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The pion in QCD
Volume 96B, number 3, 4
PHYSICS LETTERS
3 November 1980
THE PION 1N QCD ~r J. FINGER Physics Department, Massachusetts Institute o¢ Technology, Cambridge, MA, USA J.E. MANDULA Physics Department, Washington University, St Louis, MO, USA and J. WEYERS D@artement de Physique, Universit~ de Louvain, Louvain-la-Neuve, Belgium Received 27 August 1980
A model for chixal symmetry breaking in QCD patterned on the BCS theory of superconductivity is given.
The work of Nambu and Jona-Lasinio [1] has given a picture of spontaneous breaking of chiral symmetry which is now part of the common lore of particle physics. In this picture the vacuum contains a coherent condensate of fermion-antifermion pairs which is chirally asymmetric and the pion is the Goldstone boson associated with this spontaneous breaking of chiral symmetry. In this note we will argue that this picture is a likely consequence of QCD [2] and leads to a quantitatively satisfactory account of some of the pion's properties. Specifically, it has been shown [3] by a variational calculation that the QCD vacuum can contain a coherent, isospin symmetric, quark-antiquark condensate with the chiral transformation properties of the N a m b u Jona-Lasinio ground state. This condensate has massive fermion pseudoparticle excitations -- the constituent quarks. As a result the pion - Goldstone boson state obtained by an infinitesimal chiral rotation of the vacuum is explicitly seen to be a constituent quark-antiquark bound state. Various physical parameters follow from the condensate wave function. We compute the constituent quark mass (Mq), the ratio of ¢' Supported in part by the US Department of Energy under contract no. DR-78-S-02-4915.
2 to the current mass (mq), and the pion decay ~t,r coupling (f~). This letter is organized as follows: First we describe the condensate ground state and its pseudoparticle excitations. Second we describe the pion, making explicit its dual nature as a Goldstone boson and as a consistent q~l bound state. Finally we calculate MQ, p2/mq and f~. 1. The condensate. A perturbative calculation [4] of the energy o f quark-antiquark pair states indicates that for a c = ¼g2 > 9 pair states can have lower energy than the no-particle state, which is a signal of a pair formation instability. Such an instability would lead to a condensation of qgl pairs. A q~ condensate may be described, by analogy with the BCS theory of superconductivity as a coherent state of pairs. A variational calculation indicates that for couplings a c ~> 89-these coherent states lie below the ordinary vacuum. As a model for the ground state wave function [3] we take an isospin (flavor) symmetric coherent state of quark-antiquark pairs. In this note we are concerned only with the spontaneous breaking of chiral symmetry, hence we will ignore any possible gluonic condensation, since gluons are invariant under chiral transformations. 367
Volume 96B, number 3, 4
PHYSICS LETTERS
Specifically we take the ground state wave function to be
]~)=exp{~s fd3p4(p)b;,sd+_p,s}lO),
(1)
where @,s and d~s are the creation operators for quarks and antiquarks of'momentum p and helicity s and there is an implied sum over colors and flavors. The state 14) is rotationally and translationally invariant. Since we are ignoring gluons, we are free to work in the radiation gauge, which has the advantage that the states created by b + and d + automatically satisfy Gauss' law and so lie entirely in the physical sector of Hilbert space. It is of course impossible to maintain manifest Lorentz covariance in the radiation gauge, but this is in any case a potential problem with any fixed time description of the vacuum state. A gauge and Lorentz invariant description of the condensate will be given elsewhere. The wave function 4(P) is determined by requiring that the expected value of the hamiltonian in the state 14) be a minimum with respect to variations in 4(p). The non-linear integral equation for 4(P) so obtained is equivalent to a three dimensional Schwinger-Dyson equation for the quark self mass:
~(p) =~g2f
(27r) 3d3q
( _ / q ~ 3"O[S(q)_
(2)
where
Y.(p) = A(p)tpl + B(p)p" V,
(3)
_(dP0 i s ( p ) - . , 2rr p%O _ p . ' t - Z(p)
=
A(p) - [1 + 8(t,)lt~.~ 2{A2~) + [1 +B(p)]2) 1/2'
(4a)
(4b)
with ~b= p/Ipl, s(O)(q) is S(q) with ~(q) = 0 and the factor of ~ is associated with the SU(3) of color. The connection between the self mass ~ ( p ) and the wave function 4(P) is
A(p)/[1 + B(p)] = 24(p)/[i -
42(p)] •
(5)
3 November 1980
The mass of these pseudoparticle excitations, which we take to be the expectation value of the hamiltonian in the state B~,sl~), is given by . . . . 4(0)_. MQ:]P[A(p)]p=o=4g2f(~3q21+qa2(q )
(7)
Were it not for confinement, these pseudoparticle excitations would be physical quark particles, and so it is appropriate to identify them with the constituent quarks. MQ is the constituent quark mass.
2. The pion. The equation for the pion is obtained by making an infinitesimal chiral rotation of eq. (2) for the quark self mass E(p). We define the pion vertex by e(p)
= d e a3'5 Z(p) e~3's Is=o,
(8)
where the implicit isospin structure a ~ at" x is understood. Making this infinitesimal rotation on eq. (2) gives 4 2/" d4q
i
P(P) = gg J (~n)4 ( p - q ) 2 × p(q)
1
q%0_q.~_2(q)
3'0
1
qO3"O_q.y_ ~(q)
3'0 •
(9)
We recognize that eq. (9) is exactly the Bethe-Salpeter equation for a zero mass pseudoscalar bound state of the constituent quarks! Since eq. (9)has been derived from eq. (2) it follows that the chiral transform of the ground state is a massless solution of the BetheSalpeter equation, as expected from Goldstone's theorem [5]. From eq. (3) we see that the pseudoscalar vertex which solves the Bethe-Salpete r equations is:
P(p) = IPlA(p)3"5 .
(10)
The fact that the Nambu-Goldstone pion is indeed a constituent quark-antiquark state can be made explicit by examining the chiral rotation of the ground state [4): with Q5 = f d3x i: 7q(x) 3,03,5 q(x): the chiral generator, one obtains from eqs. (1) and (6)
As in the BCS model, there are pseudoparticle excitations of the state 14) with the same quantum numbers as the bare quarks. They are created by the operators
Q5[~)
B;,s = [b;,s_ s4(p)d p,s]/[1 + ~2(p)] 1/2.
It is straightforward to include the effects of chiral symmetry breaking terms in the QCD lagrangian.
368
(6)
(
Z) d3p~
1 +2~(p) 42(g) ~,,,D+p,,l¢>.
(11)
Volume 96B, number 3, 4
PHYSICS LETTERS
A quark mass term mq~q modifies eq. (4a) by adding a term - m q to the denominator and, mutatis mutandis, this modifies eqs. (4b) and (2). The Bethe-Salpeter equation, eq. (9), is modified by adding quark mass terms to the propagators and adjusting the kinematics to allow for a non-zero energy transfer to the bound state, since when chiral symmetry is explicitly broken in the lagrangian the pion bound state will not remain at zero mass,
3. Basic pion properties. The pion mass squared ~t2 (i.e. its relation to mq) and decay constant fTr can be directly computed from the Bethe-Salpeter wave function. The ratio la2/mq can be calculated by taking the pion to vacuum transition amplitude of the QCD field equation for the divergence of the axial current:
~uAu = 2rnq~t75q.
(12)
The pion decay amplitude is given by (0[Au[rr)
=f~Pu •
(13)
Note that eq. (13) is only useful for moving and/or massive pions. Since eq. (12) is useful only for massive pions we will compute f~ by taking the zero mass limit of eq. (13) for massive pions at rest. When the current quark mass is non-zero, the BetheSalpeter wave function of the pion is more complicated than eq. (10). To first order i n / 1 , for a pion at rest, it has the form
P(p) = Pp(p) 75
3 November 1980
t" d3p
rprA(p)P(p)
f" : N .) (2rr)3 p3 ;A2ip) -+ii + Bip)] 2)3/2 (01~T5qlrr) _
xf- d3p
Only the linear combination
(14) (15)
P(p) = Pp(p) + 21p[ (A(P)PA(p)
+ [1 +B(p)IPT(P)) enters into f,~ and it is determined in terms of Pp(p) = Ip[A(p) by the integral equation
P(p) = 4g2 f
d3q
1
(27r)3 (p _q)2
X pqA(p)A(q) + p ' q [ l +B(p)] [1 + B(q)] p(q) 2 q3 (A2(q) + [1 + B(q)] 2)3/2 - Pp(v).
(17)
2i N
PP(P)
(18)
a(27r)3 IPl (A2(p) + [1 + B2(p)] 2)1/2 ' where the normalization factor [6] for the pion wave function is
N 2 = 1-f -d3p 2
PP(P)P(P)
(19)
(2n)3p3 (A2(p) + [1 +B(p)]2)3/2
The energy scale with respect to which all the preceding dimensionful quantities are measured presumably comes, in QCD, via dimensional transmutation [7], from the variation of the coupling constant with distance. In order to introduce this scale, while respecting asymptotic freedom, we replace g2 by an effective running coupling constant which is a function of the momentum transferred to the virtual gluon and vanishes at large momentum transfer. Nothing in the foregoing depends sensitively on how g2(p) vanishes, since 4, A and B all go rapidly to zero for [pl above that value at which g2/4n = ~. 9 For convenience we take
gZ(p _ q) =g2 A2/[A2+ (p - q ) 2 ] . The best values of the physical parameters were obtained for g2o/4n = 2.23 and A = 443 MeV. They are fTr = 95 MeV, MQ = 244 MeV,
+U [PA(P)'Y03 ~5 + PT(P)P " ¥ 7 0 @ ) "
'
i.t2/2mq = 1380.
We conclude this note with two remarks: (i) The dynamical argument for condensation applies to gluons as well as to quarks. Since gluons are chirally neutral, their presence in the vacuum will not directly affect chirality breaking although it will obviously change the quark condensate's wave function and so will somewhat modify the results given here. (ii) A Lorentz invariant generalization of our formulation of quark condensation is provided by the covariant self-consistent Schwinger-Dyson equations for the quark self mass, which can in fact be derived variationally from an effective action [8]. Their analysis is in progress.
(16)
f~r and the pseudoscalar transition element are given by
It is a pleasure to acknowledge useful conversations on the subject of this letter with R. Brout, F. Englert and A.J.G. Hey. 369
Volume 96B, number 3, 4
PHYSICS LETTERS
References [1] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345. [2] H. Fritzsch, M. Gell-Mann and H. Leutwyler, Phys. Lett. 47B (1973) 365; S. Weinberg, Phys. Rev. Lett. 31 (1973) 494; D.V. Nanopoulos, Lett. Nuovo Cimento 8 (1973) 873; H.D. Politzer, Phys. Rev. Lett. 26 (1973) 1346; D. Gross and F. Wilczek, Phys. Rev. Lett. 26 (1973) 1343. [3] J. Finger and J.E. Mandula, to be published. [4] A.J.G. Hey, D. Horn and J.E. Mandula, Phys. Lett. 80B (1978) 90; J. Finger, D. Horn and J.E. Mandula, Phys. Rev. D20 (1979) 3253; R. Fukuda, Phys. Lett. 73B (1978) 33;
370
3 October 1980
R. Fukuda and T. Kugo, Prog. Theor. Phys. 60 (1978) 565; V.P. Gusynin and V.A. Miransky, Phys. Lett. 76B (1978) 585. [5] J. Goldstone, Nuovo Cimento 19 (1961) 154. [6] S. Mandelstam, Proc. Roy. Soc. (London) A233 (1955) 248. [7] S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888. [8] See, e.g., H.D. Dahmen and G. Jona-Lasinio, Nuovo Cimento 52A (1967) 807; A.N. Vasil'ev and A.K. Kazanskii, Teor. Mat. Fiz. 12 (1972) 352; [Theor. Math. Phys. 12 (1972) 875]; J.M. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev. D10 (1974) 2428.
PHYSICS LETTERS
3 November 1980
THE PION 1N QCD ~r J. FINGER Physics Department, Massachusetts Institute o¢ Technology, Cambridge, MA, USA J.E. MANDULA Physics Department, Washington University, St Louis, MO, USA and J. WEYERS D@artement de Physique, Universit~ de Louvain, Louvain-la-Neuve, Belgium Received 27 August 1980
A model for chixal symmetry breaking in QCD patterned on the BCS theory of superconductivity is given.
The work of Nambu and Jona-Lasinio [1] has given a picture of spontaneous breaking of chiral symmetry which is now part of the common lore of particle physics. In this picture the vacuum contains a coherent condensate of fermion-antifermion pairs which is chirally asymmetric and the pion is the Goldstone boson associated with this spontaneous breaking of chiral symmetry. In this note we will argue that this picture is a likely consequence of QCD [2] and leads to a quantitatively satisfactory account of some of the pion's properties. Specifically, it has been shown [3] by a variational calculation that the QCD vacuum can contain a coherent, isospin symmetric, quark-antiquark condensate with the chiral transformation properties of the N a m b u Jona-Lasinio ground state. This condensate has massive fermion pseudoparticle excitations -- the constituent quarks. As a result the pion - Goldstone boson state obtained by an infinitesimal chiral rotation of the vacuum is explicitly seen to be a constituent quark-antiquark bound state. Various physical parameters follow from the condensate wave function. We compute the constituent quark mass (Mq), the ratio of ¢' Supported in part by the US Department of Energy under contract no. DR-78-S-02-4915.
2 to the current mass (mq), and the pion decay ~t,r coupling (f~). This letter is organized as follows: First we describe the condensate ground state and its pseudoparticle excitations. Second we describe the pion, making explicit its dual nature as a Goldstone boson and as a consistent q~l bound state. Finally we calculate MQ, p2/mq and f~. 1. The condensate. A perturbative calculation [4] of the energy o f quark-antiquark pair states indicates that for a c = ¼g2 > 9 pair states can have lower energy than the no-particle state, which is a signal of a pair formation instability. Such an instability would lead to a condensation of qgl pairs. A q~ condensate may be described, by analogy with the BCS theory of superconductivity as a coherent state of pairs. A variational calculation indicates that for couplings a c ~> 89-these coherent states lie below the ordinary vacuum. As a model for the ground state wave function [3] we take an isospin (flavor) symmetric coherent state of quark-antiquark pairs. In this note we are concerned only with the spontaneous breaking of chiral symmetry, hence we will ignore any possible gluonic condensation, since gluons are invariant under chiral transformations. 367
Volume 96B, number 3, 4
PHYSICS LETTERS
Specifically we take the ground state wave function to be
]~)=exp{~s fd3p4(p)b;,sd+_p,s}lO),
(1)
where @,s and d~s are the creation operators for quarks and antiquarks of'momentum p and helicity s and there is an implied sum over colors and flavors. The state 14) is rotationally and translationally invariant. Since we are ignoring gluons, we are free to work in the radiation gauge, which has the advantage that the states created by b + and d + automatically satisfy Gauss' law and so lie entirely in the physical sector of Hilbert space. It is of course impossible to maintain manifest Lorentz covariance in the radiation gauge, but this is in any case a potential problem with any fixed time description of the vacuum state. A gauge and Lorentz invariant description of the condensate will be given elsewhere. The wave function 4(P) is determined by requiring that the expected value of the hamiltonian in the state 14) be a minimum with respect to variations in 4(p). The non-linear integral equation for 4(P) so obtained is equivalent to a three dimensional Schwinger-Dyson equation for the quark self mass:
~(p) =~g2f
(27r) 3d3q
( _ / q ~ 3"O[S(q)_
(2)
where
Y.(p) = A(p)tpl + B(p)p" V,
(3)
_(dP0 i s ( p ) - . , 2rr p%O _ p . ' t - Z(p)
=
A(p) - [1 + 8(t,)lt~.~ 2{A2~) + [1 +B(p)]2) 1/2'
(4a)
(4b)
with ~b= p/Ipl, s(O)(q) is S(q) with ~(q) = 0 and the factor of ~ is associated with the SU(3) of color. The connection between the self mass ~ ( p ) and the wave function 4(P) is
A(p)/[1 + B(p)] = 24(p)/[i -
42(p)] •
(5)
3 November 1980
The mass of these pseudoparticle excitations, which we take to be the expectation value of the hamiltonian in the state B~,sl~), is given by . . . . 4(0)_. MQ:]P[A(p)]p=o=4g2f(~3q21+qa2(q )
(7)
Were it not for confinement, these pseudoparticle excitations would be physical quark particles, and so it is appropriate to identify them with the constituent quarks. MQ is the constituent quark mass.
2. The pion. The equation for the pion is obtained by making an infinitesimal chiral rotation of eq. (2) for the quark self mass E(p). We define the pion vertex by e(p)
= d e a3'5 Z(p) e~3's Is=o,
(8)
where the implicit isospin structure a ~ at" x is understood. Making this infinitesimal rotation on eq. (2) gives 4 2/" d4q
i
P(P) = gg J (~n)4 ( p - q ) 2 × p(q)
1
q%0_q.~_2(q)
3'0
1
qO3"O_q.y_ ~(q)
3'0 •
(9)
We recognize that eq. (9) is exactly the Bethe-Salpeter equation for a zero mass pseudoscalar bound state of the constituent quarks! Since eq. (9)has been derived from eq. (2) it follows that the chiral transform of the ground state is a massless solution of the BetheSalpeter equation, as expected from Goldstone's theorem [5]. From eq. (3) we see that the pseudoscalar vertex which solves the Bethe-Salpete r equations is:
P(p) = IPlA(p)3"5 .
(10)
The fact that the Nambu-Goldstone pion is indeed a constituent quark-antiquark state can be made explicit by examining the chiral rotation of the ground state [4): with Q5 = f d3x i: 7q(x) 3,03,5 q(x): the chiral generator, one obtains from eqs. (1) and (6)
As in the BCS model, there are pseudoparticle excitations of the state 14) with the same quantum numbers as the bare quarks. They are created by the operators
Q5[~)
B;,s = [b;,s_ s4(p)d p,s]/[1 + ~2(p)] 1/2.
It is straightforward to include the effects of chiral symmetry breaking terms in the QCD lagrangian.
368
(6)
(
Z) d3p~
1 +2~(p) 42(g) ~,,,D+p,,l¢>.
(11)
Volume 96B, number 3, 4
PHYSICS LETTERS
A quark mass term mq~q modifies eq. (4a) by adding a term - m q to the denominator and, mutatis mutandis, this modifies eqs. (4b) and (2). The Bethe-Salpeter equation, eq. (9), is modified by adding quark mass terms to the propagators and adjusting the kinematics to allow for a non-zero energy transfer to the bound state, since when chiral symmetry is explicitly broken in the lagrangian the pion bound state will not remain at zero mass,
3. Basic pion properties. The pion mass squared ~t2 (i.e. its relation to mq) and decay constant fTr can be directly computed from the Bethe-Salpeter wave function. The ratio la2/mq can be calculated by taking the pion to vacuum transition amplitude of the QCD field equation for the divergence of the axial current:
~uAu = 2rnq~t75q.
(12)
The pion decay amplitude is given by (0[Au[rr)
=f~Pu •
(13)
Note that eq. (13) is only useful for moving and/or massive pions. Since eq. (12) is useful only for massive pions we will compute f~ by taking the zero mass limit of eq. (13) for massive pions at rest. When the current quark mass is non-zero, the BetheSalpeter wave function of the pion is more complicated than eq. (10). To first order i n / 1 , for a pion at rest, it has the form
P(p) = Pp(p) 75
3 November 1980
t" d3p
rprA(p)P(p)
f" : N .) (2rr)3 p3 ;A2ip) -+ii + Bip)] 2)3/2 (01~T5qlrr) _
xf- d3p
Only the linear combination
(14) (15)
P(p) = Pp(p) + 21p[ (A(P)PA(p)
+ [1 +B(p)IPT(P)) enters into f,~ and it is determined in terms of Pp(p) = Ip[A(p) by the integral equation
P(p) = 4g2 f
d3q
1
(27r)3 (p _q)2
X pqA(p)A(q) + p ' q [ l +B(p)] [1 + B(q)] p(q) 2 q3 (A2(q) + [1 + B(q)] 2)3/2 - Pp(v).
(17)
2i N
PP(P)
(18)
a(27r)3 IPl (A2(p) + [1 + B2(p)] 2)1/2 ' where the normalization factor [6] for the pion wave function is
N 2 = 1-f -d3p 2
PP(P)P(P)
(19)
(2n)3p3 (A2(p) + [1 +B(p)]2)3/2
The energy scale with respect to which all the preceding dimensionful quantities are measured presumably comes, in QCD, via dimensional transmutation [7], from the variation of the coupling constant with distance. In order to introduce this scale, while respecting asymptotic freedom, we replace g2 by an effective running coupling constant which is a function of the momentum transferred to the virtual gluon and vanishes at large momentum transfer. Nothing in the foregoing depends sensitively on how g2(p) vanishes, since 4, A and B all go rapidly to zero for [pl above that value at which g2/4n = ~. 9 For convenience we take
gZ(p _ q) =g2 A2/[A2+ (p - q ) 2 ] . The best values of the physical parameters were obtained for g2o/4n = 2.23 and A = 443 MeV. They are fTr = 95 MeV, MQ = 244 MeV,
+U [PA(P)'Y03 ~5 + PT(P)P " ¥ 7 0 @ ) "
'
i.t2/2mq = 1380.
We conclude this note with two remarks: (i) The dynamical argument for condensation applies to gluons as well as to quarks. Since gluons are chirally neutral, their presence in the vacuum will not directly affect chirality breaking although it will obviously change the quark condensate's wave function and so will somewhat modify the results given here. (ii) A Lorentz invariant generalization of our formulation of quark condensation is provided by the covariant self-consistent Schwinger-Dyson equations for the quark self mass, which can in fact be derived variationally from an effective action [8]. Their analysis is in progress.
(16)
f~r and the pseudoscalar transition element are given by
It is a pleasure to acknowledge useful conversations on the subject of this letter with R. Brout, F. Englert and A.J.G. Hey. 369
Volume 96B, number 3, 4
PHYSICS LETTERS
References [1] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345. [2] H. Fritzsch, M. Gell-Mann and H. Leutwyler, Phys. Lett. 47B (1973) 365; S. Weinberg, Phys. Rev. Lett. 31 (1973) 494; D.V. Nanopoulos, Lett. Nuovo Cimento 8 (1973) 873; H.D. Politzer, Phys. Rev. Lett. 26 (1973) 1346; D. Gross and F. Wilczek, Phys. Rev. Lett. 26 (1973) 1343. [3] J. Finger and J.E. Mandula, to be published. [4] A.J.G. Hey, D. Horn and J.E. Mandula, Phys. Lett. 80B (1978) 90; J. Finger, D. Horn and J.E. Mandula, Phys. Rev. D20 (1979) 3253; R. Fukuda, Phys. Lett. 73B (1978) 33;
370
3 October 1980
R. Fukuda and T. Kugo, Prog. Theor. Phys. 60 (1978) 565; V.P. Gusynin and V.A. Miransky, Phys. Lett. 76B (1978) 585. [5] J. Goldstone, Nuovo Cimento 19 (1961) 154. [6] S. Mandelstam, Proc. Roy. Soc. (London) A233 (1955) 248. [7] S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888. [8] See, e.g., H.D. Dahmen and G. Jona-Lasinio, Nuovo Cimento 52A (1967) 807; A.N. Vasil'ev and A.K. Kazanskii, Teor. Mat. Fiz. 12 (1972) 352; [Theor. Math. Phys. 12 (1972) 875]; J.M. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev. D10 (1974) 2428.