- Email: [email protected]
Dynamical symmetry breaking as the origin of the pomeranchuk singularity
Volume 62B, number 3
PHYSICS LETTERS
7 June 1976
DYNAMICAL SYMMETRY BREAKING AS THE ORIGIN OF THE POMERANCHUK
S I N G U L A R I T Y ¢~
H.D.I. ABARBANEL* Stan]ord Lmear Accelerator Center, Stanford University, Stanford, CA. 94305, USA
Received 20 Aprd 1976 We consider the viewpoint that Reggeons are built out of Reggelzed quarks and thatan O (2) symmetry between two kinds of quarks ("heavy" and "light") is dynamically broken by their interaction. The Goldstone boson thus generated is associated with the Pomeron, and it is argued that it naturally has a smaller slope than the other Reggeons which are ordinary bound states of the quarks. The spectrum of boson Regge singularities has the peculiar feature of containing a number of "ordinary" Reggeons (p, A 2, f,. ) with Intercepts around J = 1/2 or lower and slopes close to 1 (GeV/c) -2 and one peculiar object, the Pomeron, with intercept J = 1 and slope about 1/4 to 1/3 of the ordinary trajectories. Chew and Rosenzweig [1 ] have proposed a model which identifies the Pomeron (P) with the next leading vacuum trajectory (f) and explains its intercept and small slope by significant curvature of the P-f object near t = 0. Quigg and Rabinovici [2] have made an analysis of meson-baryon cross section data from 6 to 280 GeV/c and find no good evidence for his P-f coincidence. From the point of view of Reggeon field theory the P is a gapless singularity with Reggeon energy, E = 1 - J , vanishing at zero Reggeon momentum [3]. All other Reggeons have a nonzero energy gap A > 0, and trajectory: E(q) = a'q2 + A. Such a situation is highly suggestive of a bound state spectrum with a Goldstone boson arising from the breaking of an underlying continuous symmetry and a series o f ordinary bound states to be identified with the usual Reggeons. Several people have taken up this theme by studying the Reggeon field theory for the pomeron suggested originally by Gribov [4], and asking how the P field operator might acquire a nonzero vacuum expectation value and give rise to Goldstone excitations [5 ]. A thorough study of this possibility in ordinary Reggeon field theory reveals no reason why the Pomeron gap Ap = 1 - ap(0) should be zero except for special values of the parameters in the Reggeon lagrangian * 1,2. This suggests that one ought to look into the origin o f Regge singularities as bound states of some constituents. In the same spirit as one thinks of the ordinary hadrons as bound states o f quark degrees of freedom, we would like to think o f Reggeons as bound states of the same degrees of freedom which are now realized off the mass shell and for complex angular momentum, as is appropriate for a Reggeon. Since fremion degrees of freedom appear to Reggeize m non-Abelian gauge theories [8], such a theory may well underhe our approach. So let us consider a set of quark Reggeon fields Xi(X, r) which are two component spinor fields in rapidity, T, and impact parameter, x, space. The label i encompasses color and some set of flavors. We want to imagine that among these flavor labels there is an O(2) symmetry. For example, it could mix an SU(3) triplet o f u, d, s quarks with some triplet of U, D, S quarks. We'll call these two sets rotated into one another by the O(2), heavy and light. A free Reggeon lagrangian invariant under 0 ( 2 ) rotations U(0) = exp (it20); r2( 0 01) is
¢' Work supported by Energy Research and Development Administration. * On leave from Theoretical Physics Department, Fermilab, Batavia, Illinois 60510. ,1 See ref. [6]. The broken symmetry turns out to be time-reversal mvarmnce. ~:2 Dyatlov [7] has looked into a P-f sponteneous symmetry breaking which does not violate T invariance. It requtres an unnatural but possible coincidences among Reggeon couplings and has no triple ~Pcoupling. 308
Volume 62B, number 3
PHYSICS LETTERS
7 June 1976
+~ Z?x= ~i ~ t o~~ x + i ~20 Xto3o.OX-Ao~tX,
(I)
where ~ is the transpose in heavy-hght space. The o matrices operate in the spin space of two component spinors and g = (Ol, 02) operates in the two dimensional momentum space of the Reggeons. The flee propagator *a
S(E,q) = 1/E - / ~ 0 o 3 ~ - q
- A0 ,
(2)
describes the usual doublet of positive and negative parity fermion Reggeons with intercept c~0 = 1 - A 0. As a simple interaction, which is just renormahzable in two dimensional transverse space, we choose a scalar Reggeon q~(x, 7-) to provide ~I
k0 = _ f ~*X(0-t2+ q~2),
(3)
plus its free propagation ~=
2,1" ~ / ~ -
~V,t.
7~-~O~t~ .
(4)
Now we seek a solution to this 0 ( 2 ) symmetric Reggeon field theory which has a proper self energy for the fermien propagator N(E, q) = ZI(E,q) + r3X3(E, q) where E l and ~3 are invariant under 0 ( 2 ) rotations .4 . A term like ~3 breaks the heavy-light symmetry and gives rise to a Goldstone boson. To see this consider the proper vertex function of two quarks and the conserved 0 ( 2 ) current ( p , ~ ) = ( ~ t r2x , x t o3ar2x) • The vertex function satisfies the Ward identity
Erp(E, q; El, ~ql) - q F(E, q; E 1 , q~l) = r 2 S - 1 (El, ql)
_ S -1 (E 1 _ E, q~l- q~)~2 •
(5)
As E, q ~ 0 , this means lim EFo - q r = - 2 i r l Z 3 ( E l , q l ) , E,~0
(6)
and that the proper vertex may be written as
(re, g) = (rReg, F Reg) + _ _(1,aq) ~ (-irlF1) E - aq2
(7)
'
where F l (E, q; E l, q 1)IE q=0 = 2 ~3(E1, q l ) ,
(8)
and 1-'Reg has no pole at E, q =0. So if a symmetry breaking solution exists, then a Goldstone boson appears in the ~t 7-1X channel. It has Regge trajectory t~(t) = 1 + at, and we identify it to be the Pomeron. To find out whether such a solution is possible we look at the proper self energy ~3 to lowest order in k 0. In D transverse dimensions this is _ f d E 1 dDkl dE2dDk 2 ~3(E1, kl) [(El -/30 ?3 ~O"~kl- AO- ~I )2 - ~2 ]_-I ~3(E, q) - 2(4~r2)D d
[ E 2 - °~0~k22- 60+ie] [ E - E l - E 2 - u0(q~ _ ~kl- ~k2)2 -60+ie]
,
(9)
, a This is the two component version of the fermion propagator which is given by Gribov et al. [9]. ,4 In quite a different context a dynamically broken 0 ( 2 ) symmetry has been studied by Cornwall and Norton [10].
309
Volume 62B, number 3
PHYSICS LETTERS
7 June 1976
We seek a solution o f the form E 3 (E, qq) = A (E+ b0q2 )-r~ for large E and q 2 with r7 > 0 [ 11 ]. Similarly for Y~I"This only works for D = 2, where the coupling k~/c~ is dimensionless, and requires to lowest order in ;k0 t
77= X0/4rrc~0
and
P
S
b 0 = c~0 47rc~0/~0 .
(10)
At least m this crude approximation there is a symmetry breaking solution for every )to > 0 at the physical dimensions D = 2. The Goldstone boson associated with this symmetry breaking couples only to the off diagonal channel ~ t rl × = X~HXL+ ×fL×H where H and L label eigenvalues of the proper self energy. All other Reggeons couple to general mixtures of light and heavy quarks. The spectrum of ordinary Reggeons near t = 0 (q2 = 0) should be strongly determined near E ~ 112 (J ~ 1/2) by both heavy and light quarks so that the hadrons which he on those trajectories will couple strongly to the P trajectory there. Away from t = 0 and J ~ 1/2 perhaps only light or only heavy quarks will be important. As we expect the mass m H at which the heavy trajectory c~tt reaches J = 1/2 to be much larger than m E P where otL = 1/2, the slopes of these trajectories satisfy a]f < aL, since their intercepts ought to be nearby. Since the P is determined as a mixture of heavy arid light only, its slope will be determined by the usual combination t
t
t
~
C~p= a H a L / a n + a L ,
(11)
while other Reggeons wdl have their slope determined by the hght quarks primarily, so a ~ = a [ / 2 . Thus
and should be small. If we take ct~,/a~ ~ 1/3 and ask that m E ~ 300 MeV and m n ~ 1.5 GeV, then A L ~ 0.7 and A H ~ 0.9 making our picture of nearby singularities in the E plane for q2 ~ 0 consistent. From this picture also emerges a constant triple P vertex and, yet, through a field theory which satisfies Reggeon unitanty a special (broken) symmetry allows a simple pole [ E - c~,q 2 ] -1 to emerge for the Pomeron. The details of this and several technical issues connected with the building of Reggeons from quarks and symmetry breaking will appear m an expanded version of this note.
References [1] [21 [3] [4] I5] [6] [7] [8] [9] [10] [11]
310
G.F. Chew and C. Rosenzweig, Phys. Lett. B 58 (1975) 93. C. Quigg and E. Rabinovlci, Fermilab preprint 75/81-THY. H.D.I. Abarbanel et al., Physics Reports C21 (1975) 119. V.N. Gribov, Soviet Physics-JETP 26 (1968) 414. H.D.I. Abarbanel, Phys. Lett. B49 (1974) 61. H.D.I. Abarbanel, J.B. Bronzan, A. Schwimmer, and R.L. Sugar, Rutgers Preprmt (January 1976). I.T. Dyatlov, Zh. Eksp. Teo. Fiz. 69 (1975) 1127. M.T. Gnsaru, H.J. Schmtzer and H.-S. Tsao, Phys. Rev. D8 (1973) 4498. V.N. Gnbov, E.M. Levm, and A.A. Mlgdal, Soy. J. Nucl. Phys. 12 (1971) 93. J.M. Cornwall and R.E. Norton, Phys. D8 (1973) 3338. M. Baker and K. Johnson, Phys. Rev. D3 (1971) 2516.
PHYSICS LETTERS
7 June 1976
DYNAMICAL SYMMETRY BREAKING AS THE ORIGIN OF THE POMERANCHUK
S I N G U L A R I T Y ¢~
H.D.I. ABARBANEL* Stan]ord Lmear Accelerator Center, Stanford University, Stanford, CA. 94305, USA
Received 20 Aprd 1976 We consider the viewpoint that Reggeons are built out of Reggelzed quarks and thatan O (2) symmetry between two kinds of quarks ("heavy" and "light") is dynamically broken by their interaction. The Goldstone boson thus generated is associated with the Pomeron, and it is argued that it naturally has a smaller slope than the other Reggeons which are ordinary bound states of the quarks. The spectrum of boson Regge singularities has the peculiar feature of containing a number of "ordinary" Reggeons (p, A 2, f,. ) with Intercepts around J = 1/2 or lower and slopes close to 1 (GeV/c) -2 and one peculiar object, the Pomeron, with intercept J = 1 and slope about 1/4 to 1/3 of the ordinary trajectories. Chew and Rosenzweig [1 ] have proposed a model which identifies the Pomeron (P) with the next leading vacuum trajectory (f) and explains its intercept and small slope by significant curvature of the P-f object near t = 0. Quigg and Rabinovici [2] have made an analysis of meson-baryon cross section data from 6 to 280 GeV/c and find no good evidence for his P-f coincidence. From the point of view of Reggeon field theory the P is a gapless singularity with Reggeon energy, E = 1 - J , vanishing at zero Reggeon momentum [3]. All other Reggeons have a nonzero energy gap A > 0, and trajectory: E(q) = a'q2 + A. Such a situation is highly suggestive of a bound state spectrum with a Goldstone boson arising from the breaking of an underlying continuous symmetry and a series o f ordinary bound states to be identified with the usual Reggeons. Several people have taken up this theme by studying the Reggeon field theory for the pomeron suggested originally by Gribov [4], and asking how the P field operator might acquire a nonzero vacuum expectation value and give rise to Goldstone excitations [5 ]. A thorough study of this possibility in ordinary Reggeon field theory reveals no reason why the Pomeron gap Ap = 1 - ap(0) should be zero except for special values of the parameters in the Reggeon lagrangian * 1,2. This suggests that one ought to look into the origin o f Regge singularities as bound states of some constituents. In the same spirit as one thinks of the ordinary hadrons as bound states o f quark degrees of freedom, we would like to think o f Reggeons as bound states of the same degrees of freedom which are now realized off the mass shell and for complex angular momentum, as is appropriate for a Reggeon. Since fremion degrees of freedom appear to Reggeize m non-Abelian gauge theories [8], such a theory may well underhe our approach. So let us consider a set of quark Reggeon fields Xi(X, r) which are two component spinor fields in rapidity, T, and impact parameter, x, space. The label i encompasses color and some set of flavors. We want to imagine that among these flavor labels there is an O(2) symmetry. For example, it could mix an SU(3) triplet o f u, d, s quarks with some triplet of U, D, S quarks. We'll call these two sets rotated into one another by the O(2), heavy and light. A free Reggeon lagrangian invariant under 0 ( 2 ) rotations U(0) = exp (it20); r2( 0 01) is
¢' Work supported by Energy Research and Development Administration. * On leave from Theoretical Physics Department, Fermilab, Batavia, Illinois 60510. ,1 See ref. [6]. The broken symmetry turns out to be time-reversal mvarmnce. ~:2 Dyatlov [7] has looked into a P-f sponteneous symmetry breaking which does not violate T invariance. It requtres an unnatural but possible coincidences among Reggeon couplings and has no triple ~Pcoupling. 308
Volume 62B, number 3
PHYSICS LETTERS
7 June 1976
+~ Z?x= ~i ~ t o~~ x + i ~20 Xto3o.OX-Ao~tX,
(I)
where ~ is the transpose in heavy-hght space. The o matrices operate in the spin space of two component spinors and g = (Ol, 02) operates in the two dimensional momentum space of the Reggeons. The flee propagator *a
S(E,q) = 1/E - / ~ 0 o 3 ~ - q
- A0 ,
(2)
describes the usual doublet of positive and negative parity fermion Reggeons with intercept c~0 = 1 - A 0. As a simple interaction, which is just renormahzable in two dimensional transverse space, we choose a scalar Reggeon q~(x, 7-) to provide ~I
k0 = _ f ~*X(0-t2+ q~2),
(3)
plus its free propagation ~=
2,1" ~ / ~ -
~V,t.
7~-~O~t~ .
(4)
Now we seek a solution to this 0 ( 2 ) symmetric Reggeon field theory which has a proper self energy for the fermien propagator N(E, q) = ZI(E,q) + r3X3(E, q) where E l and ~3 are invariant under 0 ( 2 ) rotations .4 . A term like ~3 breaks the heavy-light symmetry and gives rise to a Goldstone boson. To see this consider the proper vertex function of two quarks and the conserved 0 ( 2 ) current ( p , ~ ) = ( ~ t r2x , x t o3ar2x) • The vertex function satisfies the Ward identity
Erp(E, q; El, ~ql) - q F(E, q; E 1 , q~l) = r 2 S - 1 (El, ql)
_ S -1 (E 1 _ E, q~l- q~)~2 •
(5)
As E, q ~ 0 , this means lim EFo - q r = - 2 i r l Z 3 ( E l , q l ) , E,~0
(6)
and that the proper vertex may be written as
(re, g) = (rReg, F Reg) + _ _(1,aq) ~ (-irlF1) E - aq2
(7)
'
where F l (E, q; E l, q 1)IE q=0 = 2 ~3(E1, q l ) ,
(8)
and 1-'Reg has no pole at E, q =0. So if a symmetry breaking solution exists, then a Goldstone boson appears in the ~t 7-1X channel. It has Regge trajectory t~(t) = 1 + at, and we identify it to be the Pomeron. To find out whether such a solution is possible we look at the proper self energy ~3 to lowest order in k 0. In D transverse dimensions this is _ f d E 1 dDkl dE2dDk 2 ~3(E1, kl) [(El -/30 ?3 ~O"~kl- AO- ~I )2 - ~2 ]_-I ~3(E, q) - 2(4~r2)D d
[ E 2 - °~0~k22- 60+ie] [ E - E l - E 2 - u0(q~ _ ~kl- ~k2)2 -60+ie]
,
(9)
, a This is the two component version of the fermion propagator which is given by Gribov et al. [9]. ,4 In quite a different context a dynamically broken 0 ( 2 ) symmetry has been studied by Cornwall and Norton [10].
309
Volume 62B, number 3
PHYSICS LETTERS
7 June 1976
We seek a solution o f the form E 3 (E, qq) = A (E+ b0q2 )-r~ for large E and q 2 with r7 > 0 [ 11 ]. Similarly for Y~I"This only works for D = 2, where the coupling k~/c~ is dimensionless, and requires to lowest order in ;k0 t
77= X0/4rrc~0
and
P
S
b 0 = c~0 47rc~0/~0 .
(10)
At least m this crude approximation there is a symmetry breaking solution for every )to > 0 at the physical dimensions D = 2. The Goldstone boson associated with this symmetry breaking couples only to the off diagonal channel ~ t rl × = X~HXL+ ×fL×H where H and L label eigenvalues of the proper self energy. All other Reggeons couple to general mixtures of light and heavy quarks. The spectrum of ordinary Reggeons near t = 0 (q2 = 0) should be strongly determined near E ~ 112 (J ~ 1/2) by both heavy and light quarks so that the hadrons which he on those trajectories will couple strongly to the P trajectory there. Away from t = 0 and J ~ 1/2 perhaps only light or only heavy quarks will be important. As we expect the mass m H at which the heavy trajectory c~tt reaches J = 1/2 to be much larger than m E P where otL = 1/2, the slopes of these trajectories satisfy a]f < aL, since their intercepts ought to be nearby. Since the P is determined as a mixture of heavy arid light only, its slope will be determined by the usual combination t
t
t
~
C~p= a H a L / a n + a L ,
(11)
while other Reggeons wdl have their slope determined by the hght quarks primarily, so a ~ = a [ / 2 . Thus
and should be small. If we take ct~,/a~ ~ 1/3 and ask that m E ~ 300 MeV and m n ~ 1.5 GeV, then A L ~ 0.7 and A H ~ 0.9 making our picture of nearby singularities in the E plane for q2 ~ 0 consistent. From this picture also emerges a constant triple P vertex and, yet, through a field theory which satisfies Reggeon unitanty a special (broken) symmetry allows a simple pole [ E - c~,q 2 ] -1 to emerge for the Pomeron. The details of this and several technical issues connected with the building of Reggeons from quarks and symmetry breaking will appear m an expanded version of this note.
References [1] [21 [3] [4] I5] [6] [7] [8] [9] [10] [11]
310
G.F. Chew and C. Rosenzweig, Phys. Lett. B 58 (1975) 93. C. Quigg and E. Rabinovlci, Fermilab preprint 75/81-THY. H.D.I. Abarbanel et al., Physics Reports C21 (1975) 119. V.N. Gribov, Soviet Physics-JETP 26 (1968) 414. H.D.I. Abarbanel, Phys. Lett. B49 (1974) 61. H.D.I. Abarbanel, J.B. Bronzan, A. Schwimmer, and R.L. Sugar, Rutgers Preprmt (January 1976). I.T. Dyatlov, Zh. Eksp. Teo. Fiz. 69 (1975) 1127. M.T. Gnsaru, H.J. Schmtzer and H.-S. Tsao, Phys. Rev. D8 (1973) 4498. V.N. Gnbov, E.M. Levm, and A.A. Mlgdal, Soy. J. Nucl. Phys. 12 (1971) 93. J.M. Cornwall and R.E. Norton, Phys. D8 (1973) 3338. M. Baker and K. Johnson, Phys. Rev. D3 (1971) 2516.