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Goldstone-Nambu bosons in dual and parton theories
Nuclear Physics B32 (1971) 75-92. North-Holland Publishing Company
GOLDSTONEoNAMBU
BOSONS IN DUAL AND PARTON
THEORIES
Aharon CASHER, Simon Henri NOSKOWlCZ and Leonard SUSSKIND* Tel-A viv University, Ramat- Aviv, Israel
Received 12 April 1971
Abstract: We conjecture general criteria for a dual or patton theory to have a Goldstone-Nambu boson and prove the consistency of the criteria by demonstrating an exactly solvable model. The amplitudes constructed from this model satisfy all soft-pion theorems or Adler zero conditions. A second unsolvable model based on a ferromagnetic state of the spins of the parton distribution is shown to satisfy the general criteria conjectured on the basis of the first model.
1. INTRODUCTION On the basis o f several successful applications o f PCAC, it appears likely that Nambu's theory o f a spontaneously broken chiral symmetry realized with massless bosons (pions) may be a good approximation [1]. In this paper we explore the relations between the dual [2] and parton models and spontaneous breakdown o f symmetry. In the models we discuss, no a t t e m p t is made to incorporate current algebra. In our view the current algebra SU(3) X SU(3) is purely kinematic being determined b y the properties o f the bare constituent partons. Spontaneous symmetry breakdown is a much more dynamical effect which manifests itself as a rearrangement o f the symmetry o f the many-parton wave function. In sect. 2 we use the Fubini-Furlan m e t h o d [4] to express chiral current conservation in a form which is especially suited to parton and dual models. The main result of this section is the formulation o f conditions in the dual model which we conjecture to be equivalent to the soft-pion theorems implied b y spontaneous symmetry breakdown. These soft-pion theorems can be summarized as the set o f all Adlerzero conditions [3]. Sect. 3 dealt with an exactly solvable model in which the conditions formulated in sect. 2 are satisfied. Although we demonstrate Adler zeros only in four-point functions, a completely general p r o o f exists which will be presented elsewhere. * On leave from Yeshiva University.
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A.Casher et al., Goldstone-Nambu bosons
In sect. 3 a model based on a ferromagnetic state of the patton spin wavefunction is shown to satisfy the conditions of sect. 2. In this case we can not solve the model and the existence of Adler zeros remains a conjecture. The existence of a massless boson is proved.
2. SPONTANEOUS BREAKDOWN, WEE PARTONS AND THE DUAL MODEL The parton theory of hadrons is really nothing but the representation of the states of field theory by Fock space. Every state is expressed as a superposition of many particle configurations in which the number of each type of bare parton is specified. Without entering into the discussion of the mathematical existence of such a description for interacting systems the general Fock space description has some very unintuitive aspects related to the existence of virtual partons in the vacuum. The method of infinite momentum frames has been employed in a variety of situations [7]. It is assumed that when a system is boosted to a frame in which its momentum is very large, all of its constituent partons have momenta much greater than any vacuum parton. In this way it was found that the system partons (those which carry a finite fraction of the momentum of the whole system) decouple from the vacuum partons in perturbation theory. Feynman, however, has given reasons why this is probably not quite correct for real hadrons. Several features of hadronic processes led Feynman to conjecture that the number of partons with longitudinal momentum p in a fast moving hadron goes like dN -- dp/p, for p large but small in comparison with the total hadron momentum. This expression is expected to hold down to some finite values o f p where the parton begin to overlap the vacuum-parton momenta. Furthermore, since the expression is boost-invariant the distribution of the slowest partons is independent of the total momentum of the hadron. Therefore, by boosting the hadron, it is not possible to decouple it from the vacuum partons. The partons in the fuzzy region which begin to overlap the vacuum partons, Feynman calls "WEE". Because of this unavoidable dynamical contact between system and vacuum partons, the system partons are not dynamically isolated. In fact, it is precisely this contact which allows the system to feel the spontaneous breakdown of symmetry. Without it, it would be difficult to understand how the system of fast moving partons remembers the direction of the symmetry breaking of the vacuum state. Consider the chiral charge in the quark model: Q5 = f ~
3'075 ra if" d3x.
It is easy to show that the contribution of the fast moving system-partons is simply To~ O"z system-partons
A.Casher et al., Goldstone-Nambu bosons
77
where "ca is the isospin or SU(3) matrix of the parton and o z its spin along the direction of motion of the hadron. Since the vacuum and WEE partons must not be symmetric under SU(2) × SU(2), we must assume the existence of a second term QSa(WEE) in Qs, whose exact nature is unimportant, except that it is formed from the low momentum components of ~I, and # t . (Low momentum means an infinitesimal fraction of the momentum of the hadron which carries an infinite momentum.) Chiral conservation expressed in terms of ZOzr ~ and QS(WEE) states that losses of chiral current from the system can only occur by a transfer of current to the WEE and vacuum degrees of freedom. We shall need to know a bit more about Q5(WEE) and its time derivative. For this purpose we transform the equations of PCAC to the infinite momentum frame using the method of Fubini and Furlan. To start off, it is helpfulto break the chiral symmetry limit and give the pion a small mass. PCAC is then expressed by the formal equality (1)
aAu - m2rb ax u
Integrating over space gives:
Os=~fAod3x=m2f,I,d'x.
(2)
Taking matrix elements between states of equal three-momentum reduces eq. (2) to:
(3)
i ( A I Q I B ) ( l e A - leB) = m2
In eq. (3) leA and leB are the energies of the states A and B. Now, using non-invariant state normalization and separating the pion pole in (AIqSIB) gives m2 FAB i (AI QsIB) (leA - WB)=
1
(WA - WB)2- m2 2W~/W-~AWB+ m 2 C '
(4)
where C represents continuum contributions to (qb) and is assumed to behave smoothly as m 2 -~ 0. Allowing the momentum of A and B to go to infinity and m 2 to zero simplifies the expression considerably. The difference WA - leB goes to ( m 2 - rn2)/2p and x/-WA WB tends to p. The result is i (AI Q5 [B) (m 2 - m 2) = FAB,
(5)
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A.Casher et al., Goldstone-Nambu bosons
or
i[Q5, m 2] = _ r .
(6)
In the infinite momentum frame all internal and relative motions are time dilated by a factor of the total momentum over the mass of the system. It is convenient in such a frame to rescale the time so that motions take place at a finite rate. We define [7] r=t/p. Then for a system with purely longitudinal momentum, the important part of the Hamiltonian which generates the internal and relative motions is the (mass) 2. We may write (see ref. [7] ): i a/ar = M 2 ,
which, combined with eq. (6) gives
aQS/ar
=
-
r.
(7)
The matrix P is the on-shell matrix element of the pion source. Accordingly, eq. (7) states that the loss of chiral current into the WEE degrees of freedom is equivalent to the emission of a soft pion. Let us next consider the relation of these ideas to the dual theory. It is well known that the dual theory is equivalent to the continuum mechanics of a twodimensional "world sheet" embedded in four-space [2, 5]. Nielsen has speculated [5, 6] that the world sheet is really an approximation to an "average" Feynman diagram with very many vertices and propagators. Subsequently, Kramer, Nielsen and Susskind [5] discovered very intimate relations between the Feynman parton distribution and the dual theory. Kramer, Nielsen and Susskind speculate that the most important diagrams in hadron dynamics can be drawn on an infmite strip
-oo
0<0<~r,
with a locally isotropic distribution of vertices of density p = cosec 2 0. From this it was proved that the average momentum per patton line at 0 is Phadron sin 0. Thus, the WEE partons with very low momentum are found near the boundary, 0 = 0, lr. Now, for a given complicated Feynman diagram, we can define a two- dimensional flow of electric charge. This flow is a two-vector with components J o , ' I x def'med as follows: At a point 0, X consider a line element d~ orthogonal to the unit vector n. The component of J in the n direction is obtained by counting the total
A.Casher et aL, Goldstone-Nambu bosons
79
charge crossing dlZ in the form of charged propagators. Each line of charge e crossing dl/contributes a term e. The component of J is, then, given by the charge crossing d£ divided by d£. We also suppose that by counting Q5 (helicities times isospin) a current o f chiral charge ./5, ,/5 can be defined. The current defined in this manner refers to a particular Feynman graph. By the time we sum over graphs, each with a different J, we will have defined a kind of path sum over current distributions and J will presumably become an operator. The conservation of axial and vector currents requires:
OoJo +
= o,
oJSo
= o.
The vector currents are conserved in the ordinary manner so that the vacuum charge is identically zero. Thus, we shall suppose that the system cannot lose vector charges, which means that at 0 = 0 and rr, the flow must be parallel to the boundaries:
Jo=O
at
O=O, rr.
We shall have no more to say about the vector current. From now on we will discuss only j 5 and the superscript 5 will be dropped. The loss o f axial charge into the WEE parton edges is just dQ(system)
dX
Jo(O ) - Jo(,O =- Sol °
Finally, we note that in the dual theory, l~ is conjugate to m 2 so that it may be identified with r. The result is that Jo[0 may be identified as the on-shell soft pion matrix P. This circumstance leads us tO conjecture conditions in the dual theory which will be equivalent to a spontaneously-broken symmetry realized with a GoldstoneNambu boson. The conditions are not specific to the case of chiral asymmetry and the models which support the conjecture have simpler symmetries.
Conjecture If (i) There exists a flow of current Jo, x on the world sheet and OoJo + a},Jx = O, (ii) There exists a particle whose vertex for emission in the soft limit is the operator J010 . Then (i) The mass o f that particle is zero. (ii) All soft-pion theorems or Adler zeros are satisfied. The next section gives an example which supports the conjecture.
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A.Casher et aL, GoMstone.Nambu bosons
3. SOLVABLE MODEL We consider the harmonic rubber-string model [2], in which four space-time coordinates xu(O , r) satisfy
a2x a2x
= 0,
(8)
00 = O. aX 0 =O,rr
(9)
ar 2
002
The 0, r space is the usual strip 0 ~< 0 ~< 7r, - oo ~ r ~ oo. For the purposes of symmetry and duality it is often better to use r = tg~ so that eq. (8) becomes elliptic
[21. By virtue of eqs. (8) and (9), x satisfies a normal mode decomposition
xu(O, r) = xC~'m'(r)+ ix~2 ~ a+(r' ~) --a-(r, ~) cos ~0
(10)
a±(r, £) = a-+(O, £) e ±i~ ,
(11)
xcan. = xcan(0) + p C a n . r .
(12)
V~
where
The quantities
ax/ar=p~,
0x/00 =Po,
(13)
form a two-dimensional momentum flow vector on the world sheet and the equation of motion (8) is the local conservation of flow. The boundary condition, eq. (9), saysP 0 = 0 at the boundaries, expressing the further condition that no flow of momentum crosses the boundary. Our model for the flow of axial current in this section is borrowed from the theory of momentum flow. In hydrodynamic terms it is potential flow governed by a flow potential xI,(0, r). The flux of Q will be Jo = - a ~ / O 0 ,
Jr = 0~/0r.
(14)
Local conservation (condition (i) of the conjecture of sect. 2) requires xI, to obey: a2~ ar 2
02xI• --
ao 2
-0.
(15)
A. Casher et aL, Goldstone-Nambu bosons
81
The total charge Q(r) at an "instant" r is:
t2=f g~do =f g~ dO .
(16)
According to the Gauss theorem:
-~ =Jo(O, "r)- Jo(., "0.
(17)
Although the ~ field is similar to x, the four-position, the boundary conditions must be different or else no current will flow into the boundaries. Instead, we must define the boundary conditions so that condition (ii) o f our conjecture is satisfied. This requires a particle whose soft coupling is Jo I0- For cases in which this particle carries non-zero momentum its emission vertex must include the momentum shift e g~x (see ref. [2] ). The complete vertex for emission must be
7r(k) = Jo(O) : eU'x(O) : -,lo(Tr) : e/gX('O.
(18)
For the position field x the dual theory uses the Neuman boundary condition. We have found that by using the Dirichlet condition • = 0 at the boundaries, eq. (18) is satisfied for one of the particles in the spectrum. It will then be easily seen that the mass of the particle coupled to Jo is zero and somewhat less easily seen that all the Adler zeros are present. In the present paper we restrict our study of Adler zeros to four-point functions. The boundary condition and equation of motion for * imply a normal mode expansion tm
=i ~ b+(~, r ) - b-(£, r) ~=1 ~ sin£O
(19)
and the contribution of ~o to the Hamiltonian or (mass) 2 of the hadron is Zb+(£)b-(£)£. As in ref. [6], we shall assume that the trajectories which are unexcited in the modes are the vacuum-like system of pomeron and its family. As long as we con, sider all external particles to be members if this family, the xI, mode plays no role whatever and the theory is identical to the old rubber string model which required a trajectory height a(0) = 1 for the leading trajectory. As before, the ground state o f this trajectory is a tachyon of m 2 = - 1 . This makes the theory only of formal significance. The first excited states o f the pomeron at m 2 = 0 is completely spurious as it is easy to show that the pomeron family is purely positive signature and the "odd
a2
A.Casher et al.. Go&tone-Nambu bosom
daughters” all decouple. In any case this m 2 = 0 particle could not be the Goldstone boson we are looking for, since it does not involve \k in its coupling. The correct state turns out to be: In) = eikxCm’b+(1) 10) .
(20)
The factor ek@n* is needed in order to give the state a momentum k. The internal excitation is contained in b+( 1). To find the coupling scheme for rr when it couples to the edge, 0 = 0, the method of conformal mappings must be used [2] . The idea is that eq. (20) is the prescription for a pion which enters the world sheet at one end of the infinite strip. By mapping the point at = to a point on the 0 = 0 edge, the prescription (20) may be translated into a coupling matrix r. Carrying out this plan requires replacing r by A in order to utilize the conformal invariance of the elliptic equation 2 a+?-&.
ax2
2
(21)
w
The factor eihcJn* was studied in ref. [2] and it merely becomes the momentum shift: eibc(0): acting at the 0 = 0 boundary. The more interesting factor b+(l) is mapped by use of the identity
b+(l) IO)= hly
+m
*(A, in) e+h 10) .
To see this, note first of all that the b operators in \k do not contribute on 10). The contribution from the Qth normal mode is therefore lim b’(Q) ecu sin Q0 e+h 10) , A-+-+-
(22)
when acting
(23)
which goes to zero if Q# 1. If Q = 1, we get b’( 1) sin $rlO) = b+( 1)lO) = la). Now, when the point h = + m is mapped to a point 0 = 0 and finite X, say h = 0, the limit h -+ + 00 maps into a limit f3+ 0. The factor e+h behaves as 8-l under the mapping and so the vertex becomes
lh
:
i,?kX(O) :
Tp ,
(24)
O-t0 which because \k(O) = 0, is : .&X(O) : 2
= : $X(O) :
Jo(O)
.
(25)
A.Casher et aL, Goldstone-Nambu bosons
83
A similar calculation for 0 = ,r shows that the full vertex satisfies eq. (18) so that conditions (i) and (ii) are satisfied. That 17r)is massless follows immediately from the fact that 10) has (mass) 2 --- ( - 1) and b+(1) adds one unit of excitation. This could also be shown in another way. Suppose some field couples to the xI, flow, and imagine the field to be absorbed by a line in the planar diagram at point 0. The matrix element for this is (e~X(O)~(O)) .
(26)
Now, e/kx(°) can be written in terms of the normal ordered form as eikx(o) = exp { - k 2 I ~
_ (c°s~0)2 ]1 : e/kx(o) : .
(27)
The sum E (cos £0)2/£ diverges, but a consistent cut-off procedure based on the finite lattice spacing of the diagram was constructed in ref. [6]. The cut-off procedure replaces the first factor in eq. (27) by (sin 0) k2. Hence the matrix element becomes sinOk2(~(O) : e/kx(O) : ).
(28)
The actual form factor should be summed over all points, 0, in the diagram with a weight factor (sin 0 ) - 2 representing the density of propagators at a point 0. Thus we consider the integral
(fsin 0k2- 2 4 ( 0 )
: e #¢x(O) : dO).
(29)
This integral will have poles in k 2 coming from the region 0 = 0 or 0 = rr where : e ~x(°) : is a regular function of O. For example near 0 = O, the integral looks like fsinOk2_l
a~
ao-:
e ikx(O) dO) :
•
(30)
There is a pole in k 2 at k 2 = 0, proving the existence of the massless particle. The pole residue is obviously a~ e~X(O) a0:0=0 : : '
(31)
which yields another proof that the mass shell coupling of ~ is the operator Jo 15=0. The Adler zeros do not follow so easily. The froofwe have found for the general case is long and all of the main features appear in the four-point case. Consider first the scattering of a pion by a ground-state scalar. There are three
84
A.Casher et al., Goldstone-Nambu bosons a)
S
b)
kl
k2
c)
+
Fig. 1. Kinematics for the four-point function for the scattering of a pion by a ground-state scalar. terms; the st, tu, and su contributions. They are schematically represented in fig. 1. For example, the term in fig. 1a is given by an integral over the distance between the two points of contact of the pions with the strip. The vertex for the emission of a pion from point (0, 0) is :exp [iklx(O, 0)] : a ~ ( 0 , o)/ao. Similarly, emission from the point (0, X) is described by :exp [ik2x(0, X)] : a ~ ( o , x)/ax. Fig. la consists of the ground-state expectation value of the product of the two vertices, integrated from X = 0 to X = oo. q-e~
(Ol f
e/klx(O,O)a@(O, O) e~2x(O,x) a~(O, X) dX 10). ao ao
(32)
Before integration on ~,, the expression completely factors in the x-degrees of freedom and the b-degree of freedom. For the x-degree of freedom, we use the standard methods of ref. [2] to get a factor eh(S+ 1) ( 1 - e - h) 2klk2 .
(33)
The remaining part of the calculation is the computation of
a~(o, o) a~(o, x) ao
ao
IO).
(34)
A. Casher et aL, Goldstone-Narnbu bosons
85
Using eq. (19), we have ~xI, aT = i ~
b÷(~) e -~x - b-(~) e +~x x/~ £ cos£0 ,
(35)
and at 0 = 0 a_~ OU
=i~
[b*(£) e - ~ x - b-(£) e +~x] x/~-.
0=0
Then the factor (34) becomes: ~ ] ~e_~X = _ d
~]
dX ~
e_~X = e_X[ 1 _ e_X]_ 2
(36)
Therefore, defining e - x = W, the entire expression for fig. la is:
f w-,-l(1-w)2kl~2-2dW=fw-s-l(1-W)-,-2dW.
(37)
Similarly, fig. lb is given by
f w-.-l(1-uoEkleE-2dW=fw-.-I(1-W)-t-2OW.
(38)
Fig. lc consists of two terms given by
-
0 f (01 e/klx(0'0) ~(0,a0 0) egC2x0r,X)OxI,(rr,00X) 10) dX, _oo
0
f
(01 e/k2x(n'O) ~ ( T r , O) e~lX(O,x ) axP(O, ~) DO 80 [0) d;k,
_oo
which is computed to be fw-s-
1(1 _ W ) - u - 1 dW.
(39)
The Adler point is s = u = - 1, and t = 0. The constraint on s, t, and u is s+t+u=-2. In the vicinity o f the Adler point the s, t term behaves like - (s + t + 1)It + F , where F is finite and goes to zero along the line s + t + 1 = 0. The u, t term is like - ( u + t + 1)It, and the s, u term is 1. Using the constraint on s, t, u, we get a total of zero at the Adler point.
86
A.Casher et al., GoMstone-Nambu
bosom
For the four-pion amplitude there is an s, t; a U, t; and an s, u term as before. This time crossing symmetry requires that they be identical forms. The s, t term is
sw-s-2(1- W)-t-2[W2+
(1 -
W)2+ W2(1- @)I
dW.
(40)
Near the Adler point this is
Combining
all three terms gives +y+y
+
(y+y)
+($?+y).
(41)
In this case s + t + u = 0, so that eq. (41) vanishes. The computations we have performed for the general N-point function are straightforward generalizations of the above. To see what all this has to do with spontaneous symmetry breaking, we consider the action of the generator
Q=j$X. Since
IQ, ‘WV = i ,
(42)
the group generated by Q acts thus: eiaQ 9(e) e-iaQ = q(e)
The symmetry Lagrangian
is the translation
_
a
.
of \k into ?Ir -II which leaves invariant the
(44) and the equation of motion. It is only the boundary condition \k = 0 which breaks the symmetry. This model can be thought of as a dual realization of theories such as the gradient coupled pseudoscalar theory where the axial current generates a simple translation of the pion field. The spontaneous breakdown of the vacuum symmetry is
87
A.Casher et al., Goldstone-Nambu bosons
transmitted through the WEE parton sea and manifests itself as a freezing of the boundary value of xI,. In sect. 4 we study a case in which the broken symmetry is a non-abelian group. For simplicity we use SU(2), although a realistic model would use the chiral SU(2) X SU(2).
4. FERROMAGNETIC HADRON MODEL The importance of the results of sect. 3 is that they support the conjecture that conditions (i) and (ii) are equivalent to the usual soft pion theorems. They do not show us when and why to expect this kind of behavior. The fact that the chiral charge Q5 of a system of fast moving partons is composed of Z Ozr a suggests that we study the dynamics of the discrete spin-like variables of the partons. If the spin-isospin couplings are near-neighbor in 0, then the hadronic string is actually a kind of spin lattice and we suggest that the origin of spontaneous symmetry breaking is related to a phenomenon similar to ferromagnetism in the onedimensional spin lattice composing a particle. Surprisingly, we have been able to prove that the specific form of coupling required by the parton model invalidates the theorem which says that ferromagnetism can not occur in a one-dimensional lattice. To construct a simple intuitive model we have replaced the set of discrete quantum numbers of a parton by a Pauli spin o. The interaction between spins is the SU(2) symmetric Heisenberg Hamiltonian Ho = ~
i
-
Gioi'oi+
(45)
1 •
Each spin i is located at a point 0 i in such a way that the density of points is proportional to (sin 0 ) - 1. The couplings G i can be obtained from considerations of the infinite momentum limit. Consider the energy of a two-particle system moving with very large momentum P1,2 along the z-axis. The energy of such a system is always M2 E = ~/P2,2 + M 2 = e l , 2 + - + ... 2P1, 2
(46)
where M is the mass of the state. On the other hand, we can write E as a sum of two kinetic energies plus coupling energy. K 2 + m2
K21 + m 2
E=x/~21+~-l+x/~+m2+
V12:P' +P2+ 2P~
÷
2P----~+ V12
(47)
where Pi is the z-component of momentum of the ith particle and is assumed to be
88
A. Casher et al., Goldstone-Nambu bosons
a finite fraction 7/i of P12. Hence we have
2P1--~=P12 \
~1
+ 2~-2--2 ] + V12"
(48)
Eq. (48) was obtained by combining eqs. (46) and (47) and momentum conservation which states P12 = P1 + P2" Thus V12 must behave like V12 V12 = PI2 '
(49)
where V is independent of the longitudinal momentum. Now, if we consider a many-body system we have
_
2Ptotal
+
i
2Pi
i/ Pi + Pi "
(50)
If we define Pi/Ptotal = r/i, then M2 = ~
K/2 + m 2 - - n~+
~
2
~q ,,v.,7i7D
(51)
The origin of the factor (Pi + P])- 1 in Vii is time dilation. The faster the partons move, the slower their internal motions so that the Hamiltonian governing these motions must grow small with increasing momentum. Now, Kramer, Nielsen and Susskind have shown that the fraction of momentum carried by partons at point 0 is roughly sin 0. If we combine this with a near neighbor coupling then Vii will be V(sin 0)-1 and will presumably have a contribution Hspin = - ~
oioi+ 1 G/sinO i .
(52)
We will consider a continuum approximation in which o i is replaced by the average spin per parton. Then, re-writing Hspin as Hspin = G ~ i
( O i - Oi- 1)2 sin 0 i
~}o 0o ° i - ° i - 1 = ~-~ ( O i - Oi- 1) ~ ~-~ sin 0 , we obtain H in the continuum limit. From the fact that the density is sin 0 - 1 the
A.Casher et al., Goldstone-Nambu bosons
89
sum ~i is replaced by fdO/sin O:
,%m = a
f(°°l ~
dO.
(53)
o
From the commutation relation
[o~, 4 ] = 6iiea¢q o7,
(54)
[o~(O), oo(o')l = sinO~(O - o') e,,,~ o ( o ) ,
(55)
we get
combining eqs. (55) and (53) we find
6(0) = G sine ~o o(O) x
aa(O) ~0
(56)
Eq. (56) has the form of a local conservation law. Since the total spin is
o(0), we define o(0)/sin 0 to be Jr and - o X Oo/aO to be Jo. Then eq. (56) is just ~--J ~,/- 7" +~0 Jo = 0
(57)
Thus condition (i) is satisfied. Eq. (56) indicates that o(0) and o(zr) are constant and therefore must be chosen as initial conditions which will be preserved for all r. This represents a spontaneous breakdown since the boundary conditions will choose a particular direction for o(0) and o(zr) (recall 02 = 1). As in the previous example, the spontaneous asymmetry is in the form of a boundary condition. We will take the boundary condition to be frozen in the three-direction. Define ~ to be the components of o in the 1, 2 direction. Then,
Jo,1 = a~2/ao ,
Jo,2 = a/h/ao
•
(58)
The g field plays the same role as the • field in the previous model. By definition, ~ goes to zero at the boundary, and so we will linearize the equations of
A. Casher et aL, Goldstone-Nambu bosons
90
motion in this region. For a normal mode of frequency 6o 02~2 icon1 = G sin 0 - a02 ' (59) ~)2~1 ico~ 2 = -- G sin 0 - a02 These equations show that ~ goes to zero as sin 0 near the boundary and that Jo is finite near the boundary. Thus, the boundary acts as a source of the two components of o orthogonal to the boundary value. To see the existence of the two massless particles coupled to Jo,1 and Jo,2, we imagine a three-component field rra coupled to the spins o. The argument follows the same line as the argument of eq. (26)-(31). Assume that the vertex for the absorption of this field by a parton located at (0, r) is
o~(0) e ikx(°) .
(60)
Following ref. [6], we obtain the full vertex by integrating eq. (60) with conformally invariant measure d0/sin 20. The actual on-shell hadron couplings will be the residues of the poles in k 2. To obtain these, we express [6] e ikx(°) as e - k 2 ~ 1/~cos2Qo : eikx(O) : ,
(61)
and use the conformally invariant cut-off procedure of ref. [6] to approximate the first factor of eq. ( 6 1 ) b y (sin 0) k2. The matrix elements of the normal ordered exponential are regular near the boundary. Hence, the vertex function is (62)
f s . ~ 2 0 (sinO)k2°a(O) : e~X(O) : .
The poles in k arise from the region of integration near the boundary. For a = 1, 2 o~(0) behaves as +- sin 0 a ~ J a 0 and a~,,/ao remains t'mite. The contribution to the pole can be expressed as
dO(sinO)k2-1 O0 : e~X(°): -0
f /1"--4
dO(sinO)k2-1 b-O-: e~"x(°): "
A.Casher et aL, GoMstone-Nambu bosons
91
The first pole occurs at k 2 = 0 and its residue is
ao
0=0 : egCX(°) : - ao
0 =rr : egCX0r) :
= ec~#[J~(O = 0) : e/kx(O) : -Jfl(O = It) : e ikx0r) :] .
(63)
The pole at k 2 = 0 and eq. (63) show that condition (ii) is satisfied. For a = 3, o(O) goes to a constant and the first pole in k 2 will be the pomeron pole k 2 = 1. Despite the initial rotational symmetry in the ( 1 , 2 , 3)-space, the particle coupled to the third component o f o is not degenerate with the other two. This is a manifestation of the spontaneous symmetry breakdown of the ferro-magnetic state. The situation is very similar to that o f the o-model in which a four-dimensional chiral multiplet consisting o f a n-field and a o-field interact in a rotationally invariant manner, but the spontaneous breakdown of symmetry splits one particle, the o, from the three pions. This model does not exhibit full duality between the particles described as external quanta and those described as hadronic strings, because the equation of motion is not completely conformally invariant. We believe a more realistic model can be based on a study of the flow o f spin and isospin in a conformally invariant class of dense planar Feynman diagrams [5, 6]. We hope that if the couplings o f these diagrams are SU(2) × SU(2) invariant, then a realistic spontaneous breakdown of chiral symmetry will occur. One final comment about the possibility of ferromagnetism in a one-dimensional system is in order. For a homogeneous system it is well known that no spontaneous magnetization occurs in one-dimensional systems. This theorem does not apply to our system because o f the divergence o f the couplings G i near the boundaries. Indeed we have solved the Ising model with G i = sin 0 - 1 and have found spontaneous magnetization for all temperatures. The reason is not too hard to see. In the homogeneous case just the smallest amount of energy density is enough to disorder the ground state. For example, in the Ising case, we can have very long chains of spins, all pointing up, followed by very long chains with spin down and so on. These states cost very little average energy density and the number o f them is very large. Hence entropy beats energy and the non-magnetic properties continue down to zero temperature. In our case the couplings diverge near the boundary and most of the disordered states cost too much energy, since they have "fiipovers" very near 0 = 0 and ~r. As a result, energy beats entropy and the magnetic state persists at all temperatures. We would like to thank R.Brout for explaining the method of section 2 to us and S.Nussinov for a discussion which led to the model of section 3. The idea that a hadron may be a spin lattice arose during a discussion between one of us (L.S.) and Y.Aharonov, and in a private communication from H.B.Nielsen. It also appears in another form in unpublished notes o f Nambu.
92
A.Casher et al., GoMstone-Nambu bosons
N o t e added: A dual model for pion N-point functions with Adler zeros was recently given by R.Brower [9].
REFERENCES [1] Y.Nambu and D.Lurie, Phys. Rev. 125 (1962) 1429; Y.Nambu and G.Jona-Lasinio, Phys. Rev. 122 (1961) 345. [2] L.Susskind, Nuovo Cimento 69A (1970) 457; Y.Nambu, Proc. int. conf. on symmetries and qiark models, Wayne State Univ. (1969); S.Fubini, D.Gordon, G.Veneziano, Phys. Letters 29B (1969) 679. [3] S.L.Adler, Phys. Rev. 139B (1965) 1638. [4] S.Fubini and G.Furlan, Physics 1 (1965) 229; S.Weinberg, Phys. Rev. 177 (1969) 2604. [5] H.B.Nielsen, "An almost physical interpretation of the N-point veneziano formula", Nordita Preprint (1969). [6] A.Kraemer, H.B.Nielsen and L.Susskind, A parton theory based on the dual resonance model, to be published in Nuclear Physics; S.H.Noskowicz and L.Susskind, "The distribution of partons in a hadron", Tel-AvivReprint. [7] L.Susskind, Infinite momentum frames and particle dynamics, lectures in theor, phys. (1968), Univ. of Colorado, Boulder. [81 F.Keffer, Spin waves, Handbuch der Phys, XVIII/2. [9] R.C.Brower, Phys. Letters 34B (1971) 143.
GOLDSTONEoNAMBU
BOSONS IN DUAL AND PARTON
THEORIES
Aharon CASHER, Simon Henri NOSKOWlCZ and Leonard SUSSKIND* Tel-A viv University, Ramat- Aviv, Israel
Received 12 April 1971
Abstract: We conjecture general criteria for a dual or patton theory to have a Goldstone-Nambu boson and prove the consistency of the criteria by demonstrating an exactly solvable model. The amplitudes constructed from this model satisfy all soft-pion theorems or Adler zero conditions. A second unsolvable model based on a ferromagnetic state of the spins of the parton distribution is shown to satisfy the general criteria conjectured on the basis of the first model.
1. INTRODUCTION On the basis o f several successful applications o f PCAC, it appears likely that Nambu's theory o f a spontaneously broken chiral symmetry realized with massless bosons (pions) may be a good approximation [1]. In this paper we explore the relations between the dual [2] and parton models and spontaneous breakdown o f symmetry. In the models we discuss, no a t t e m p t is made to incorporate current algebra. In our view the current algebra SU(3) X SU(3) is purely kinematic being determined b y the properties o f the bare constituent partons. Spontaneous symmetry breakdown is a much more dynamical effect which manifests itself as a rearrangement o f the symmetry o f the many-parton wave function. In sect. 2 we use the Fubini-Furlan m e t h o d [4] to express chiral current conservation in a form which is especially suited to parton and dual models. The main result of this section is the formulation o f conditions in the dual model which we conjecture to be equivalent to the soft-pion theorems implied b y spontaneous symmetry breakdown. These soft-pion theorems can be summarized as the set o f all Adlerzero conditions [3]. Sect. 3 dealt with an exactly solvable model in which the conditions formulated in sect. 2 are satisfied. Although we demonstrate Adler zeros only in four-point functions, a completely general p r o o f exists which will be presented elsewhere. * On leave from Yeshiva University.
76
A.Casher et al., Goldstone-Nambu bosons
In sect. 3 a model based on a ferromagnetic state of the patton spin wavefunction is shown to satisfy the conditions of sect. 2. In this case we can not solve the model and the existence of Adler zeros remains a conjecture. The existence of a massless boson is proved.
2. SPONTANEOUS BREAKDOWN, WEE PARTONS AND THE DUAL MODEL The parton theory of hadrons is really nothing but the representation of the states of field theory by Fock space. Every state is expressed as a superposition of many particle configurations in which the number of each type of bare parton is specified. Without entering into the discussion of the mathematical existence of such a description for interacting systems the general Fock space description has some very unintuitive aspects related to the existence of virtual partons in the vacuum. The method of infinite momentum frames has been employed in a variety of situations [7]. It is assumed that when a system is boosted to a frame in which its momentum is very large, all of its constituent partons have momenta much greater than any vacuum parton. In this way it was found that the system partons (those which carry a finite fraction of the momentum of the whole system) decouple from the vacuum partons in perturbation theory. Feynman, however, has given reasons why this is probably not quite correct for real hadrons. Several features of hadronic processes led Feynman to conjecture that the number of partons with longitudinal momentum p in a fast moving hadron goes like dN -- dp/p, for p large but small in comparison with the total hadron momentum. This expression is expected to hold down to some finite values o f p where the parton begin to overlap the vacuum-parton momenta. Furthermore, since the expression is boost-invariant the distribution of the slowest partons is independent of the total momentum of the hadron. Therefore, by boosting the hadron, it is not possible to decouple it from the vacuum partons. The partons in the fuzzy region which begin to overlap the vacuum partons, Feynman calls "WEE". Because of this unavoidable dynamical contact between system and vacuum partons, the system partons are not dynamically isolated. In fact, it is precisely this contact which allows the system to feel the spontaneous breakdown of symmetry. Without it, it would be difficult to understand how the system of fast moving partons remembers the direction of the symmetry breaking of the vacuum state. Consider the chiral charge in the quark model: Q5 = f ~
3'075 ra if" d3x.
It is easy to show that the contribution of the fast moving system-partons is simply To~ O"z system-partons
A.Casher et al., Goldstone-Nambu bosons
77
where "ca is the isospin or SU(3) matrix of the parton and o z its spin along the direction of motion of the hadron. Since the vacuum and WEE partons must not be symmetric under SU(2) × SU(2), we must assume the existence of a second term QSa(WEE) in Qs, whose exact nature is unimportant, except that it is formed from the low momentum components of ~I, and # t . (Low momentum means an infinitesimal fraction of the momentum of the hadron which carries an infinite momentum.) Chiral conservation expressed in terms of ZOzr ~ and QS(WEE) states that losses of chiral current from the system can only occur by a transfer of current to the WEE and vacuum degrees of freedom. We shall need to know a bit more about Q5(WEE) and its time derivative. For this purpose we transform the equations of PCAC to the infinite momentum frame using the method of Fubini and Furlan. To start off, it is helpfulto break the chiral symmetry limit and give the pion a small mass. PCAC is then expressed by the formal equality (1)
aAu - m2rb ax u
Integrating over space gives:
Os=~fAod3x=m2f,I,d'x.
(2)
Taking matrix elements between states of equal three-momentum reduces eq. (2) to:
(3)
i ( A I Q I B ) ( l e A - leB) = m2
In eq. (3) leA and leB are the energies of the states A and B. Now, using non-invariant state normalization and separating the pion pole in (AIqSIB) gives m2 FAB i (AI QsIB) (leA - WB)=
1
(WA - WB)2- m2 2W~/W-~AWB+ m 2 C '
(4)
where C represents continuum contributions to (qb) and is assumed to behave smoothly as m 2 -~ 0. Allowing the momentum of A and B to go to infinity and m 2 to zero simplifies the expression considerably. The difference WA - leB goes to ( m 2 - rn2)/2p and x/-WA WB tends to p. The result is i (AI Q5 [B) (m 2 - m 2) = FAB,
(5)
78
A.Casher et al., Goldstone-Nambu bosons
or
i[Q5, m 2] = _ r .
(6)
In the infinite momentum frame all internal and relative motions are time dilated by a factor of the total momentum over the mass of the system. It is convenient in such a frame to rescale the time so that motions take place at a finite rate. We define [7] r=t/p. Then for a system with purely longitudinal momentum, the important part of the Hamiltonian which generates the internal and relative motions is the (mass) 2. We may write (see ref. [7] ): i a/ar = M 2 ,
which, combined with eq. (6) gives
aQS/ar
=
-
r.
(7)
The matrix P is the on-shell matrix element of the pion source. Accordingly, eq. (7) states that the loss of chiral current into the WEE degrees of freedom is equivalent to the emission of a soft pion. Let us next consider the relation of these ideas to the dual theory. It is well known that the dual theory is equivalent to the continuum mechanics of a twodimensional "world sheet" embedded in four-space [2, 5]. Nielsen has speculated [5, 6] that the world sheet is really an approximation to an "average" Feynman diagram with very many vertices and propagators. Subsequently, Kramer, Nielsen and Susskind [5] discovered very intimate relations between the Feynman parton distribution and the dual theory. Kramer, Nielsen and Susskind speculate that the most important diagrams in hadron dynamics can be drawn on an infmite strip
-oo
0<0<~r,
with a locally isotropic distribution of vertices of density p = cosec 2 0. From this it was proved that the average momentum per patton line at 0 is Phadron sin 0. Thus, the WEE partons with very low momentum are found near the boundary, 0 = 0, lr. Now, for a given complicated Feynman diagram, we can define a two- dimensional flow of electric charge. This flow is a two-vector with components J o , ' I x def'med as follows: At a point 0, X consider a line element d~ orthogonal to the unit vector n. The component of J in the n direction is obtained by counting the total
A.Casher et aL, Goldstone-Nambu bosons
79
charge crossing dlZ in the form of charged propagators. Each line of charge e crossing dl/contributes a term e. The component of J is, then, given by the charge crossing d£ divided by d£. We also suppose that by counting Q5 (helicities times isospin) a current o f chiral charge ./5, ,/5 can be defined. The current defined in this manner refers to a particular Feynman graph. By the time we sum over graphs, each with a different J, we will have defined a kind of path sum over current distributions and J will presumably become an operator. The conservation of axial and vector currents requires:
OoJo +
= o,
oJSo
= o.
The vector currents are conserved in the ordinary manner so that the vacuum charge is identically zero. Thus, we shall suppose that the system cannot lose vector charges, which means that at 0 = 0 and rr, the flow must be parallel to the boundaries:
Jo=O
at
O=O, rr.
We shall have no more to say about the vector current. From now on we will discuss only j 5 and the superscript 5 will be dropped. The loss o f axial charge into the WEE parton edges is just dQ(system)
dX
Jo(O ) - Jo(,O =- Sol °
Finally, we note that in the dual theory, l~ is conjugate to m 2 so that it may be identified with r. The result is that Jo[0 may be identified as the on-shell soft pion matrix P. This circumstance leads us tO conjecture conditions in the dual theory which will be equivalent to a spontaneously-broken symmetry realized with a GoldstoneNambu boson. The conditions are not specific to the case of chiral asymmetry and the models which support the conjecture have simpler symmetries.
Conjecture If (i) There exists a flow of current Jo, x on the world sheet and OoJo + a},Jx = O, (ii) There exists a particle whose vertex for emission in the soft limit is the operator J010 . Then (i) The mass o f that particle is zero. (ii) All soft-pion theorems or Adler zeros are satisfied. The next section gives an example which supports the conjecture.
80
A.Casher et aL, GoMstone.Nambu bosons
3. SOLVABLE MODEL We consider the harmonic rubber-string model [2], in which four space-time coordinates xu(O , r) satisfy
a2x a2x
= 0,
(8)
00 = O. aX 0 =O,rr
(9)
ar 2
002
The 0, r space is the usual strip 0 ~< 0 ~< 7r, - oo ~ r ~ oo. For the purposes of symmetry and duality it is often better to use r = tg~ so that eq. (8) becomes elliptic
[21. By virtue of eqs. (8) and (9), x satisfies a normal mode decomposition
xu(O, r) = xC~'m'(r)+ ix~2 ~ a+(r' ~) --a-(r, ~) cos ~0
(10)
a±(r, £) = a-+(O, £) e ±i~ ,
(11)
xcan. = xcan(0) + p C a n . r .
(12)
V~
where
The quantities
ax/ar=p~,
0x/00 =Po,
(13)
form a two-dimensional momentum flow vector on the world sheet and the equation of motion (8) is the local conservation of flow. The boundary condition, eq. (9), saysP 0 = 0 at the boundaries, expressing the further condition that no flow of momentum crosses the boundary. Our model for the flow of axial current in this section is borrowed from the theory of momentum flow. In hydrodynamic terms it is potential flow governed by a flow potential xI,(0, r). The flux of Q will be Jo = - a ~ / O 0 ,
Jr = 0~/0r.
(14)
Local conservation (condition (i) of the conjecture of sect. 2) requires xI, to obey: a2~ ar 2
02xI• --
ao 2
-0.
(15)
A. Casher et aL, Goldstone-Nambu bosons
81
The total charge Q(r) at an "instant" r is:
t2=f g~do =f g~ dO .
(16)
According to the Gauss theorem:
-~ =Jo(O, "r)- Jo(., "0.
(17)
Although the ~ field is similar to x, the four-position, the boundary conditions must be different or else no current will flow into the boundaries. Instead, we must define the boundary conditions so that condition (ii) o f our conjecture is satisfied. This requires a particle whose soft coupling is Jo I0- For cases in which this particle carries non-zero momentum its emission vertex must include the momentum shift e g~x (see ref. [2] ). The complete vertex for emission must be
7r(k) = Jo(O) : eU'x(O) : -,lo(Tr) : e/gX('O.
(18)
For the position field x the dual theory uses the Neuman boundary condition. We have found that by using the Dirichlet condition • = 0 at the boundaries, eq. (18) is satisfied for one of the particles in the spectrum. It will then be easily seen that the mass of the particle coupled to Jo is zero and somewhat less easily seen that all the Adler zeros are present. In the present paper we restrict our study of Adler zeros to four-point functions. The boundary condition and equation of motion for * imply a normal mode expansion tm
=i ~ b+(~, r ) - b-(£, r) ~=1 ~ sin£O
(19)
and the contribution of ~o to the Hamiltonian or (mass) 2 of the hadron is Zb+(£)b-(£)£. As in ref. [6], we shall assume that the trajectories which are unexcited in the modes are the vacuum-like system of pomeron and its family. As long as we con, sider all external particles to be members if this family, the xI, mode plays no role whatever and the theory is identical to the old rubber string model which required a trajectory height a(0) = 1 for the leading trajectory. As before, the ground state o f this trajectory is a tachyon of m 2 = - 1 . This makes the theory only of formal significance. The first excited states o f the pomeron at m 2 = 0 is completely spurious as it is easy to show that the pomeron family is purely positive signature and the "odd
a2
A.Casher et al.. Go&tone-Nambu bosom
daughters” all decouple. In any case this m 2 = 0 particle could not be the Goldstone boson we are looking for, since it does not involve \k in its coupling. The correct state turns out to be: In) = eikxCm’b+(1) 10) .
(20)
The factor ek@n* is needed in order to give the state a momentum k. The internal excitation is contained in b+( 1). To find the coupling scheme for rr when it couples to the edge, 0 = 0, the method of conformal mappings must be used [2] . The idea is that eq. (20) is the prescription for a pion which enters the world sheet at one end of the infinite strip. By mapping the point at = to a point on the 0 = 0 edge, the prescription (20) may be translated into a coupling matrix r. Carrying out this plan requires replacing r by A in order to utilize the conformal invariance of the elliptic equation 2 a+?-&.
ax2
2
(21)
w
The factor eihcJn* was studied in ref. [2] and it merely becomes the momentum shift: eibc(0): acting at the 0 = 0 boundary. The more interesting factor b+(l) is mapped by use of the identity
b+(l) IO)= hly
+m
*(A, in) e+h 10) .
To see this, note first of all that the b operators in \k do not contribute on 10). The contribution from the Qth normal mode is therefore lim b’(Q) ecu sin Q0 e+h 10) , A-+-+-
(22)
when acting
(23)
which goes to zero if Q# 1. If Q = 1, we get b’( 1) sin $rlO) = b+( 1)lO) = la). Now, when the point h = + m is mapped to a point 0 = 0 and finite X, say h = 0, the limit h -+ + 00 maps into a limit f3+ 0. The factor e+h behaves as 8-l under the mapping and so the vertex becomes
lh
:
i,?kX(O) :
Tp ,
(24)
O-t0 which because \k(O) = 0, is : .&X(O) : 2
= : $X(O) :
Jo(O)
.
(25)
A.Casher et aL, Goldstone-Nambu bosons
83
A similar calculation for 0 = ,r shows that the full vertex satisfies eq. (18) so that conditions (i) and (ii) are satisfied. That 17r)is massless follows immediately from the fact that 10) has (mass) 2 --- ( - 1) and b+(1) adds one unit of excitation. This could also be shown in another way. Suppose some field couples to the xI, flow, and imagine the field to be absorbed by a line in the planar diagram at point 0. The matrix element for this is (e~X(O)~(O)) .
(26)
Now, e/kx(°) can be written in terms of the normal ordered form as eikx(o) = exp { - k 2 I ~
_ (c°s~0)2 ]1 : e/kx(o) : .
(27)
The sum E (cos £0)2/£ diverges, but a consistent cut-off procedure based on the finite lattice spacing of the diagram was constructed in ref. [6]. The cut-off procedure replaces the first factor in eq. (27) by (sin 0) k2. Hence the matrix element becomes sinOk2(~(O) : e/kx(O) : ).
(28)
The actual form factor should be summed over all points, 0, in the diagram with a weight factor (sin 0 ) - 2 representing the density of propagators at a point 0. Thus we consider the integral
(fsin 0k2- 2 4 ( 0 )
: e #¢x(O) : dO).
(29)
This integral will have poles in k 2 coming from the region 0 = 0 or 0 = rr where : e ~x(°) : is a regular function of O. For example near 0 = O, the integral looks like fsinOk2_l
a~
ao-:
e ikx(O) dO) :
•
(30)
There is a pole in k 2 at k 2 = 0, proving the existence of the massless particle. The pole residue is obviously a~ e~X(O) a0:0=0 : : '
(31)
which yields another proof that the mass shell coupling of ~ is the operator Jo 15=0. The Adler zeros do not follow so easily. The froofwe have found for the general case is long and all of the main features appear in the four-point case. Consider first the scattering of a pion by a ground-state scalar. There are three
84
A.Casher et al., Goldstone-Nambu bosons a)
S
b)
kl
k2
c)
+
Fig. 1. Kinematics for the four-point function for the scattering of a pion by a ground-state scalar. terms; the st, tu, and su contributions. They are schematically represented in fig. 1. For example, the term in fig. 1a is given by an integral over the distance between the two points of contact of the pions with the strip. The vertex for the emission of a pion from point (0, 0) is :exp [iklx(O, 0)] : a ~ ( 0 , o)/ao. Similarly, emission from the point (0, X) is described by :exp [ik2x(0, X)] : a ~ ( o , x)/ax. Fig. la consists of the ground-state expectation value of the product of the two vertices, integrated from X = 0 to X = oo. q-e~
(Ol f
e/klx(O,O)a@(O, O) e~2x(O,x) a~(O, X) dX 10). ao ao
(32)
Before integration on ~,, the expression completely factors in the x-degrees of freedom and the b-degree of freedom. For the x-degree of freedom, we use the standard methods of ref. [2] to get a factor eh(S+ 1) ( 1 - e - h) 2klk2 .
(33)
The remaining part of the calculation is the computation of
a~(o, o) a~(o, x) ao
ao
IO).
(34)
A. Casher et aL, Goldstone-Narnbu bosons
85
Using eq. (19), we have ~xI, aT = i ~
b÷(~) e -~x - b-(~) e +~x x/~ £ cos£0 ,
(35)
and at 0 = 0 a_~ OU
=i~
[b*(£) e - ~ x - b-(£) e +~x] x/~-.
0=0
Then the factor (34) becomes: ~ ] ~e_~X = _ d
~]
dX ~
e_~X = e_X[ 1 _ e_X]_ 2
(36)
Therefore, defining e - x = W, the entire expression for fig. la is:
f w-,-l(1-w)2kl~2-2dW=fw-s-l(1-W)-,-2dW.
(37)
Similarly, fig. lb is given by
f w-.-l(1-uoEkleE-2dW=fw-.-I(1-W)-t-2OW.
(38)
Fig. lc consists of two terms given by
-
0 f (01 e/klx(0'0) ~(0,a0 0) egC2x0r,X)OxI,(rr,00X) 10) dX, _oo
0
f
(01 e/k2x(n'O) ~ ( T r , O) e~lX(O,x ) axP(O, ~) DO 80 [0) d;k,
_oo
which is computed to be fw-s-
1(1 _ W ) - u - 1 dW.
(39)
The Adler point is s = u = - 1, and t = 0. The constraint on s, t, and u is s+t+u=-2. In the vicinity o f the Adler point the s, t term behaves like - (s + t + 1)It + F , where F is finite and goes to zero along the line s + t + 1 = 0. The u, t term is like - ( u + t + 1)It, and the s, u term is 1. Using the constraint on s, t, u, we get a total of zero at the Adler point.
86
A.Casher et al., GoMstone-Nambu
bosom
For the four-pion amplitude there is an s, t; a U, t; and an s, u term as before. This time crossing symmetry requires that they be identical forms. The s, t term is
sw-s-2(1- W)-t-2[W2+
(1 -
W)2+ W2(1- @)I
dW.
(40)
Near the Adler point this is
Combining
all three terms gives +y+y
+
(y+y)
+($?+y).
(41)
In this case s + t + u = 0, so that eq. (41) vanishes. The computations we have performed for the general N-point function are straightforward generalizations of the above. To see what all this has to do with spontaneous symmetry breaking, we consider the action of the generator
Q=j$X. Since
IQ, ‘WV = i ,
(42)
the group generated by Q acts thus: eiaQ 9(e) e-iaQ = q(e)
The symmetry Lagrangian
is the translation
_
a
.
of \k into ?Ir -II which leaves invariant the
(44) and the equation of motion. It is only the boundary condition \k = 0 which breaks the symmetry. This model can be thought of as a dual realization of theories such as the gradient coupled pseudoscalar theory where the axial current generates a simple translation of the pion field. The spontaneous breakdown of the vacuum symmetry is
87
A.Casher et al., Goldstone-Nambu bosons
transmitted through the WEE parton sea and manifests itself as a freezing of the boundary value of xI,. In sect. 4 we study a case in which the broken symmetry is a non-abelian group. For simplicity we use SU(2), although a realistic model would use the chiral SU(2) X SU(2).
4. FERROMAGNETIC HADRON MODEL The importance of the results of sect. 3 is that they support the conjecture that conditions (i) and (ii) are equivalent to the usual soft pion theorems. They do not show us when and why to expect this kind of behavior. The fact that the chiral charge Q5 of a system of fast moving partons is composed of Z Ozr a suggests that we study the dynamics of the discrete spin-like variables of the partons. If the spin-isospin couplings are near-neighbor in 0, then the hadronic string is actually a kind of spin lattice and we suggest that the origin of spontaneous symmetry breaking is related to a phenomenon similar to ferromagnetism in the onedimensional spin lattice composing a particle. Surprisingly, we have been able to prove that the specific form of coupling required by the parton model invalidates the theorem which says that ferromagnetism can not occur in a one-dimensional lattice. To construct a simple intuitive model we have replaced the set of discrete quantum numbers of a parton by a Pauli spin o. The interaction between spins is the SU(2) symmetric Heisenberg Hamiltonian Ho = ~
i
-
Gioi'oi+
(45)
1 •
Each spin i is located at a point 0 i in such a way that the density of points is proportional to (sin 0 ) - 1. The couplings G i can be obtained from considerations of the infinite momentum limit. Consider the energy of a two-particle system moving with very large momentum P1,2 along the z-axis. The energy of such a system is always M2 E = ~/P2,2 + M 2 = e l , 2 + - + ... 2P1, 2
(46)
where M is the mass of the state. On the other hand, we can write E as a sum of two kinetic energies plus coupling energy. K 2 + m2
K21 + m 2
E=x/~21+~-l+x/~+m2+
V12:P' +P2+ 2P~
÷
2P----~+ V12
(47)
where Pi is the z-component of momentum of the ith particle and is assumed to be
88
A. Casher et al., Goldstone-Nambu bosons
a finite fraction 7/i of P12. Hence we have
2P1--~=P12 \
~1
+ 2~-2--2 ] + V12"
(48)
Eq. (48) was obtained by combining eqs. (46) and (47) and momentum conservation which states P12 = P1 + P2" Thus V12 must behave like V12 V12 = PI2 '
(49)
where V is independent of the longitudinal momentum. Now, if we consider a many-body system we have
_
2Ptotal
+
i
2Pi
i/ Pi + Pi "
(50)
If we define Pi/Ptotal = r/i, then M2 = ~
K/2 + m 2 - - n~+
~
2
~q ,,v.,7i7D
(51)
The origin of the factor (Pi + P])- 1 in Vii is time dilation. The faster the partons move, the slower their internal motions so that the Hamiltonian governing these motions must grow small with increasing momentum. Now, Kramer, Nielsen and Susskind have shown that the fraction of momentum carried by partons at point 0 is roughly sin 0. If we combine this with a near neighbor coupling then Vii will be V(sin 0)-1 and will presumably have a contribution Hspin = - ~
oioi+ 1 G/sinO i .
(52)
We will consider a continuum approximation in which o i is replaced by the average spin per parton. Then, re-writing Hspin as Hspin = G ~ i
( O i - Oi- 1)2 sin 0 i
~}o 0o ° i - ° i - 1 = ~-~ ( O i - Oi- 1) ~ ~-~ sin 0 , we obtain H in the continuum limit. From the fact that the density is sin 0 - 1 the
A.Casher et al., Goldstone-Nambu bosons
89
sum ~i is replaced by fdO/sin O:
,%m = a
f(°°l ~
dO.
(53)
o
From the commutation relation
[o~, 4 ] = 6iiea¢q o7,
(54)
[o~(O), oo(o')l = sinO~(O - o') e,,,~ o ( o ) ,
(55)
we get
combining eqs. (55) and (53) we find
6(0) = G sine ~o o(O) x
aa(O) ~0
(56)
Eq. (56) has the form of a local conservation law. Since the total spin is
o(0), we define o(0)/sin 0 to be Jr and - o X Oo/aO to be Jo. Then eq. (56) is just ~--J ~,/- 7" +~0 Jo = 0
(57)
Thus condition (i) is satisfied. Eq. (56) indicates that o(0) and o(zr) are constant and therefore must be chosen as initial conditions which will be preserved for all r. This represents a spontaneous breakdown since the boundary conditions will choose a particular direction for o(0) and o(zr) (recall 02 = 1). As in the previous example, the spontaneous asymmetry is in the form of a boundary condition. We will take the boundary condition to be frozen in the three-direction. Define ~ to be the components of o in the 1, 2 direction. Then,
Jo,1 = a~2/ao ,
Jo,2 = a/h/ao
•
(58)
The g field plays the same role as the • field in the previous model. By definition, ~ goes to zero at the boundary, and so we will linearize the equations of
A. Casher et aL, Goldstone-Nambu bosons
90
motion in this region. For a normal mode of frequency 6o 02~2 icon1 = G sin 0 - a02 ' (59) ~)2~1 ico~ 2 = -- G sin 0 - a02 These equations show that ~ goes to zero as sin 0 near the boundary and that Jo is finite near the boundary. Thus, the boundary acts as a source of the two components of o orthogonal to the boundary value. To see the existence of the two massless particles coupled to Jo,1 and Jo,2, we imagine a three-component field rra coupled to the spins o. The argument follows the same line as the argument of eq. (26)-(31). Assume that the vertex for the absorption of this field by a parton located at (0, r) is
o~(0) e ikx(°) .
(60)
Following ref. [6], we obtain the full vertex by integrating eq. (60) with conformally invariant measure d0/sin 20. The actual on-shell hadron couplings will be the residues of the poles in k 2. To obtain these, we express [6] e ikx(°) as e - k 2 ~ 1/~cos2Qo : eikx(O) : ,
(61)
and use the conformally invariant cut-off procedure of ref. [6] to approximate the first factor of eq. ( 6 1 ) b y (sin 0) k2. The matrix elements of the normal ordered exponential are regular near the boundary. Hence, the vertex function is (62)
f s . ~ 2 0 (sinO)k2°a(O) : e~X(O) : .
The poles in k arise from the region of integration near the boundary. For a = 1, 2 o~(0) behaves as +- sin 0 a ~ J a 0 and a~,,/ao remains t'mite. The contribution to the pole can be expressed as
dO(sinO)k2-1 O0 : e~X(°): -0
f /1"--4
dO(sinO)k2-1 b-O-: e~"x(°): "
A.Casher et aL, GoMstone-Nambu bosons
91
The first pole occurs at k 2 = 0 and its residue is
ao
0=0 : egCX(°) : - ao
0 =rr : egCX0r) :
= ec~#[J~(O = 0) : e/kx(O) : -Jfl(O = It) : e ikx0r) :] .
(63)
The pole at k 2 = 0 and eq. (63) show that condition (ii) is satisfied. For a = 3, o(O) goes to a constant and the first pole in k 2 will be the pomeron pole k 2 = 1. Despite the initial rotational symmetry in the ( 1 , 2 , 3)-space, the particle coupled to the third component o f o is not degenerate with the other two. This is a manifestation of the spontaneous symmetry breakdown of the ferro-magnetic state. The situation is very similar to that o f the o-model in which a four-dimensional chiral multiplet consisting o f a n-field and a o-field interact in a rotationally invariant manner, but the spontaneous breakdown of symmetry splits one particle, the o, from the three pions. This model does not exhibit full duality between the particles described as external quanta and those described as hadronic strings, because the equation of motion is not completely conformally invariant. We believe a more realistic model can be based on a study of the flow o f spin and isospin in a conformally invariant class of dense planar Feynman diagrams [5, 6]. We hope that if the couplings o f these diagrams are SU(2) × SU(2) invariant, then a realistic spontaneous breakdown of chiral symmetry will occur. One final comment about the possibility of ferromagnetism in a one-dimensional system is in order. For a homogeneous system it is well known that no spontaneous magnetization occurs in one-dimensional systems. This theorem does not apply to our system because o f the divergence o f the couplings G i near the boundaries. Indeed we have solved the Ising model with G i = sin 0 - 1 and have found spontaneous magnetization for all temperatures. The reason is not too hard to see. In the homogeneous case just the smallest amount of energy density is enough to disorder the ground state. For example, in the Ising case, we can have very long chains of spins, all pointing up, followed by very long chains with spin down and so on. These states cost very little average energy density and the number o f them is very large. Hence entropy beats energy and the non-magnetic properties continue down to zero temperature. In our case the couplings diverge near the boundary and most of the disordered states cost too much energy, since they have "fiipovers" very near 0 = 0 and ~r. As a result, energy beats entropy and the magnetic state persists at all temperatures. We would like to thank R.Brout for explaining the method of section 2 to us and S.Nussinov for a discussion which led to the model of section 3. The idea that a hadron may be a spin lattice arose during a discussion between one of us (L.S.) and Y.Aharonov, and in a private communication from H.B.Nielsen. It also appears in another form in unpublished notes o f Nambu.
92
A.Casher et al., GoMstone-Nambu bosons
N o t e added: A dual model for pion N-point functions with Adler zeros was recently given by R.Brower [9].
REFERENCES [1] Y.Nambu and D.Lurie, Phys. Rev. 125 (1962) 1429; Y.Nambu and G.Jona-Lasinio, Phys. Rev. 122 (1961) 345. [2] L.Susskind, Nuovo Cimento 69A (1970) 457; Y.Nambu, Proc. int. conf. on symmetries and qiark models, Wayne State Univ. (1969); S.Fubini, D.Gordon, G.Veneziano, Phys. Letters 29B (1969) 679. [3] S.L.Adler, Phys. Rev. 139B (1965) 1638. [4] S.Fubini and G.Furlan, Physics 1 (1965) 229; S.Weinberg, Phys. Rev. 177 (1969) 2604. [5] H.B.Nielsen, "An almost physical interpretation of the N-point veneziano formula", Nordita Preprint (1969). [6] A.Kraemer, H.B.Nielsen and L.Susskind, A parton theory based on the dual resonance model, to be published in Nuclear Physics; S.H.Noskowicz and L.Susskind, "The distribution of partons in a hadron", Tel-AvivReprint. [7] L.Susskind, Infinite momentum frames and particle dynamics, lectures in theor, phys. (1968), Univ. of Colorado, Boulder. [81 F.Keffer, Spin waves, Handbuch der Phys, XVIII/2. [9] R.C.Brower, Phys. Letters 34B (1971) 143.