One-loop amplitudes for the bosonic open string: A first quantized approach

Volume 178, number 4 PHYSICS LETTERS B 9 October 1986

ONE-LOOP AMPLITUDES FOR THE BOSONIC OPEN STRING: A FIRST QUANTIZED APPROACH

Jo~o P. RODRIGUES

Physics Department, University of the Witwatersrand, Johannesburg 2001, South Africa

Received 23 June 1986

The N-point scalar and four-point vector amplitudes for the bosonic open string in the critical dimension are obtained using a first quantized path integral treatment of Polyakov's string that assumes scale independence.

1. Introductio~ One of the approaches to string theories is to consider them as two-dimensional models whose basic variables are the string coordinates x , , /a = 1,2,. . . , d, tr = (tr 1 , 0 2) [1] .1. In this formalism, the understanding of the correct path integral measure is crucial.

Hsue, Sakita and Virasoro [4] were the first to use this first quantized approach to obtain tree and one-loop N- point tachyon amplitudes in the old dual resonance model (bosonic open string). However, the partition function for the one-loop amplitudes was still computed with operator methods.

Polyakov's path integral formulation of the bosonic string [5] provided the major step in the understanding of the correct string functional integral measure and provided the basis of much further work (see for instance refs. [6-8]). However, one-loop amplitudes had to wait for a fuller understanding of moduli and of the nature of Teichmiiller space [7,9-11 ]. A one-loop analysis of the dosed bosonic string has now been carried out [9], and the question of fermionie degrees of freedom has also been addressed [12].

The applications that I have in mind in this communication are, however, related to the possibility of using string theories to obtain the effective action of gauge fields [13,14]. For this, one needs to consider the open string, and in particular vector amplitudes if one wants to go beyond the string tree level contribution.

In the next section the one-loop partition function of the bosonic open string is obtained. In section 3 the N- point scalar amplitude is computed and compared to known one-loop results. The four-point vector amplitude is presented in section 4.

The method used is that 0f Polchinski's [9], where scale invariance is assumed at every step. Moore and Nelson [I0] have given it a sound theoretical foundation and I refer the reader to their paper. The new aspect of the treatment is the presence of boundaries and the need to impose Neumann boundary conditions. The question of boundaries in string theory has been considered by Alvarez [7], and in the context of the present approach, in ref. [15]. For Neumann (as opposed to Dirichelet) boundary conditions, certain non-trivial issues arise in the computation of the propagator which are often not mentioned in the literature.

To the best of my knowledge, the vector amplitude is a new result and a further example of the use and power of the method.

2. Partition function. The starting point is Polyakov's path integral [5] :

C dgab dx t~ w=j oxp (-f d2ov~[(1/41rc~)gabbaxuOb x~+ XR+//2]+ Sb).

.1 For the connection between string theories and sigma models, see refs. [2,3] and refs. [12,13] from ref. [3].

(1)

350


One integrates over all euclidean metrics gab (a) on a surface of a given Euler number (fixed topology) and all embeddings xU(a) into R d. R is the scalar curvature. S b is a boundary dependent term required for quantum Weyl invariance [7].

In their original treatment of the one-loop bosonic string, Hsue, Sakita and Virasoro [4] considered as the two- surface the annulus

q < l z l < l , 0 < q < l , z E Z . (2)

I will work with the strip 0 <~ Im a ~< 1 covering the annulus with projection z = exp [2ni(a 1 + iro2)], where r is the modulus of the annulus [16], a positive real number related to the inner radius of the annulus by

r = - (1/2zr) In q . (3)

Then, both gab (o) and xu (a) are periodic in the real direction

gab(O 1 + 1, o2)=gab(al, o2), x u ( a l + 1, o2)=xu(ol , o2). (4)

In the imaginary direction, it is well known that Neumann boundary conditions must be imposed for xU(o), i.e.,

axU (o 2 = O) =axu ( 2 = 1) = 0. (5a) ~° 2 aa 2 .a

This is a sufficient condition for the existence of a classical extremum of the action [7]. The condition to be imposed on the metric is the following [15] : i f t a is the tangent vector to the boundary and n a an arbitrary normal vector to the boundary, then

tanbgab = 0 on the boundary. (5b)

By a general coordinate transformation, any metric can be transformed, while satisfying (4), to

ds 2 =gabdaada b = exp[¢(a)] [(do1) 2 + r2(da2)2]. (6)

Any variation of the metric connected to unity can be decomposed as

dgab (a) = gab (o) ~ ~ (o) + ~ ~'a ;b (O) + ~ ~" b ;a (O) + gab,r dr. (7)

The method of ref. [9] consists in finding the jacobian J(¢, r) defined by

dg = (de d~')' dr J(~b, r), (8)

where the prime denotes variations orthogonal to the zero translational mode

6~'1(a) = e l , 6 ¢ ( a ) = - e l 0 1 ¢ ( a ) . (9 )

Following ref. [9], and with the same notation, I obtain

/ 1 " 2 \1/2 J(¢, r)=(det 'c~)l /2 [Q - Jd ox,~) , (10)

where Q = fd2ax/g(a l~ a l e + 1) and

2 + 4 C 0 0 \

c~= 0 2 Ad -2De Xe .J (11)

0 2XedDe Xe fX eli

In the above A d = d 2 1 --6 c D - DdDc + DcD d and Xab = gab,r -- ~ gab gCd gcd, r" The partition function of the strip then takes the form (the system has been put in a hypereube of side L)

351


Wst ip = f (dl" d4,)' La(det, )l,2tfd2c, V )(d+l)i2Q-li2(2,r)-al2[det,(_(il2 ra)(ilv,'d)a:g<,b ab)]-dl2.'--\ d VGc V W

(12) It is well known [5] that the bulk ¢ dependence in the above expression can be made to vanish in 26 dimensions

by a suitable adjustment of / f t . It was explicitly shown in ref. [15] what further local, boundary dependent terms S b are required so that all scale dependence is renormalized away. One can then set ~ = 0.

The condition (5b) implies, for the metric (6), that

a2~l(o 2 = o) = az~l(t72 = 1) = o, ~2(02 = o) = ~2(tr2 = 1) = o. (13)

Since f d 2 o x/g = r, Q = r, one obtains:

(det'C~) 1/2 = [det(2 + 4C)] 1/2 (x/~/r) [det'(--26dcgabaaab)] 1/2

= [det(2 + 4C)] 1/2 (x/2/r) [det~ ( - 2gabOaab)] 112 [det D (_ 2gabaaab)] 1/2

= [det(2 + 4C)] 1/2 [detN(2) ] 1/2 [detD(2) ] 1/2 ( l / r ) [det~ (A)] 1/2 [det o (A)] 1/2, (14)

det' ( 1 - (1/2rr tQ (1/x/rg) Oa gab X/~-ab) = det~ ((1/2rta) A) = (2rra) det N (I/2zta) det~ (A), (15)

where A is the scalar laplacian and the subscripts D and N refer to Dirichlet (13b) and Neumann ((5a) and (13a)) boundary conditions, respectively. They are given by

det~l(A): I ] ' [(2rr)2n 2 +n22/r2], (16) n2~0

?

detD(A ) : l-I [(2,) 2 n 2 + n2/r21. (17) n 2 ~ l

The prime indicates that the term n 1 = n 2 = 0 in the product must be removed. Then

detD(A ) det~ (A) = 1-I' [(2.) 2 n 2 + n~lr2], (18)

]1' ÷ det~(A) = [(27r)2 n1211/2 ~ [(27r)2 n2/r2] 1/2 (19)

The r-dependent product has been evaluated in ref. [9] with zeta function regularization and yields

(D ; [(2.) 2 n 2 + n2/ ' r 2] 27" exp( ~ 1 7r't') 1 = [1 - exp ( - 4rrnr)] . (20)

The r-independent product in (19) is exactly 1 if zeta function regularization is used. Collecting all factors to- gether in eq. (12), and neglecting all the determinants [9] I obtain

L d ? (D s dr exp(41rr) [1 - exp ( - 47mr)] • (21) Wstrip - (81r2°013 0 1

In the above I have included a factor of ½ corresponding to the orientation preserving symmetry (respecting (4) and (5)) o 2 ~ 1 - tr 2, o 1 ~ - o 1 .

Using (3), Wstri p can be written as 1 ** ))-24

L d 1 ( d q [ 11 (1 _q2n Wstrip - 2rr (81r2ot)13 ~ q3 l,n = 1 " (22)

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The q-3 divergence near q = 0 is the well-known divergence for the open bosonic string .2. Notice that, as it is the case for the one-loop sum over surfaces for the dosed string, the term containing X in (1) does not contribute: the Euler characteristic is given by [7] X = 2 - 2h - b, where h is the number of handles and b the number of boundaries, yielding 0 in the present topology. The overall normalization is completely fixed.

3. Scalar amplitudes. The N-point scalar (tachyon) amplitudes are well known to be generated by (see refs. [13, 14,18])

(exp(go f dsA(xU(s)))) , (23)

where bM is the boundary and ds is the invariant line element (in our case, ds = do 1 when ¢ = 0). Fourier transforming,

d26k A(x u) = f (2~r)26 exp(ikuxU)A(kU), (24)

and for the one-point amplitude one needs to evaluate

N

fdxexp(-~fd2o~ax~aXU ) exp(i i~=lk~Xu(Si)). (25)

As usual, one shifts x " = x ~ + x~ + x t~ where x " is the constant configuration, x~ the solution of Poisson's qz equation with Neumann boundary conditions and the normal component 3n x at the boundary vanishes. What it

should be remembered is that Neumann boundary conditions for Poisson's equation correspond to fixing the value of the normal derivative of the Green's function at the boundary, but that the value of this derivative cannot be ar- bitrarily set due to Gauss' law [19]. This is particularly important if we consider a finite region. Therefore, whether or not the normal derivative of x~q at the boundaries is zero, depends on the source. Integrating in (25), I obtain

N

(2rr)26~(i__~l/C~/)exp ( - l fdsx~qanx~q)exp(~dd2crx~(-OaOa)X~q) 4--~ Oi

(1 ) ( ) × fdx'"exp -~-~-d fa x'.a.x exp -~--~fd2oaax'U~ax'U. (26) OM

A formula similar to the above one has been used in ref. [4]. The momentum conservation delta function in the equation is a result of the integration over the constant configuration. By integration by parts, the first factor in- side the integral sign yields 1. It is easy to verify that 3nX~q at the boundary is proportional to the sum of that component of momenta. Due to the presence of the delta function, the first factor in eq. (26) can be set to 1.

The Green's function for our problem

(r/2rta) ( - Oaba) G(o, o') = 6 (o - o') (27)

is

4:2 See, e.g., the discussion in ref. [17].

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Volume 178, number 4 PHYSICS LETTERS B 90ctolmr 1986

G(a, 0 ' ) = - a ~ r l a 2 - 0 ' 2 1 - awrla 2 + 0'21

- t x In

- a l n

- a l n

I1 - exp [i2~(a 1 - 0'1)] e x p [ - 21rrl02 - 0'21]1 I1 - exp[i21r(a I - 0'1)] e x p [ - 2~rrlo 2 + 0'21]1

ao

I-I I1 - exp[i2~(o 1 - o'I)] e x p ( - D r r l o 2 - o '2 + 2nl)] I1 - exp [i21r(o 1 - o'l)] e x p ( - 27trio 2 + a '2 + 2n01 n = l

oo

H I1 - exp [i2~r(o 1 - 0'1)] e x p ( - Dr r l a 2 - a '2 - 2n[)l I1 - exp [i21r(01 - a ' l )] e x p ( - 21trio 2 + 0 '2 - 2n I)1 n= l

f o r 0 :#: 0 ~,

G(a, o') = - 2a~r 02 + a/2t + a ln(eV/47r) - a ln[1 - e x p ( - 4rrra2)]

oo oo

- a In H [1 - e x p ( - 4nrn)] 2 _ a In I-I {1 - exp [ - 4~r(n + 02)]} (1 - exp [ - 4 . r ( n - 02)]} n = l n= l

(28a)

for 0 = 0' and 0 < a 2 ( 1, (28b)

oo

a(o, o') = - 2alrr + a/ t + 20 ln(e~/4~r) - 20 In l-I [1 - e x p ( - 4~rrn)] 2 for 0 = 0' and 02 = 1, n = l

G(o,o ' )=a/ t+201n(eV/41r) -2a ln f i [ 1 - e x p ( - 4 r r r n ) ] 2 f o r o = a ' a n d a 2 = 0 . / * = 1

(28c)

(28d)

Several comments are in order. The above propagator has been obtained using heat kernel regularization, since it is not def'med for o = a' . I have used an dectrostafic analogue in the derivation of the above result. Details will be presented elsewhere [18]. I note that 02G = 0 at a 2 = 0 and 0 2 G = - 2rro~r at a 2 = 1, in agreement with Gauss' law.

Using eqs. (23), (25), and (26) as well as the results of the previous section, I obtain for theN-point amplitude

1 . . N 1 . 1 . N '- "' (20) .xi--1 0 I 0 q ti=~ 2 i ~ /

where

Oi} = a ( ° , , as') - } a(°i ' o~)-½ a(OS, °S)" (30)

In order to have agreement with ref. [17], the renormalized coupling constantg R is defined as

gR = lim 2g0( t ) exp(1 /2 t - 7). t--+ oo

Then, on mass shell (k 2 = or- 1), 1 / N - 1 1 1 N _

A(kl'k2 ..... kN) = 8( i ~ ki)gRN2~" (-~( i--I'll O f 0(O:+I- o:) do:)f d_~q(q, \i =I'I1(1-q2n)) 24

X /I~>/(Xllp o r X I / N p ) 2 r ' k ~ k / ' ~ ,

(31)

(32)

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where

l_ 2 cos(27r(oli _ o:))q2n + q4n ~P=Isin(rr(al-°/)) y I (l_q2n)2 '

1 q-l/4 H 1-2cos(27r(°:-°: ))q2n-l+q4n-2 ~NP ='2-~ n=l (1 _q2n)2 '

(33)

1 for two points on the same (planar) and opposite (non-planar) boundaries, respectively. For a - ~, these results agree with those of ref. [17] but for the factor of ½ due to the discrete gauge invariance.

4. Vector amplitude. The N-point vector amplitudes are generated by (refs. [13,14,18])

(exp(go afM dSdX~(S)Au(xV(s)))) . (34)

For simplicity, I will consider the four-point planar amplitude. The generalization is straightforward and will be presented elsewhere [I 8]. Fourier transforming and using the propagator of the previous section, I obtain

( 2 ~ ) 13 (N_[_I] 0 ~ 1 N ~ / ' d q ( I - I ( l _ q 2 n ) ) -24 A t~ltzz'''t~N (kl, k2," .., kN)= , ( ~ ki ) gN2zr 0(tT:+ 1 -- o : ) d o : , ~ q-~\i= 1

>( I-I " 2~k~k~ [(QUlU:QUaU4 +...) _//"~/,tl/.t219u3Du4 .) .1. D/.tlDU2DU3D/'L4] Vp ' J (35) ~.~ ~3 ~4 + " -1 ~2 ~3 ~4 J i> /

(the dots indicate permutations), where (1 - q2n)2 _ (1 + q4n) sin2(Tr(o: _ 4 )

1 Z_J 8q 2n QUiU/= 21r2 ~i~. (s in2(Tr(o:_o/1) ) +n=l ['l'---2~00S(-2~/l~oj-~))q-~-~q4n] 2 ) '

( n~l 4q2nsin(27r(°:--°:)) ) e " i = -21ra ~ k~" cot(Tr(o 1 - 4 ) ) + . . . . . . . . . (36)

i~/ = 1 -- 2 cos(27r(a/1 -- o : ) )q 2n +q4n In eq. (36), I have defined

1 d d~iG'(a:,o:)=5 l i m : G;(a:,ol). t ~ 0 do 1

Although another regularization was used, the above definition is in agreement with that of ref. [8], when O = 0.

This letter originated in a visit to Brown University, which became possible as a result of an FRD travel grant. I thank the Nuclear Theory Research Group and the Deputy Vice-Chancellor for Research at Wits for partial support. I wish to also thank the hospitality and support of the High Energy Group at Brown. I have benefited from discussions with C-I Tan and in particular with Barry Fridling, who patiently introduced me to the subject.

References

[ 1] See e.g.S. Weinberg, talk Sixth Workshop on Grand unification (Minneapolis, April 1985), University of Texas preprint UTTG- 28-85, and references contained therein.

[2] C. CaUan et al., Nucl. Phys. B 262 (1985) 593; A. Sen, Phys. Rev. Lett. 55 (1985) 1856.

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[3] A. Sen, Phys. Re('. D 32 (1985) 2102. [4] C.S. Hue, B. Sakita and M.A. Virasoro, Phys. Rev. D 2 (1970) 2857. [5] A.M. Polyakov, Phys. Lett. B 103 (1981) 207. [6] D. Friedan, in: Recent advances in field theory and statistical mechanics (Les Houches, 1982) eds. J.-B. Zuber and R. Stora

(North-Holland, Amsterdam, 1984). [7] O. Alvarez, Nucl. Phys. B 216 (1983) 125. [8] H. Aoyama, A. Dhar and M.A. Namazie, Nucl. Phys. B 267 (1986) 605;

D. Friedan and S. Shenker, UC-EFI preprint (1985). [9] J. Polchinski, Commun. Math. Phys. 104 (1986) 37.

[10] G. Moore and P. Nelson, Nucl. Phys. B 266 (1986) 58. [11] E. D'Hoker and D.H. Pong, Columbia preprint CU TP323, Nucl. Phys. B, to be published. [12] G. Moore, P. Nelson and J. Polchinski, Phys. Lett, B 169 (1986) 47;

P. Nelson and G. Moore, Harvard preprint HUTP-86/A014. [13] E.S. Fradkin and A.A. Tseytlin, Phys. Lett. B 158 (1985) 316;B 160 (1985) 69; JETP Lett. 41 (1985) 206. [ 14] B. Fridling and A. Jevicki, Brown University preprint Brown HET-566/85. [15] A. Cohen, G. Moore, P. Nelson and J. Polchinski, Nucl. Phys. B 267 (1986) 143. [ 16] S.L. Krushkal, Quasi conformal mappings and Riemann Surfaces (Wiley, New York, 1979). [17] J.H. Schwartz, Phys. Rep. 89 (1982) 223. [ 18] J.P. Rodrigues, in preparation. [ 19] See e.g.J. Jackson, Classical electrodynamics (Wiley, New York, 1975) ch. 1.,

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One-loop amplitudes for the bosonic open string: A first quantized approach