Quark spin and the Θ-term for the QCD string

Physics Letters B 311 (1993) 98-102 North-Holland

PHYSICS LETTERS 13

Quark spin and the O-term for the QCD string * Jacek Pawetczyk Institute of Theoretical Physics, University of Warsaw, Ho2a 69, PL-O0-681 Warsaw, Poland Received 14 April 1993 Editor: P.V. Landshoff

We describe a way in which the spin of quarks can enter a consistent QCD string theory. We show that the spin factor of the 4D massless, spin-½ fermions is related to the self-intersection number of a 2D surface immersed in the 4D space. We argue that the latter quantity should appear in a consistent description of the QCD string. We also calculate the chiral anomaly and show that the self-intersection number corresponds to the topological charge F/~ of QCD.

Since the formulation o f q u a n t u m mechanics in terms of path integrals #l there exists a challenging problem in describing the particle spin within the formalism. Recent papers on the subject [2,3] elaborate on the so-called spin factors which were introduced by Polyakov [4] (see also [5]). The subject is interesting by its own but can also shed more light on the possibility of including similar factors in string theory. Here we want to focus on the relation between the spin factor of the 4D massless, spin-½ fermion and the self-intersection n u m b e r of a 2D surface (with b o u n d a r y ) immersed in the 4D (euclidean) s p a c e time. The self-intersection number, denoted by I, appeared in the context of the rigid string. In [6] it was proposed to consider 2D dynamics of the rigid string with a so-called 0-term, 2~i0I. The hope was that for 0 = ~ this term will help to preserve the rigidity as a relevant degree o f freedom of the low energy vibrations of the string. There were also suggestions that the 0-term corresponds to the Q C D 0-term [7,8]. In this paper we shall discuss the way in which the spin of quarks can enter a consistent Q C D string theory. W e show that after proper identification the spin factor o f the massless 4D fermion equals exp (2~zi I ) , where I is the self-intersection number of a 2D surface M i m m e r s e d in 4D space with the fermion living Work supported, in part, by Polish Government Research Grant KBN 2 0165 91 01. #1 There exists extensive literature on the subject contained e.g. in [1-3].

on the boundary 0 M . This suggests that inclusion of the 0-term at 0 = zc is equivalent to taking into account 4D spin quantum numbers. In other words one can say that the boundary of the string become 4D spin-½ fermions. This is a very desirable property because we believe that quarks live on the b o u n d a r y o f the Q C D string. We also introduce new anticommuting variables on the boundary 0 M and derive the result for the chiral anomaly. This result was obtained in [7] in a different, not purely stringy way. We shall consider the first quantized massless spinning particle with spin-½ moving in d-dimensional space. The relevant action has the following form [9,4]: 1

s = ½

at

i0

,

'

~,o,~,

)

(1)

where e denotes the 1D graviton field, Z the anticommuting gravitino, X the position o f the particle and qt is the supersymmetric partner of X. The crucial role is played by the supersymmetry transformations o f the fields:

(

*

98

(o,x) 2 - j

1

)

e ~Z = - 2 0 t a ,

~e = - a Z .

(2)

0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.

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With help of the transformations (2) we shall show that one o f the components of the field ~ can be gauged away. Moreover, the gravitino works as a Lagrange multiplier, again reducing the number of anticommuting fields ~ by one. Integration over the remaining anticommuting degrees of freedom leads to the spin factors o f interest. Altogether we shall see that the description o f the d-dimensional massless spinor leads to the same spin factor as for the massive ( d - 1 )dimensional spinor [ 2 - 4 ] . The Minkowski space is the most natural frame for the description o f massless particles with spin in the standard q u a n t u m field theory approach. However, here we want to concentrate on closed paths thus we choose to work in the euclidean space. At each point o f the path there is a natural decomposition o f the tangent space to the s p a c e - t i m e into 1D space spanned by OtX and ( d - 1 )-dimensional space spanned by the vectors r ( t ) , n i ( t ) , i = 1..... d - 2. The basis of the linearly independent vectors {OtX, r, n ~} respects

OtXr~=O,

OtXn i = r n i = 0,

nin j = 6 ij.

(3)

The only restriction on the OtX, r sector o f the basis is the requirement that these vectors should be linearly independent and that OtXr t= 0 everywhere. One can take for convenience e.g. OtXr = 1. The a n t i c o m m u t i n g field ~ can be decomposed in this basis in the following manner: n-2

qt = 4oOtX + 4 , r + Z

4~ n ~.

(4)

i=1

With the help o f the supersymmetry transformations (2) we can immediately get the transformation rules for the components q?. F r o m (2) and (3) we get

6# = 60oa,2 + 4o6(0,2) + 64,7 + 4r0,2 +

64i ~i

+ 4i

Z ~OtX = ~Z [400tX 2 + 4r(rOtX)]. -~

(7)

After the gauge fixing 00 = 0, this term reduces to Z4r OtX r/e 2. After redefinition of the gravitino Z -~ e2z/OtX r we simply get Z4r. The crucial point here is that OtXr ~ O. The functional integration over the gravitino introduces the functional Dirac delta function 6 (4r) under the path integral, which effectively puts 4r = 0 everywhere. In this way the kinetic term of the fermionic part o f the path integral takes the form ~'Ot~' = 0~Otq~i + C i J 4 ~ # ,

(8)

where the sum over i, j = 1..... d - 2 is understood and C ij = niOtn j. Before we do the functional integral over the fermions 4 we have to change the functional measure: Dig = D O J [ n ] . The Jacobian ( J ) o f the transformation is det-ll2(gu~Ot + CU~), where gU~ = nun ~ is the metric in the new coordinate system, C u" = nUOtn ~ and now n # denotes all the basis vectors (3), # = 1..... d. The Jacobian is J[n] = ±1 due to global gauge anomaly [10] ~2 The alternation of sign o f J In] results from topologically nontrivial configurations o f C u~. Now we can perform a functional integral over 4i, i = 1..... d - 2. We consider a closed path with the 4's respecting antiperiodic boundary conditions (AP):

t,,,,,O}.

= (640 + ...)o,2 + (64r + ...)7 (5)

where dots denote field dependent terms. Comparison with (2) plus linear independence o f the basis vectors yields 640 = a + "field dependent term" .

The above equation means that the field 4o can be completely gauged away. The F a d e e v - P o p o v determinant is trivial in this case. Now we go to the discussion o f the gravitino term. In the new basis it has the following form:

AP

6~i

+ (6g + ...)~,

29 July 1993

(6)

,9)

where T denotes the path ordered product and the y's are Dirac matrices. We also redefined 4 ~ ~ e4 i. This expression represents the so-called spin factor o f the path integral for the massless particle with spin

~¢2 1 would like to thank Dr. G. Korchemsky for bringing this reference to my attention. 99

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½. The spin factor is the trace over the SO(d - 2) group element, which is known to be relevant for the classification o f massless spin states. It is invariant under small and big gauge transformations of the basis of the normal vectors {n i} due to the Jacobian J [n ]. From now on we shall consider only the 4D spacetime. In the case d = 4 we can take the following Dirac matrices of interest: ~ = 1 @ a 1, ~ 2 1 (~0"2, SO [yi, ?j] = 2ieiJl ® a3. The spin factor is now

st.]

Tr [exp(-¼if

ciJ
(lO)

We suppress the path ordering T because the spin factor (10) is abelian. It is the 4 × 4 matrix which elements equal exp(+¼i f niOtn j~ij). The integral in the exponential factor measures the rotation angle (twist) of the frame {n ~, n2}. This spin factor is analogous to the well-known Wigner phase for states of given (±½) helicity [ l l ] . In the case of a massive particle the factor under the trace of (10) belongs to a non-abelian group, which for the 4D space-time is SO(3) [2]. Now we want to show that the above spin factor can be represented as a surface functional integral. It is known that for 2D surfaces immersed in the 4D space there exists a topological invariant counting selfintersections of the given surface. The invariant is given by the formula [6,7]

I -

1 f d2~v~gaboat~vOb~Uu '

16n J

(11)

M

AiaJ(¢) = x i ( ¢ ) OaNi(~).

(12)

We also define their field strength tensor: F ~ = OaN~ObNj - ObN' OaNj • The field-strength tensor F~ defines an element of the second coh6mology class H 2 (M). Now the intersection number is

= ~ ,1, , f M

100

If the surface M has a boundary 0M, thus H 2 (M) = 0, the expression for I simplifies to

I = -~I f

dteiJNiOtNJ •

(14)

0M

and now I is no more a topological invariant. Thus, I is just the sum over the twists of the normals of all the boundaries. Polyakov [6] proposed to add to the rigid string action a P-term which is 2i01, i.e. he proposed to consider

f DXexp(-S[X])exp(2iOI),

(15)

where S[X] is a string theory action (e.g. action for the rigid string). From what we said it is clear that for the open strings (i.e. surfaces with boundary) at 0 = rt the quantity exp(2niI) can be identified with the spin factor of the boundary (10), after identification o f C ij with Aij Ot~~. In this way we arrive at the interesting interpretation of the self-intersection number: if the surface is bosonic the 2i0I-term at 0 = makes its boundary fermionic in the 4D sense. This suggests that for 0 = n the P-term can simulate quarks at the ends of the Q C D string. At 0 = 0 the boundary term does not give any contribution, i.e. it stays to be bosonic. All what was said till now can be summarized by the formula which describes kinematics of a string theory with certain spin factor on the boundary of the string:

f DX exp ( - S [X] )

where X = X(~ ~) defines the immersion, gab is a 2D induced metric on the surface M, gab = OaX ObX and tu~ =eab OaX ObX/x/~. The formula for the selfintersection number I can be put into a different, more suitable for our purpose, form. We introduce gauge fields of the vectors normal to the surface N' (i = 1,2), defined by

I

29 July 1993

ij . d2g ( ab£ ij Fab

(13)

xTr

[exp ( -~-~ iO /dtAiJOt~a(t)~iJ)]

(16)

OM where the trace under the functional integral is interpreted as the spin factor o f a massless particle (quark, if 0 = n) living on the boundary 0 M of the surface M. The functions X u define immersion of the surface M into 4D euclidean space. The field A~J (t) is a natural gauge field (of normals) which appear on the string world-sheet M. The formula (16) contains also the interaction of the string boundary with the bulk because normal vectors are determined by the surface M. It is hard to say if this is all we need. Below we find out the expression for the chiral anomaly using new anticommuting

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PHYSICS LETTERS B

variables on the boundary of the string. We denote them by Ta, a = 1, 2. With the help of them we introduce a new interaction of the fermionic path discussed above with the string world-sheet. The proposed modification arises naturally as an interaction of T ' s with the external gauge field (here it is the gauge field of normals A~j) if one treats them as superpartners of 2D coordinates ~ . In this way we get 1D locally supersymmetric action for the interaction with A~j [9 ]. Thus the proposed action for the boundary is 1

SOM :--½i dl(~gab~aOt ~J8 0 I ij a b

(17)

where e is 1D metric of the boundary 0M. The action is l D reparametrization invariant so we can fix the symmetry setting e = L (L is the moduli parameter of the boundary). In a string theory with the boundary contribution (17) one can find out the chiral anomaly• We shall consider M with single boundary, i.e. 0 M is a connected manifold. The anomaly is given by the trace of the 75 with evolution operator in the limit L ~ 0 [12]. In the first quantized language it corresponds to the evaluation of the functional integral with 1D ferm ions respecting periodic (P) boundary conditions in the limit L -~ 0. The anomaly comes from the zero modes qJ0~ (i.e. constant fields) of ~ua which appear for the periodic (P) boundary conditions. The expression for the anomaly reads

limTr(.f d~pld~Fo2fdtis~eiJF~Jb(~o)~kVob )

L~O

0

× / DT'exp(-S~M),

(18)

zrNc

29 July 1993

e Fabexp £

-~-~

d

~Fab£

£

,

(19)

M

where OM = O. In the above formula the factor Nf is the number of different fermion species of T's*3. The factor 1/No was introduced by hand due to the arguments from large Arc expansion which says that amplitudes with fermion loops are suppressed by 1/Nc [141. Chiral anomaly gives a change of the functional integral under infinitesimal (chiral) change of the phases of the space-time fermion fields. We see that this is equivalent to the change of the 0 in eq. (15) (but with 0 M = 0). Nf) 0 ~ 0(1 + c o n s t . ~c .

(20)

For 0 = 0 the anomaly vanishes. This suggests that in this case we deal with a bosonic boundary. The formula (20) differs from the standard QCD formula where we have 0oco ~ 0ocD + const. This result holds for 0eco close to zero, which we want to identify with 0 close to ~. In this region one can take 0 = ~ - 0QCD and now both formulas agree for small enough 0OCO. We conclude the paper with several remarks and speculations concerning application of the results for the QCD string. If one believes that the low energy 4D Q C D is described by a kind of string theory then the 0term at 0 = ~ maybe useful as the one which provides spin degrees of freedom for the string boundary as well as a natural interaction of the string world-sheet with the boundary. This is good news because there are some arguments which say that one needs such a term in order to assure that the rigidity is relevant for low energy vibrations of the string [ 6 ]. We have added new anticommuting variables living on the boundary of the string. We have coupled them to the gauge field of normals and reproduced expression for the chiral anomaly with the self-intersection number playing the role of topological charge of the QCD. Of course, much more work has to be done in order to decide if these ideas will work properly.

Nf/Nc.

Fff

where the "prime" denotes omission of the zero modes, C0 is the position of the loop in the limit L --* 0. The only nontrivial contribution from S~M which survive this limit is the topological term corresponding to the spin factor. Finally we get that the anomaly is proportional to

~3 One can discuss the many-flavour case introducing an-

other type of anticommuting variables [ 12,13] or considering the matrix valued action for the boundary. 101

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PHYSICS LETTERSB

I want to thank R. Budzyfiski, Z. Lalak,M. Spalifiski and A. Niemi for their kind interest in this work.

References [1] M. Henneaux and C. Teitelboim, in: Quantum mechanics of fundamental systems 2, eds. C. Teitelboim and J. Zanelli (Plenum Press, New York, 1988). [2] J, Grundberg, T.H. Hansson and A. Karlhede, Nucl. Phys. B 347 (1990) 420. [3] M.A. Nowak, M. Rho and I. Zahed, Phys. Lett. B 254 (1991) 94; T. Jaroszewicz and P. Kurzepa, Ann. Phys. (NY) 210 (1991) 255; I.A. Korchemskaya and G.P. Korchemsky, Phys. Lett. B 257 (1991) 125. [4] A.M. Polyakov, Gauge fields and strings (Harwood Academic, New York, 1987); A.M. Polyakov, in: Fields, strings and critical phenomena, Proc. Les Houches Summer School, Vol. IL (1988), eds. E. Brdzin and J. Zinn-Justin (NorthHolland, Amsterdam, 1990).

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[5} A. Strominger, Phys. Lett. B 101 (1981) 271. [6] A.M. Polyakov, Nucl. Phys. B 268 (1986) 406. [7] P.O. Mazur and V.P. Nair, Nucl. Phys. B 284 (1986) 146. [8] A.P. Balachandran, F. Lizzi and G. Sparano, Nucl. Phys. B 263 (1986) 608. [9] A. Barducci, R. Casalbuoni and U Lusanna, Nuovo Cimento A35 (1976) 377; L. Brink, S. Deser, B. Zumino, P. Di Vecchia and P. Howe, Phys. Lett. B 64 (1976) 435; P.A. Collins and R.W. Tucker, Nucl. Phys. B 121 (1977) 307; F.A. Berezin and M.S. Marinov, Ann. Phys. (NY) 104 (1977) 307. [10]S. Elitzur, E. Ravinovici, Y. Frishman and A. Schwimmer, Nucl. Phys. B 273 (1986) 93. [ 11 ] J. Lopuszafiski, Rachunek spinorow (PWN, Warsaw, 1985). [ 12] U Alvarez-Gaum6 and E. Witten, Nucl. Phys. B 234 (1984) 269. [13] N. Marcus and A. Sagnotti, Phys. Lett. B 188 (1987) 58. [14] G. 't Hooft, Nucl. Phys. B 72 (1974) 461; B 75 (1974) 461.