thereby avoiding some of the algebraic complications and hopefully illustrating the main ideas of the method. Of course, the problem we are tackling is somewhat academic; since the zero intercept * Research supported by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under Contract number F44620-70-C-0028. This document has been approved for public release and sale: its distribution is unlimited.
K. BardakEi, Dual models
398
orbital model is known to have negative norm states, they are expected to survive the spontaneous symmetry breaking. The particular problem treated here then serves as a training ground, and we hope to apply the techniques developed to the more interesting and important case of the model with unit intercept described in (I). In the following sections, we shall briefly review the zero intercept orbital model, the idea of the lowering of the energy of the original vacuum by spurion emission and finally the construction of the effective potential for the spurions. Although some of this is a repetition of the material covered in (I), it is included here to make the present paper self contained. The new ingredient of this paper is the development of Feynman graph techniques for the calculation of the effective potential. In terms of appropriately defined fields, the interaction turns out to consist of only a trilinear term. The problem of finding a non-trivial minimum of the effective potential is then shown to reduce to a non linear integral equation. We solve this equation in a very crude approximation and obtain a non-trivial minimum. Finally, we derive the propagator for the model in the usual operator formalism. The propagator depends on the solution to the integral equation described above, and apart from involving this one unknown function, it has simple explicit form. The trajectories stay straight, but in general the intercept shifts.
2. The effective potential
for the zero-intercept
model
The zero intercept orbital model is defined by the following formula for the Npoint amplitude in the tree approximation: 1 BN(Pl’
t perm. p2> .‘., pN) = s du X J n u?ii-' 11
(2.1)
ij
0
The symbols have the same meaning as in (I), and we briefly repeat the definitions for the sake of completeness. Fig. 1 shows the indices 1 to N arranged in planar
N-1 N 1
*
3
Fig. 1. The N-point
2 amplitude.
K. Bardakgi, Dual models
Fig. 2. A typical
399
tree graph
cyclic order, and the Mandelstam variables Sii are given by the formula sji = &i + pi+l + .. . t pi)2, where the indices are again ordered in the cyclic configuration of fig. 1. The Koba-Nielsen variables uji are defined in (I) or in any standard reference on dual models [2]. The integration in eq. (2.1) is over any set ofN-3 variables corresponding to a complete set of non-overlapping channels. The volume element J then depends on the choice of the variables of integration. To determine it, one has first to draw a planar tree graph as shown in fig. 2. The various internal lines are then labelled by the corresponding U’Saccording to the channel they represent. By the rules of the game, only trilinear vertices are allowed. Each propagator is then taken to be unity, and a typical vertex where three lines carrying variables ~1, ~2 and u3 meet is given by l/V, where, W1’
U2’ u3> =
[4 UI#2U3(2-U1-U2-U3)
+(I-utu2-u2u3-~~u3)2]~.
(2.2)
The overall volume element J is then the product of the trilinear couplings defined by eq. (2.2), so that we can write the following symbolic equation: /=+.
(2.3)
If one of the lines of the vertex is external, the corresponding set equal to zero, and eq. (2.2) simplifies to the following form V(0,
U2’
u3) = l-U2U3.
variable, say ul, is
(2.4)
The factorizability of the volume element, explained above, will be important in what follows. Finally, we have to take care of Bose symmetry since the dual amplitude, as it stands, has only cyclic symmetry. This means that (N-l)! terms, corresponding to all the permutations of the external lines axcept the cyclic ones, have to be added by hand. The terms represented by “perm” in eq. 1 are supposed to take care of the Bose symmetry. The lowest mass state in the model is a zero mass scalar, and it is an ideal candidate to serve as a spurion, since it can be emitted at zero four momentum, staying
K. BardakEi, Dual models
400
on the mass shell and without violation Lorentz invariance. We remind the reader that since there exists no completely satisfactory off mass shell continuation in dual theories, staying on the mass shell seems to be a technical requirement. We now very briefly review the construction of a new set of amplitudes from the old ones by emitting spurious into vacuum, and refer the reader to (1) for more details. A spurion is taken to be the zero mass scalar at zero four momentum, and a “mixed” amplitude BN, is defined through the following sum: B,(k,,
k,, .. . . kN; c) = c
cmS BN+m(kl,
k,, .. .. kN; 0, 0, .. . . 0).
m=O
(2.5)
zzz
The m zero four momenta stand for the m spurions emitted into vacuum, and S is the appropriate Bose symmetrization symbol, and c is the spurion to vacuum transition factor. If a Goldstone solution exists, BN is a multivalued function of c, and one is supposed to take the limit c + 0 after going around the cut in the complex c-plane to a different sheet *. Rather than tackle this problem directly, it is covenient to approach it through the effective potential or the efJfective Lagrangian [4]. Given the amplitude BN, one defines a truncated amplitude BN from which all the zero mass one particle poles have been subtracted. One then sets all the external momenta equal to zero to define&,(O). The effective potential is then given by the following expression.
where
L-z= ga$Lq2 - V(@).
(2.6)
The field 4 in the above equation represents the spurion, and the coefficients c, are determined by the requirement that in the tree approximation the above Lagrangian reproduces all the BN(O). Setting 6 c 3 = g, where g is the coupling constant of the model, we have, (QNEN(0) =N! cN .
(2.7)
The Goldstone solution corresponds to a non-trivial minimum of V, if such exists. To investigate the model more completely, and to determine the mass spectrum etc., one has to extend the effective Lagrangian to states other than the spurion, and this will be dealt with shortly. We first adress ourselIes, however, to the problem of determining I’. As explained in (l), the amplitudes BN can be determined by a subtraction procedure, and the cases N = 4 and 5 are dealt with in the appendix, using this procedure. However, this method is not practical for large N, and in sect. 3 we shall develop an alternative method.
* For a treatment
of a field theory
model along these lines, see Lee
[ 31.
K. Bardakci, Dual models
401
3. The graph expansion Consider eq. (2.1) in the limit of all Sii + 0. We are prevented from directly setting sjj = 0 since the integral then becomes divergent. This is, of course, a reminder that the single particle poles at zero mass have to be eliminated from all channels before letting sii + 0. The problem is further complicated by the appearance of various overlapping channels in eq. (2.1), whereas we know that they cannot simultaneously develop poles. To sort this problem out, we invoke the fundamental identity that the Uii’s satisfy [2] : uii +
l-I
Uk,l =
I,
(3.1)
k,l
where the pair of indices (k, 2) run over all the channels that overlap with the channel (i, j). The pole in the variable Sii comes from the region of integration when uii is near zero, and the above identity then requires all the overlapping Uk,J’s to be near one, and prevents poles from developing in the overlapping channels. We wish now to use the above identity to write eq. (2.1) as a sum over its non-overlapping channel poles, rather than the apparent product from of (2.1). Ordering the momenta as in fig. 1, we have the following set of identities as special cases of (3.1), k=i I=N-I
I=N-1 5,2
+
r-I I=3
=l, u2,1
Uli+
n
+J
= l,
(3 < i
k=2, I=i+l
I=N-2 Ulfl
+
n I=2
qJ=
1.
(3.2)
If we multiply the first term of each identity by the left-hand side of the following identity and sum, we obtain the equation below, I=N- 1
N- 2 m=i-2
n 1=3
i=3
in=2
k=i, I=N-1 Ul,m
n k=2, I=i+l
I=N-2 Uk,[ +
n
ul,J = 1.
(3.3)
1=2
This is the final form the fundamental identity needed in what follows. We make use of the above identity as follows. Let us multiply eq. (2.1) by unity, and reexpress unity as a sum over products of u’s through eq. (3.3). This is equivalent to multiplying (2.1) by the left-hand side of (3.3). The expression so obtained can be simplified further as follows. Since we are only interested in the limit sii + 0, we can cancel various factors of u against the singular terms in the integrand; for example, we can write, lim
(uuu~sij_‘)= 1.
(3.4)
We are permitted to do this because the resulting expression is non-singular in the relevant variable and the corrections vanish in the limit sii + 0. On the other hand, all the factors that are not cancelled in this manner must be kept as they are;
K. Bardakfi, Dual models
402
i.e., the dependence
on Sii should not be dropped. We then have the following result:
N-l
lim BN = c
u~!,‘-~, (3.5) ’ where the symbol Gi stands for a set of pair of indices (k,l) and is defined as follows. For i = 2, both k and 1 range from 3 to N. For i = N-l, they range from 2 to N-l. For any i in between these values, k and 1 both range either simultaneously from 2 to i, or from i + 1 to N. We now pause to explain this result and to provide some delayed motivation for the identity of (3.3). Consider the external lines labeled from 1 to N. The line 1 is treated on a different footing from the rest. (Of course, the choice of line 1 for this privileged status is arbitrary, and any other line would do as well.) The rest of the external lines, labeled from 2 to N, are divided into two bunches, one running from 2 to i, and the other from it1 to N, where i is the index of summation on the right hand side of eq. (3.5). The main point is that the ith term then involves only those channels which lie completely either in one bunch or the other, (see fig. 3). For an alternative but equivalent description consider the following three collections of external lines: the first is the line 1 only, the second includes lines 2 to i, and the third it1 to N. In the ith term on the right hand side of (3.3), factors involving momentum transfers connecting any two of the three sets of lines are suppressed. We shall call this result a partial tree decomposition; this nomenclature is justified in fig. 3. The solid line 1, which stands for an external leg, goes into two wavy lines, representing internal propagators. The wavy lines then branch into various external legs, and one then sums over all possible partitions of the external legs keeping the cyclic order intact. The important point is that the wavy internal line is a scalar propagator; it carries no momentum transfer. This then partially solves the problem of overlapping channels, and is the beginning of a Feynman graph expansion. To get a full graph expansion, it is clear that we have iterate (3.5) till a complete decomposition into tree graphs is achieved. This involves, however, first generalizing S-+0
i=2
Jdu
J
n
k,ltGi
Fig. 3. Partial
tree decomposition.
K. Bardakgi, Dual models
403
m
F
4
II
Fig. 4. Partial
tree decomposition
with one internal
”
line.
the identity of (3.5) to amplitudes which contain one internal wavy line in addition to the external legs. The amplitude in question, depicted in fig. 4, has the external legs numbered from m to n in cyclic order, and the wavy line is inserted between the legs numbered by m and n respectively as shown. We denote the invariant energy of the wavy line by s,,, E s, and the corresponding internal variable by u,,,~ e U, and the amplitude itself by B n_m+l(S;P,v Pm+13 '..3P,>.
The new amplitude
satisfies an integral representation
B n_m+l(s; p,. ... . p,) = ]duXJm
n F
identical to eq. (2.1)
uii-sii-l,
(3.6)
11
0
where i and j range from m to n. The only difference between (3.6) and (2.1) is in the range of indices i and j; the variables s and u that appear in eq. (3.6) are a subset of the variables that appear in (2.1). Similarly, Jm,n is that part of the volume factor which applies to the subset in question. For the sake of completeness, explicit expressions for the U’Sand the volume factor are given below in terms of the independent variables u and Ui,n, where 1 +m
=(1-u
‘l,k
=(I-
m+l,n
...Uk,n)(l--uUm+l,n”‘Uk,n
‘k+l,n)
Um+l,n...Uk,.)(l-Um+l,n”‘Uk,nUk+l,n)
’
Cl-q+1,n-.+ ,n)(l-Ul,n-Uk+l,n) where mtl lows:
J
Ul+l,n...Uk+l,n)(l-Uln..,Uk,n)
< k < n-l,
l
m.n = l-14 urn+r n
(3.7)
’
m+l < 1< n- 1 and I < k. The expression
x
1 I-%+l,n
x %+2,n
1
x ...
for .I, n is as fol-
l-T-23
%-l,n
.
(3.8)
K. Bardakfi, Dual models
404
Now we are ready to derive identities similar to (3.1) and (3.2) for the above set of u’s The first identity is the following: (3.9) where, as before, u, n G u, and u, n = 0. This identity is easily verified using eq. (3.7). Proceeding as before, we can derive the analogue of eq. (3.5)
lim Bn_m+1(S;Pm9 .... P,)
(3.10)
s-0 =
i=E1 jdu
i=m
,
J,,,,
0
where the Gimn of pairs of indices (k,Z) is defined by the requirement that either m~kkiandm~l~i,oritl~k~nandit1~I~nholds.Thisisthepartial tree decomposition for the amplitude with the wavy line, and it is pictorially represented in fig. 4. We note that the only difference between eq. (3.5) and (3.9) is the extra factor on the left of eq. (3.9), which disappears as u + 0. This is as it should be, since at u = 0, the wavy line becomes an external line of mass zero, and the subset of u’s that appear in (3.9) become a complete set of Koba-Nielsen variables in their own right for the subamplitude. One slightly unpleasant feature is the left-right asymmetry of the extra factor in (3.9). Clearly, this must disappear in the final answer after the integrations are carried out. However, it is convenient, although not essential, to eliminate this asymmetry by writing the mirror reflection of identity (3.9) and averaging, with the following final result,
lim Bn_m+l(S;Pm,.... p,j=‘=fj’
i=m
s-+0
l-;.,
r-f /=m+1
r-7 ’ k,lEGi.m,
Uk [-Sk,1 -1 ’
j=n-1
j=i+l
x
]duJmn 0
U&U
17 j=i
Umi , 4
(3.11)
where the symbols have the same meaning as in (3.10). Eq. (3.11) can be iterated in an obvious way to generate a complete graph expansion at the tree level. As mentioned earlier, one of the external lines has to be taken off mass shell (the wavy line), and has to be treated on a different footing then the other ones. Of course, in the end, one has to set the internal variable u of this line equal to zero; the wavy line becomes an ordinary on the mass shell external line and the symmetry is restored. There is one further problem with Bose symmetry, however. The original dual amplitude had only cyclic symmetry, and the complete Bose symmetry was achieved by explicit symmetrization of the external legs. When the external momenta all vanish, the order of the external legs is immaterial, and the net effect is multiplication by iV!/IV = (N- l)! The correction factor 1/IV takes care of the overcounting due
405
K. Bardakri, Dual models
,’
/“2
,’
Kh,,
us-1
u2,uJ
=
=
I’
u, -’
‘.\
++..&.+X______U
Fig. 5. The vertex and the propagator. to cyclic symmetry.
On the other hand, in the graph expansion, only the N-l external lines should be symmetrized, since we treat the wavy line distinct from the rest. This procedure generates non-planar tree graphs, and again introduces a factor of (N-l)!, in agreement with the previous counting. After these preliminaries, the Feynman rules are easy to read off from eqs (3.6) and (3.11). The vertex is always trilinear, and one leg of the vertex, the wavy line, is distinct from the other two legs, the dotted lines, as shown in fig. 5. We assign a field $(x, u) to the wavy line, and a field 4(x, U) to the dotted lines, where x is the usual space-time coordinate and u is an internal variable that ranges from 0 to 1. The propagator is always non diagonal. a wavy line always goes into a dotted line and vice versa, and the factor associated with the propagator is u-r-‘. The vertex, on the other hand, consists of two factors. The first factor comes from the projective volume element J and is given by eq. (2.2). The second factor is the term in parentheses on the right-hand side of eq. (3.1 I), which has to be expressed in terms of the vertex variables U, U,,i and Ui+l,n. Starting from the definitions given in eq. (3.7), and after some straightforward algebra, we have the following:
where T/is given by eq. (2.2). The discussion of the previous two paragraphs is then summerized by the following effective Lagrangian density: 1
._ie.{$(x;,,,+
1
1
dul du, dU3 K(ul; u2’ u3>
(3.12b)
K. Bardakfi, Dual models
406
where
The Lagrangian of eq. (3.12) is supposed to reproduce, in the tree approximation, the dual amplitude when all the external momenta are close to zero. However, before we can actually set them equal to zero, we must first truncate the amplitude by subtracting all zero mass one particle poles. This can easily be achieved by means of the following artifice. First, we define two new fields by scaling the old ones as follows: (3.13)
Gt = u G, $ = u@.
In addition, we introduce two new fields, Y and Z, which couple the same way as \llt and $+, except at u = 0. Their couplings are arranged so as to cancel the singularity at p2 = 0 in the propagator of the original fields. We note that the external fields are rCltand & as before, and Y and Z appear only internally. Since the one particle singularities are now eliminated, we can set all the momenta equal to zero, and drop the space-time dependence of the fields. The resulting truncated Lagrangian is given below:
x @,(u,)
#,(u,)
+ K(y
; 0, u3) G,(u,)
+ K(O; U2’ 0) YGQ) G&u,) + K(O;
0, 0) Y(y)
=w,)
-%43) -w,)l.
+ K(y;
-w,)
G,(u,)
030) qq
m4,> Z(u3) (3.14)
It is now straightforward to compute the coefficients c, of eq. (2.13) from Feynman rules. A sample calculation along these lines is carried out in the appendix. It is, however, not necessary to compute these coefficients; we can directly determine the minima (stationary points) of the Lagrangian by varying it with respect to the fields tit, @r, Y and Z. The resulting “equations of motion” are given below:
407
K. Bardakgi, Dual models
’ Z(u) = -4,(O),
Y(u) = 4,(O),
4J,(4 +;g
[
‘duz du, [-, q M(rV~,
Uj)
x G,(u,) @,(u,) ~ K(u; 0, UJ G,(O) @,(u,) -- K(% 5’ 0) @,(u,) @r(O) + K(u; 0, 0) 4,(O) 4,(0)1 = 0, G,(U) +g 11% 0 0 x @,(u,) - Wl
- GO;
5
5
CK(U1;U2, u) $&)
u2
u2’ u) G,(O) @,(u,)
; 0, u> !qq
@,(O)+ WC 0, u> YQ,K9(#@)I = 0,
(3.15)
where the kernel K is defined in eq. (3.12). The above set of non-linear integral equations are the fundamental result of this paper. In form, they are similar to Hartree-Fock equations or to the gap equations of many body physics. The remaining problem is then to ascertain whether these equations possess non-trivial solutions, and if so, determine the solution. As will be shown shortly, once tit and #+ are known, the new model can be constructed in terms of these functions. We do not know how to solve eq. (3.15) in closed form, so we must resort to approximations. Actually, with the help of a conputer, this should not be so difficult, since $r and @r are expected to be smooth non-singular functions of u in the interval 0 to 1. The kernel K can also be easily shown to have only integrable singularities in the same interval, which makes various numerical approaches feasible. Here, we will not attempt any sophisticated analysis of the above equations, since the problem under consideration is not a physically interesting one. The purpose of the whole exercise was to develop the necessary analytic machinery that would hopefully prove useful in more realistic models. For the sake of completeness, however, we shall attempt to find a solution in the following crude approximation. We expand the kernel in a power series in its arguments and retain only the lowest terms. The following approximate expression then follows: K(upT
u3)z
1 +Iu2
In this approximation, $,(u)
= a + b u, &(u)
++,r+
(3.16)
+u2u3.
rllr and @r become linear functions,
so that we can set,
= c + du, and obtain the following set of equations:
ct;g[~d2tc2+cd]=o, a+g{~bdtfactfadt~bc}=O,
dt$g~[ct$d]=o, btg[$zdtbct$zd]
=O.
(3.17)
This set of equations have the non-trivial solution gc = -2, b = d = 0, a = anything. We note that the scale of Jlt is always arbitrary, since it satisfies a homogeneous equation. In fact, it is remarkable that we were able to get a non-trivial solution at all, since the equation $, satisfies is an eigenfunction equation with a given eigenvalue! (g drops out of the problem). We consider these as unpleasent features of
K. Bardak& Dual models
408
the model, hopefully absent from more realistic models. In addition, ation scheme may be misleading.
our approxim-
4. The propagator In this section, we shall determine the propagator of the new model in terms of the functions r+Q and 4 of sect. 3, assuming, of course, that a non-trivial solution to the equations (3.15) exists. It is then possible, at least formally, to determine the vertex and hence the whole amplitude; however, this shall not be attempted here. Now consider a spray of spurions coming out of an internal line that joins a left cluster of lines to a right cluster. The two clusters may themselves contain spurions. Let us number the lines in the left cluster from 1 to 1, the lines in the right cluster from m t 1 toN, and the spurions in between from I + 1 to m. Fig. 6 shows these spurions being emitted in the multiperipheral configuration. As we let the spurion momenta go to zero, various subenergies have the following limits: si it + 0, Sij+Sim> (4.1) ‘j,k +%+l,k’ , 1 I aspj+O,where 1 Gill, 1+ 1
‘ir”(Ui,.JW + (jE uii)-si~m-l, j=I+E
j=m n
-Sm+l,k-1
(t.$k)-si’k-l
+
j=l+l
(:)::> n
uj,k
,
(4.2)
where the ranges of the indices are restricted as in (4.1). It is clear from eq. (4.2) that, in the final answer, the variables ui i appear only in the form of products that appear on the right hand side of that equation. We now express those products in terms of the independent set of variables Us,_ where 2 < r < N-2. j=m
n
j=l+l
u..=
I”
(1-xi,/ ‘1 (‘-‘l,i_l
xi,[ Z”l,m)
tlpxi,l Z”l,m)tl--ul,i_l xi,[ ‘1 ’
Fig. 6. Factorization
of the propagator.
K. Bardakgi, Dual models
‘f?f“j
k =
tl-Ym,k Z)(l-U1,&k (l-%,lym,k
j=l+l
’
‘)(l-Ym,k
’
409
Ul,k) ‘l,k) ’
(4.3)
where r=l Xi/znu,r> r=i
r=k-1 ymkE
r=m-1
n r=m
ul,,
Z=
n
Ulr.
r=l+l
’
The main lesson to be learned from eq. (4.3) and the discussions preceding it is the following. We can derive the subenergies into two classes: the “hard” subenergies that stay finite in the limit of vanishing spurion momenta, and the “soft” subenergies, or the spurion subenergies, that go to zero in the same limit. Similarly, in an obvious analogy, we can call the internal variables that have a hard subenergy as an exponent a hard variable, and those with a soft exponent, a soft variable. The important point is that the dependence of the hard variables on the soft ones is through the product z defined in eq. (4.3), and not through the individual factors. In view of this result, we can detach the middle internal line along with its spurion spray from the left and the right clusters. The propagator defined in this fashion depends only on the variables x E ul,/, y E u~,~ and z as defined in eq. (4.3). In fig. 7, the propagator is shown with the spurions emitted in the multiperipheral and tree configurations respectively, and the above comments become clearer in the tree configuration. We note that the solid lines in this figure are associated with the variables u~,~ and u~,~, and the wavy line with the variable w G u~+~ m, which are related to x,y and z as follows: x = QJ
(1 -z)( 1-xyz) YzUl,m)
WEUl+l,m
=(l_zy)(l-xy)’
(4.4)
Now let us list the various factors that make up the propagator: where r ranges from I to m. These fac(a) Subpropagators of the form u R-pZ-1, Lr tors multiply to the expression xR-p ml zR--p2-l yR-p2-1, where,
p2=Sl,l
= S1,J+1 = . . . = Sl,m,
R
=
c rz(as]+a,,, . n=l
Fig. 7. Spurion
emissions
from the propagator.
410
K. Bardakgi, Dual models
For the definition of the harmonic oscillator operators and the details of factorization in the dual models, we refer the reader to any standard review article [2]. (b) A factor of the form (1 -xzy)-l, connected with zero intercept, which again results from factorization. (c) A projective volume factor Vt(x, w, v), where Vis defined by eq. (2.2). (d) Finally the summation of the spurion lines have to be carried out. This summation is identical to the one carried out in sect. 3, and the result is again represented by the Lagrangian of eq. (3.12). This shows that, consistent with our earlier notation, the wavy line of fig. 7 stands for the field +, which has to satisfy eqs. (3.15). Putting everything together, we have the following result: 1
1
1
sj-j-dx
Ab2)=D(p2> +g
du dw I’-l&y,
0 0 x #-p2-1
$?-p2-1
w)
0
zR-p2 -1
(1 -XJJZ)_1 l//(w),
where A is the final propagator, D is the bare propagator
D(p2)=
]du
(1-u)-l
defined by
u~-+-~,
0
and $ is the solution to equations (3.15). The expression given above is somewhat formal as it stands. For example, it suffers from spurious infrared problems which require a more careful treatment. Also, there is a problem of twist invariance, which again can be taken care of fairly easily. We shall not dwell on these problems here, since our purpose was solely to develop the techniques needed in arriving at eq. (4.5), and hopefully thereby pave the way to a more realistic calculation in a better model. We end this section with the observation that the spectrum of the propagator we have just calculated consists of straight line trajectories. This follows from the fact that it only depends on the operator R. We believe this to be a general result applicable to any model constructed along the lines given in (l), and we have the following simple argument to support it. The spurions, since they are emitted at zero four momentum, cannot carry any angular momentum. Since any curvature in the trajectories must come from an angular momentum dependent interaction, the preceding argument seems to exclude it.
5. Conclusions and new directions In the preceding sactions, we have treated a somewhat artificial model, with a view to developing tools for the treatment of more interesting models. The main question is then, for example, in the case of the model defined in ref. [ 11, can we sum the spurion graphs by means of a compact Lagrangian similar to the one given by eq. (3.12)? The crucial difference between the model treated here and the one
411
K. BardakGi, Dual models
defined in (1) is the existence of a “fifth” momentum in the latter case, which is not set equal to zero at the end. This has the consequence that, in a realistic model, both the fields and the interaction in the effective Lagrangian will involve an infinite set of harmonic oscillator operators. This complicates the situation quite a bit, and converts a quantum mechanical problem into a field-theoretic one. Fortunately, there seems to be a compensating simplification available. Since the model has unit intercept, it is well known that an infinite set of Ward identities can be derived [5]. These identities seem to be sufficient in number to eliminate all the oscillator states in the effective Lagrangian and thereby reconvert the problem into a quantum mechanical one. Work is under progress on this exciting possibility.
Appendix Here we wish to compute the contact terms in the effective Lagrangian corresponding to four and five point interactions. Our aim is to compare the results of the subtraction procedure, developed in ref. [ 11, with the new technique developed in this paper. We remind the reader that in the case of a four point function, only two poles, zf the ,form 1/s12 + 1/sZ3, need be subtracted to arrive at the truncated amplitude B. Setting g = 1 for simplicity, have the following: g4
=
]&
{x-slZ-1
(1_x)-s23-1
_
x-w-l
_
(1 _x)-s23
-l}.
(A.11
0
As explained in I, we can now let s12 + 0, ~23 + 0, in the integrand obtain the following result: E4(0) = 0.
directly and
(A4
Similarly, for a five point function, we have five double poles to subtract, of which a representative one is, for example, ST; s$. The truncated amplitude is given by the following expression:
x
q3-1
q4-1
u-s45-l
u-s51-1
45
51
_
u
-slz-l -s45-1 15 u12 u45
-
. ..}.
(A.3)
where l-u45
l-U12 ‘23
= 1-u12u45
t
‘34
= 1-u12
U45’
U15 =
1&U12 U45.
The dots in the above formula stand for the other four poles which the reader can fill in easily, using cyclic symmetry. We can again set all the subenergies equal to zero, and after some algebra, obtain the following: (A.41
K. Bardakgi, Dual models
412
2,..,
+
.\ I’ z
,’ I’ 3
I ’ ___m(
2 \\ <’3 ‘\ , \. ,’
,/4
Fig. 8. Feynman
graphs
$I
’ ___-4 ‘\
,f’
+
‘%5
3..
-. / .. ,’
+ 2____
1 I 5
5
for four and five point
/4
amplitudes.
Let us now rederive (A.2) and (A.4), using the interaction of eq. (3.12). The relevant Feynman graphs for the four and five point functions are given in fig. 8. We note that, as remarked earlier, the interaction vertex always involves a wavy line and two dotted lines, and the propagator converts a wavy line into a dotted one and vice versa. The external legs are on the mass shell, with u = 0. For the calculation at hand, we only need the following special values of the kernel K,
K(z;O,y)= +$
K(O;v)=&
For the four point function, B, ‘v ]&X-S12-’ 0
K(O;O,y)=1.
(A.5)
we have the following:
+ ]dyy-%3-l.
(A-6)
0
The poles can now be subtracted tained.
trivially, and again the result of eq. (A.2) ob-
K. Bardakci, Dual models
The calculation B, u /
+ 1-i
/dx
is similar for the five point function, dy
&x-s23-1
XY x-s’2-l
Y_s4s_l
413
and the result is given below:
y-s45-1 + 1;Ax;yx-s15-1
+
1-i
xy
p15-l.y-s34-1
+ 1-i
The poles can easily be isolated and subtracted The result is.
0
0
(A.4).
References [l]
y-Q4-1
by expanding in powers of xy.
~5(0)=3]~dxdq+,, again confirming
xyp12-l
(A.7)
1-xy
1-xy
1-xy
y-%-l
K. Bardakgi,
Nucl. Phys. B68 (1974) 331. Proc. of Brandels Summer School; J.H. Schwarz, Phys. Reports 8 (1973) 270. [3] B.W. Lee, Nucl. Phys. B9 (1969) 649. [4] J. Goldstone, Nuovo Cimento 19 (1961) 154; J. Goldstone, A. Salam, and S. Weinberg, Phys. Rev. 127 (1963) [5] M.A. Virasoro, Phys. Rev. Dl (1970) 2933.
[ 21 S. Mandelstam,
965.
.